95 M16 NEW MATHEMATICAL PASTIMES BY MAJOR P. A.MACMAHON CAMBRIDGE UNIVERSITY PRESS CORNELL UNIVERSITY LIBRARY i GIFT OF Estate of William E. Patten MATHEMAT,^ ifrrw 1/ ^ ' CAYLORD PRINTED INU.S A. Cornell University Library QA 95.M16 New mathematical pas*j m «*; 3 1924 001 535 024 pi Cornell University P Library The original of this book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001535024 NEW MATHEMATICAL PASTIMES CAMBRIDGE UNIVERSITY PRESS C. F. CLAY, Manager LONDON : FETTER LANE, E.C. 4 NEW YORK • THE MACMILLAN CO. BOMBAY \ CALCUTTA L MACMILLAN AND CO., Ltd. MADRAS ) TORONTO : THE MACMILLAN CO. OF CANADA, Ltd. TOKYO : MARUZEN-KABUSHIKI-KAISHA ALL RIGHTS RESERVED NEW MATHEMATICAL PASTIMES BY Major P. A. MACMAHON, R.A. D.Sc, Sc.D., LL.D., F.R.S. st john's college, Cambridge CAMBRIDGE AT THE UNIVERSITY PRESS V / % IQ2I £ r P \ • V ....-''<■ / ... ; he put together a piece of joinery, so crossly indented and whimsically dove-tailed ; a cabinet so variously inlaid ; such a piece of diversified mosaic ; such a tessellated pavement without cement; .... Edmund Burke, 1774, on American Taxation. g-^3/3 V t. PREFACE Edwards. You are a philosopher, Dr Johnson. I have tried too in my time to be a philosopher, but I don't know how, cheerfulness was always breaking in. Boswell's Life of Johnson. THE author of this book has, of recent years, devoted much time and thought to the development of the subject of ' Permutations and Combinations ' with which all students are familiar. He has been led, during that time, to construct, for use in the home circle, various sets of pieces, of elementary geo- metrical shapes based upon these ideas, and he now for the first time brings them together with the object of introducing, in a wider sphere, what he believes to be a pleasant by-path of mathematics which has almost entirely escaped the attention of the well-known writers upon Mathematical Recreations and Amusements. The book differs in toto from their works because everything that it contains, with scarcely an exception, is the invention of the author. It is not a bringing together of materials derived from wholly different ideas. From beginning to end it proceeds along one denned path from which it never diverges. One continuous thread of thought runs through it from cover to cover. Indeed, in view of the exftllent collections of mathematical recreations that have proceeded from the pens of Ed. Lucas, W. W. Rouse Ball, W. Ahrens, H. E. Dudeney and others it would appear that there is no room for another book upon the lines upon which these have been written. I would make particular reference to the work in two volumes of W. Ahrens of Magdeburg and draw attention to the Bibliography which it contains. It involves nearly eight hundred titles. This must have necessitated much research. He has included a large number of works upon magic squares and upon the knight's tour and other chess-board amusements. These have been omitted from the short list ap- pended to this book, which had been almost completed when the list of Ahrens came into view. Many of the books appear to be rare, as they could not be found in the library of the British Museum. vi PREFACE Part I deals with sets of pieces of exactly the same size and shape, but differently coloured, numbered or otherwise dis- tinguished upon combinatory principles in such wise that no two pieces of a set are identical. It is shewn that such sets lend themselves to a great variety of pastimes, and the reader who will take the trouble to construct sets and employ suitable colours will find that he is truly in a kaleidoscope of constantly changing colour effect, the attractiveness of which will fully repay him for his trouble. The designs obtained from some of the contact systems frequently possess beautiful symmetry. It has not been found possible to produce the book in colour, and as the author has himself invariably investigated the sets in colours he must confess to a feeling of disappointment at the appearance of the pages. In colours fresh, originally bright, Preserve its portrait andreport its fate ! The Complaint. Briefly, Part I may be described as generalised dominoes. Part II follows Part I in a natural manner, for it is a mere transformation of it. The sets of Part I have the same shape but, are differently coloured. Those of Part II have the same colours but are differently shaped. The transformations can be carried out always in an infinite number of ways and give ample scope for taste and ingenuity. One merely requires squared paper (in the millimetre unit by prefeSfihce), ruler and compasses to be able to design and construct a means of endless amusement. How rich the prospect ! and for ever new ! And newest to the man who views it most; For newer still in infinite succeeds. The Consolation. Part III follows quite naturally upon Part II because the schemes of transformation prove, after a little consideration, to be a comprehensive method of designing repeating patterns for decorative work. Here we have pieces of the same size and shape which can be employed to completely cover a pavement or other flat surface. The only repeating patterns that were known in early times appear to have been the equilateral triangle, square and regular hexagon, the only regular polygons which possess the ' repeating ' property. Great advances were made in PREFACE vii the middle ages by the Arabian and Moorish architects, and in many of their buildings — ex. gr. the Alhambra at Granada — elaborate repeating patterns, based upon the square and its derivatives, are in evidence and are most effective. Each of these can be labelled with base and contact system in the classifica- tion in this book. Repeating patterns are in constant view in the home in parquet floors, carpets, paper-hangings, apparel, woven fabrics, etc. An attempt has been made to interest the reader while drawing as little as may be upon his knowledge of geometry. If he will study Part III after having become familiar with the transformations of Part II he will be able to make designs for home work to his heart's content. Beauty of form as depending upon symmetry of some kind is brought forward as being necessarily an important object of the designer. An elementary discussion of this precedes definite rules for obtaining symmetry. The fact that the number of different patterns is unlimited, in each of the categories, leaves much to the judgment and fancy of the designer who can give free play to his imagination. The patterns usually met with in public edifices and private homes are on hard and fast lines, shewing curiously little variety and cleverness in view of the fact that the theme is one of infinite scope. The subject may be regarded, it is thought, as an important recreation because the construction of the designs and assemblages possesses a distinct fascination. What we admire we praise ; and when we praise Advance it into notice, that its worth Acknowledged, others may admire it too. The Task. The subject of Part III has been carried much further than appears in these recreations, by the author and others. A work, entirely devoted to it, is in hand and may shortly appear. P. A. M. September, 1921. TABLE OF CONTENTS PART I PASTIMES BASED UPON SIMPLE GEOMETRICAL FORMS ART. PAGE i Preliminary observations upon dominoes . . . . I 2-10 Equilateral Triangle Pastimes. Set of 24 pieces with 4 colours repeatable. The different contact systems. The set of 10 pieces The set of 13 pieces 1 1 — 13 The set of 20 triangles involving 5 colours not repeatable. The contact systems and boundary types .... 14-15 The set of 12 which involves a particular colour 16 The set of 8 which involves 4 colours .... 17-19 Square Pastimes. The set of 24 which involves 3 colours repeatable . . 20-22 The set of 20 which is only symmetrical in 2 colours 23-25 A set of 16 symmetrical in 2 colours .... 26-27 A set of 15 28-29 A set of 9 30 A set of 24 involving 5 colours not repeatable but having particular colour upon each piece ... 31 Varieties of boundary types 32-33 Right-angled Triangle Pastimes. A set of 24 involving 4 colours Types of internal structure 34-35 A Cube Pastime. A set of 30 involving 6 colours not repeatable. A set of 8 which is 'contained' in any one piece. Reciprocity. Analogy with the Triangle and Square .... 36 Regular Hexagon Pastimes. A set of 24 involving 6 colours not repeatable but every piece shewing one particular colour, Three figures of assemblage 16 20 23 26 28 33 34 36 38 39 42 47 PART II THE TRANSFORMATION OF PART I 37 The general idea of transformation explained. The two natures of transformation 50 38-45 The Equilateral Triangle. Transformation of the Pastimes nos. 2-10 of Part I with diagrams 53 46-48 Transformation of nos. 11-16 62 49 Alteration of compartment boundaries that must precede the transformation of the Right-angled Triangle Pastimes . . 66 x CONTENTS ART. PAGE 50-54 The Square. Transformations of nos. 17-22 .... 68 55-57 Transformations of nos. 23-25 with a large number of specimen boundaries 71 58 A transformation of no. 30 with diagrams 79 PART III THE DESIGN OF 'REPEATING PATTERNS' FOR DECORATIVE WORK 59 General ideas shewing how the subject arises naturally from Parts I and II 80 60-64 Elementary notions concerning patterns 81 65 Removal of restrictions upon design 84 66-69 The Triangle Base. The two contact systems and methods of assembling the patterns. Principles upon which symmetry in the patterns can be secured. Specimen patterns and as- semblages. The ' Helmet ' pattern. ' Aspects ' . . . . 85 70-71 The Square Base. The category that arises from the first contact system with numerous examples. Assemblages shew- ing different numbers of aspects for the same pattern . . 91 72-73 Further principles which enable symmetry to be secured . 95 74-75 A second contact system with examples of patterns and assemblages 96 76-78 Third and fourth contact systems with examples ... 98 79-80 Fifth contact system. Remarks upon symmetry. The equi- lateral pentagon pattern 100 81 Special study of the equilateral pentagon 102 82-84 The Regular Hexagon Base. The two contact systems with remarks upon symmetry and numerous examples . . . 106 85 Concluding general remarks. The evolution of the pattern, which is a combination of the regular octagon and square, from the square base. The colouring of assemblages of re- peating patterns 112 86 The construction of the various pastimes 113 Bibliography • 115 PART I PASTIMES BASED UPON SIMPLE GEOMETRICAL FORMS Come track with me this little vagrant rill, Wandering its wild course from the mountain's breast. Doubled ay. 1 . The amusements to which this book is devoted are played with a number of cards or pieces which involve or are based upon certain regular or other polygons which are distinguished upon the sides with certain colours or numbers, in such wise that for a given number of colours and for given conditions of their occurrence there is one piece for every possible arrangement of the colours upon the sides. The principal existing amusement which embodies this idea A B B A , 5 . 6 1 6 1 2 1 2 3 1 2 2 3 4 1 2 3 4 5 1 2 4 1 1 5 2 2 6 3 3 3 4 3 5 3 6 4 4 4 5 4 6 5 5 5 6 6.6 Fig. i. is the game of dominoes which in various varieties is known all over the world. The pieces here consist essentially of straight lines — each piece being a line as AB in fig. i. In the most common set seven numbers are employed o, i, 2, 3, 4, 5, 6 and these are arranged in every possible way upon the two m. p. i 2 NEW MA THEM A TIC A L PA S TIMES halves A, B in such wise that the pieces A — B and B — A (fig. i) are regarded as identical. We thus obtain 28 pieces as numbered in fig. 1. For convenience, in actual practice, the pieces are broadened so as to consist of two equal squares joined about a side, each square being devoted to a number. Dudeney, Lucas and others have shewn that, apart from the different games, Draw, Matadore, Cyprus, etc., the pieces may be placed together so as to fulfil certain conditions and thus lend themselves to a great variety of patiences and puzzles. Other sets of dominoes have been popular at different times and in different places ; such for instance as employ ten or thirteen different numbers so as to proceed to the double-nine and double-twelve pieces respectively. EQUILATERAL TRIANGLE PASTIMES A scheme, analogy pronounced so true : Analogy, man's surest guide below. / \ \ \ / The Complaint. \ A / \ < \ / \ / 4 k A \ /y^\ /^ i^A Pi rA L ^2\\ l^2"^\ a - \ 1 a \ / X A \ A \ , A \ A X ^T- P*> pj> /^4\ \ l/v>\ //* \ A \ / X A \ / \ v A k A V /^B> l^-\ l^>\ /^3\ \ £/*> \ i -"'■* X A 4\ A V A \ A A X A X /^4~"A P^ L^2>\ pt< \ //3\\ 1/ 3^A A X A \ A \ A \ A \ A X /y^\ P^\ /y^\ /^3\ \ l^iS\ l^i^A Fig. 2. 2. In order to develop this principle take an equilateral ( triangle (fig. 2d) and join the angular points to the centre of the EQUILATERAL. TRIANGLE PASTIMES 3 circumscribing circle. This point is also the point of intersection of the three perpendiculars from the angular points upon the opposite sides and each perpendicular has the point as one of its points of trisection. The construction divides the triangle into three equal and similar parts. These parts we will call com- partments of the triangle. If we employ two colours, denoting them by the numbers 1, 2 and, as in dominoes, allow repetitions of the same numbers on the same triangle we obtain the four pieces of fig. 2b. Three colours yield eleven and four colours twenty-four pieces*. The twenty- four pieces are set forth arranged in a con- venient order in fig. 2c. * The problem of enumerating the number of ways of colouring the sides of a regular polygon of m sides with n colours, repetitions of colour on the sides of the polygon being permitted, may be studied by means of the Theory of Cyclical Permu- tations which has been very clearly set forth by Netto in his Combinatorik (Leipzig, Verlag von B. G. Teubner, 1901). If upon a set of polygons the re colours are repeated a, |3, 7, ... times respectively, where any of the numbers a, /3, 7, ... may be zero and a, /3, 7, ... do not contain any common factor greater than unity, the number of different polygons of the set is (re - 1) ! —r tt; — r — . (re ! means the factorial of re, sometimes written \u) a ! p ! 7 ! . . . l '— but if the numbers a, j3, 7, ... involve a common factor d, where dis a prime number, and a=dai, /3 = tif} 1 , y=dy 1 , ... n = dn\, the number of different polygons of the set is (re- 1) ! 11- «i («] - 1) ! a! /3! 7! ... + n ' 0l ! ft! 7^ ... ' The results are more complicated when the common divisor d is a. composite number but are given (I.e.). Utilising these results we find that the enumerations are for Triangle J«(re 2 + 2), Square \n (« + 1) (n 2 -n + i), Pentagon J re (k 4 + 4) , Hexagon \n (« + 1) (re*-« 3 + « 2 + 2), Heptagon }re (n 6 + 6). The mathematical reader will be able to establish that when the polygon has p sides, p being a prime number, the enumeration is given by ire(reP _1 +^-i). When repetitions of colour are not permitted the enumeration, in the case of a regular polygon of in sides, is 1 re ! m (re - m) ! ' If one particular colour is to occur upon each polygon the enumeration is given by (»-i)! ^ (re -?«)!' 4 NE W MA THEM A TIC A L PAS TIMES These may be set up into a regular hexagon as in fig. 3. a circumstance which supplies a useful starting-point for the study of the system of triangles. When we assemble the pieces so as to form the hexagon we may adopt some principle of contact between the compartments of different triangles. The most obvious one is to insist that a Fig- 3- compartment shall lie adjacent to a compartment similarly numbered or coloured. With this single condition the number of arrangements is so large that it proves to be possible to impose other conditions affecting the compartments which lie upon the exterior boundary of the hexagon. Meditation here May think down hours to moments. The Task. In order to realise what may be the nature of these boundary conditions we must observe that since each of the 24 triangles possesses 3 compartments there are altogether 72 compartments. The four colours are symmetrically involved so that each must appear upon one-fourth of 72 or upon 18 compartments. By reason of the specified contact condition a particular colour must occur an even number of times inside the hexagon and since 18 is an even number it follows that a particular colour must occur an even number of times upon the boundary. We see that there are 12 boundary compartments so that any particular colour must occur upon the boundary a number of times denoted by one of the even numbers o, 2, 4, 6, 8, 10, 12. E Q UILA TERA L TRIA NGL E- PA S TIMES 5 Thence it follows that, considering all four colours, the occurrence upon the boundary must be according to one of the eight schemes: Colours I 2 3 4 12 O o o IO 2 O 8 4 o O 8 2 2 O I 2 3 4 6 4 2 o 6 2 2 2 4 4 4 O 4 4 2 2 The fifth of these types indicates that the colours occur upon the boundary 6, 4, 2, o times respectively and the type is looked upon as the same when the colours are merely interchanged ; so Fig. 4. that the 24 arrangements of the numbers 6, 4, 2, o are all of the same type. Each type except the first, in which all the boundary com- partments are of the same colour, involves several varieties of Ci,i,i,i B12, 0,0,0 Ci, [,I,I B6,6,o,o \\!/V Vn 1 -^ A iV Vij A 1 3/3 Ax* l\ \\2/-/ K4/f V\2/7 /l A 3 /> A 3 />■ A 3 /> l\ \ 2 2/7 r/ 4 *$vU X 4 ' \7 4 X 2 j/y 2/ / / 2 V AI/x / 4 Sv ^, /^l^S \ 2 2 /2 2"A A 4 2\\ 1/V 2/ Fig. 5- 6 NEW MA THE MA TIC A L PAS TIMES boundary depending upon the way in which the given boundary colours are arranged. Thus for the fourth type two varieties are as in fig. 4. With the stated condition of contact inside the hexagon each of the eight types can be actually set up and many varieties of types. Some of these are difficult to obtain and make consider- able demands upon the skill and patience. It is not known if any of the varieties are in reality impossible boundary conditions. Two examples are given in fig. 5. They are framed upon what we will call the First Contact System, which we denote by C\,\,\,\. So from the first eternal order ran, And creature link'd to creature, man to man The link dissolves, each seeks a fresh embrace, Another love succeeds, another race. Essay on Man. 3. We may adopt other contact conditions inside the hexagon. We have four colours at disposal, 1, 2, 3, 4 suppose. In the first contact system we have 1 adjacent to V denoted by Ci, 1,1,1. 2 „ 2 3 » 3 4 „ 4) We take as our second contact system 1 adjacent to 1 "1 2 „ 2 Y denoted by Ci, 1,2, 3 .. 4) so that one pair of 3 and 4 compartments lie as in fig. 6. As regards the colours 1, 2, we know, from what has been said above, that each must appear an even number of times upon the boundary. The third condition necessitates the colours 3, 4 appearing an equal number of times inside the hexagon. It follows that each must appear the same number of times upon the boundary, but this number may be even or uneven. Fig. 6. EQUILATERAL TRIANGLE PASTIMES The possible types of boundary are now 16 in number: Colours I 2 3 4 12 O o o 10 2 o O 8 4 o o 6 6 o o IO o 1 I 8 2 I I 6 4 I I 8 2 "> I 2 3 4 6 2 2 2 4 4 2 2 6 o 3 3 4 2 3 3 4 O 4 4 2 2 4 4 2 O 5 5 O O 6 6 and there are varieties of every type except the first. Most of these types and many varieties have been actually set up. Probably every type is possible but nothing is known C 1,1,2 B6,6 \3^/ \\4/7 /3 4 3 " 3 A 4\ 3 74 4\ <4\\ N 2/ \\ 3 ^Y K3// ,1 3 '4 A 4 4 A A 2 2/ •^iN^ Z^i^A vA-Y V^-^Y 1 3 .4 Ya A 1 V3 A ^ 2"~ /^3^A r^^A v 1/ V\4/^ V\3/V /l 2 1T3 4 sVl 72 i\ 2 l '72 2\ ir^ ^3\\ lr^\ //2\\ Y\l/ N*/V vx 2 -^/ Ki/7 \l 4 '3 3 4 7 1 3\ 4 7 4 3\ 4 3/ ^2^ /^3\\ r 4 ^ O/ K4/7 \\3^V 1 1, 2 aV 3 A 4\ 3 3/ ^4\^ /^4\^s C2,2 B3,3,3.c V^ 2 ^- /l a\* yalaV V* 2\ /^2\^ r 2 ^ r 4 ^ V\l/^ V\l/-^ v\ 3 /^ /l 2V 74 4\ 3 Vi i\ 2 73 3\ /^2^ p^3\^ f^i^ />^3\\ V\l/^ \\ 4 ^Y V\2^? V\4/7 \l 71 3\ 4 7l i\ 2 7 4 A 3 3/ r 3 ^ r^ 4 ^ f^4^\ v\ 4 /Y v\ 3 ^Y \\ 3 -^Y \4 7 2 2\1 7 3 l\ 2 3/ A^A /^4\^ \\J^Y V\2^Y /\ 3 \ 4 74 aV 4 /3 2\ //2\\ (r*^\ r^^A vvi^v \\3/Y \\3^V /l 2V 8 A 4\ 3 l /a i\ 2 Ya 2\ r 2 ^"^ ^a^A r 4 ^) Z^2\\ v\J// \*^±/j SX3/7 Ki/7 \l 7 3 3 \ 4 V3 3 \ 4 % aV 2/ £^2V\ i^a^A £^3^> ^\i// ^\4/Y i \N$/Y \l 1/2 2W3 A3 2/ ^^N^I/^i^A C2,2 B4,4,2,2 C2,2 B5,5,I,I Fig. 8. EQUILATERAL TRIANGLE PASTIMES g It will be noticed that since the triangles are symmetrical in the four colours, 2, 2 | 4, 4 is not regarded as a type distinct from 4, 4 I 2, 2. These types and many varieties can be set up. As to the possibility of particular varieties the usual doubt remains. A number of examples are given in fig. 8. 5. We have thus, in the case of four colours, three systems of conditions of inside contact. It is convenient to denote these by 0,1,1,1, Cl,I,2, C2,2. For any number of colours a system is determined by the number of pairs of colours that are associated. The number of systems is the number of ways in which the number, which expresses how many colours are in use, can be made up of ones and twos. For 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, ... colours the systems are 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, ... in number. In other words, to obtain the number of systems we add 1 or 2 to the number of colours, according as that number is uneven or even, and then divide by 2. The scheme of triangles that has just been studied may be realised also by taking instead of four colours the blank and one, two, three pips upon the compartments. It then constitutes a b Fig. 9. set of triangular dominoes proceeding from treble blank to treble three as in fig. ga. The first contact system is similar to that adopted for the ordinary game of dominoes, whilst the third system is of the nature of that obtaining in the game of Matadore. In fact for the third system we may take, as in fig. gb, blank compartment adjacent to three-pip, one-pip „ „ two-pip, IO NE W MA THEM A TIC A L PAS TIMES so that the condition has the statement that the sum of the pips in adjacent compartments must be three. Good mother, how shall we find a pig if we do not look about for it? Will it run off o' the spit, into our mouths, think you as in Lubberland, and cry we we? B. JONSON, Barth. Fair, ill. 2. 6. We can select, from the complete set of 24, less numerous sets which are interesting to examine. There is a set of 1 1 triangles which involves only three specified colours as in fig. 10, but this set cannot be assembled into any- interesting shape ; so that we attempt to frame an additional condition with the object of reducing the number of triangles to 10. It will be noticed that 10 out of the set of 1 1 are altered by the interchange of the colours 1 and 2. The triangle which is Fig. 10. not altered by the interchange is the one which has all three compartments with the colour 3. We therefore make a new condition to the effect that every triangle is to be altered by the interchange of the colours 1, 2. We thus obtain 10 pieces which can be put together into the hexagonal shape in the lower part of fig. 10. It is important to bear in mind that this set is not sym- metrical in three colours, but notwithstanding that there is only symmetry in regard to the colours 1, 2 the set is interesting, as will appear. EQUILATERAL TRIANGLE PASTIMES n Examination of the 10 pieces shews that colours I, 2, 3 occur upon 11, 11, 8 compartments respectively. There are eight boundary compartments. For the contact system Ci,i,i, since each of the colours must occur an even number of times inside, it follows that upon the boundary the colours 1, 2, 3 must occur numbers of times which are uneven, uneven, even respectively. We obtain the types, six in number: Colours I 2 3 7 1 O 5 3 5 1 2 and varieties of each type. Examples are given in fig. 11. c 1,1,1 Bi,i ,6 V-vV'T' Vx 3 // 7 /3 3\ 3 yi \ 2 72 l\ //aM /^i^A J^*~^\ V\2/7 \\ 1/ -' \\2/v \ yi iV % 2\ 2 2/ l^3^\ /^3\\ I 2 3 3 3 2 3 1 4 1 1 6 c 1,1,1 B3, 3.2 /> V J/^ % X 1 2\ \i 2 /2 I' X 2 r 2V1 2/ /l 1^7 3 /3 \ 2 rA y 3 i\ \l A 2 X 3 2/ Ci,i,i B7,i,o Fig. 11. The contact system in which 1 is adjacent to 2 and 3 adjacent to 3 is different from that in which 1 is adjacent to 1 and 2 to 3 because of the want of symmetry in these colours. We will denote these by £2,1 and Ci,2 respectively. 7. In the case of £2,1 the colours 1, 2 must occur an equal number of times on the boundary and the colour 3 an even number of times not exceeding 6. The limit 6 is necessary because only 6 of the 10 pieces involve the colour 3. 12 NE W MA THEM A TIC A L PAS TIMES We have now four types : Colours i 2 4 3 2 I O 2 4 6 and varieties as usual. Examples are given in fig. 12. An example of every type is given as an indication that probably in other pastimes all types are possible. The reader however will notice that other considerations may rule out certain types, as in fact is seen to be the case in this particular pastime. C2,i. B4,4,o C2,I B3,3 ,2 /\ \ 2 a /lA 3 I// 7^ 3\ \ 2 2 / \to\t/ A 3 \ \ 2 5x7 7 2 \ 2 V3 \ 2 r A 2 i/7 3/ C2.1 B2,2,4 C2,i Bi,i,6 Fig. 1: 8. In the case of the remaining contact system Ci,2, viz. 1 to 1 and 2 to 3, we find colour 1 must occur an uneven number of times upon the boundary while colour 2, since it occurs inside just as often as colour 3, must occur three more times on the boundary than colour 3. Three types arise: Colours I 2 3 I 5 2 3 4 1 S 3 O Examples are given in fig. 13. EQUILATERAL TRIANGLE PASTIMES 13 Here we see that more than one variety of the type i?5,3,o Ci,2 B5,3,o Ci,2 Bs,3,o can be arranged. \<±/7 \\1// /l. iV % s\ 2 %J 2\ t^^\ (T 3 ^! // l^A \\i^y v\ 2 ^/ \\l/7 \i 73 3\ 2 73 2\ 3 2/ i^^\ l^^\ \^i/7 V^ 2 /^ /l. \ l % 3 \ 2 73 J -3\ /^1"A \L^ l^A //i\\ \\l/7 ki/7 Ki/f \l V3 s\ 2 7 a s\ 2 V l/<£^\ /^a^ Ci,2 B3,4,i Fig. 13- 9. From the complete set of 24 triangles we now isolate the set containing all of those which involve one particular colour, say the colour 4. There are thirteen as in fig. 14 which can be ^> A^ ^ / >2^ iy^> A^ A \ / \ / \ / k A \ /4 k l^\ l^\ As\ \ / /l^ /y^\ l^A A \ /"4\ \ Fig. 14. assembled into the form of a semi-regular hexagon or blunted triangle as in fig. 15. Observe that the set is symmetrical in the three colours I, 2, 3 but that it is not symmetrical in four colours. There are nine boundary compartments. The colours 1, 2, 3, 4 occur in 7, 7, 7, 18 compartments respectively. For the contact system Ci, 1,1,1 the boundary conditions H NE W MA THEM A TIC A L PA S TIMES are that the colours i, 2, 3 must each occur an uneven number of times and 4 an even number. There are thus six types : Colours I 2 3 4 I 1 1 6 3 1 1 4 3 3 1 2 1 2 3 4 5 1 1 2 5 3 1 3 3 3 and the usual varieties. The type 7, 1, 1, o is seen at a glance to be impossible. Examples are given in fig. 15. Ci,i,i,i B3,3,i,2 /\\2/7\ mi y y^U Y V4J.3 /> X 4 3\\ 3/7 l\ 1"A /2 X 3 3/7 4 A \ 2 1/7 1 / 4 i\ \ 2 t/7 7 1 r^4 4/4 \ 4 3/7 1/ /4 Y-j A 4 /7 4\ /4 iV 1/7 4/4 \ 4 S/7 7 1 4 \ \ 3 5/7 \ 4 2/7 3 A v-4 !/7 2/ Ci,i,i,i B3,3,3,o Ci,i,i,i Bi,i,i,6 Fig. 15. To guide my course aright What mound or steady mere is offered to my sight? Drayton, Polyolb. 1. p. 569. IO. For the contact system Cl,i,2, viz. : 1 to 1, 2 „ 2, 3 -. 4, we find as one boundary condition that the colour 4 must occur on the boundary 1 1 more times than colour 3, an impossibility EQUILATERAL TRIANGLE PASTIMES 15 since there are only nine boundary compartments. Hence the system Ci,\,2 is impossible. For the system Ci,2,i, viz. : 1 to 1, 2 „ 3. 4 » 4> we find as the boundary conditions 1 must occur an uneven number of times, 2 as many times as 3, 4 an even number of times, leading to the types, ten in number: Colours I 2 3 4 I 1 1 6 3 6 1 2 2 4 3 1 1 4 1 3 3 2 1 2 3 4 3 2 2 2 5 1 1 2 1 4 4 O 3 3 3 O S 2 2 O Examples are given in fig. 16. Ci ,2.1 Bi.i 1,6 /4 \4 2^S 7i 4\ A ik 3 3/V r \ 4 r 4 \ \i i/y 7 4 \ 2 8/7 2/ Ci, 2,1 63,3,3,0 \\?/7 A iV 7 4 2\ //4\\ Z^aNX \\4/7 \\3^y /3 2\ 3, f 4 A 4\ 4 ~ 7 1 i\ /^4"A r 4 ^ r 4 ^^ \\ 4 ^7 K4/y kv/ \ 3 V3 4\ 4 7 3 A 4 1/ l/$ < \ l/'i>\ Ci.2,1 Bi,4,4,o Fig. 16. i6 NE W MA THEM A TIC A L PA S TIMES This band dismiss'd, behold another crowd Preferred the same request, and lowly bow'd. The Temple of Fame. 1 1 . The next set of triangles is concerned with the equi- lateral triangle and five colours, three different colours appearing in the compartments of each triangle. This design leads to the / k / fcx A \ A Ik A \ p^\ l^> , L^\ Ao> : l/r- A \k A k A \ A \k A \\ l/i> \ /y^ 2s^ A^ ky-\ /4 5\ A \ A \ A \ A \ /^1\\ /y^\ P?> /^x\ P*>\ /y<\ A \ A \ A \ A \ /^i^\ /y<^ P^\ P^\ A^s-A Fig. 17. twenty triangles, as in fig. 17, which can be arranged in the semi-regular decagon of fig. 18. There are 12 boundary compartments. Altogether there are Fig. 18. 60 compartments and 12 of these must belong to each colour as each of the five colours is symmetrically involved. EQUILATERAL TRIANGLE PASTIMES 17 For the contact system Ci, 1,1,1,1 which is 1 to I, -> *? — » -"> 3 » 3. 4 ,. 4, 5 » 5. we gather that each colour must occur an even number of times upon the boundary. We have therefore the types, ten in number: Colours 12345 1 "6 6 4 4 4 with varieties of each type except the first. The existence of the first type is somewhat remarkable as there are only just sufficient compartments of a particular colour to go completely round the boundary. It will be noted that the 20 triangles are naturally arranged I 2 3 4 5 12 O O IO 2 O O 8 4 O O 8 2 2 6 6 O O 2 3 4 5 4 2 2 2 4 4 4 2 2 2 2 2 2 3 ^ /l \ 3 7 2 4V 2 A A \l 2 /i A 3 1/ /l A 5 3 A \l 3 / /^ \ 2 1/ \\ x -^ \2/7 \4 '9,/ 1 R/ /l 4\ y \\ 5 // \\3/ Vx 1 -^/ \l 73 s\ 5 4 '4 3\ 3 2/ r^\ -^l^A ka/7 \1/V /l a\ 3 75 4 i4 7 3 2\ r^^ r^ r 4 ^^ V\5/7 \\3/ \\Vv \ 1 ry 4 2V 1 '\ s\ 5 2/ i/}>\ ^2^ Cl, 1, i,i,i Bi2,o,o,o,o Ci,i,i,i,i B6, 6,0,0,0 Fig. 19. M. P. 1 8 NEW MA THE MA TIC A L PAS TIME S in pairs, because to every triangle (as fig. iga) there can be associated the complementary (as fig. 19^). The latter, read counterclockwise, is the same as the former read clockwise. This circumstance supplies a clue to the setting up of some of the boundary types. It is left to the reader to take advantage of this hint. Some examples are given in fig. 19. 12. For the contact system Ci, 1,1,2, viz. 1 to 1, 2 to 2, 5 to 3, 4 to 5, the colours 1, 2, 3 must each occur an even number of times upon the boundary and the colours 4, 5 the same number of times. The boundary types are 23 in number: Colours 1 2 [ 3 4 5 12 ° 10 ! 2 8 4 8 2 2 6 I 6 6 14 2 4 I 4 4 10 1 1 I 2 8~! 6 4 6 2 4 4 8 o 6 2 4 4 3 4 5 4 2 1 1 r 1 1 1 1 1 2 2 2 2 2 2 2 2 1 ; 2 6 I o 4 ! 2 2J2 4 I o 2 2 2 I o o ! o 4 5 with varieties of every type except the first. Examples are given in fig. 20. Ci,i,i,2 Bo,o,o,6,6 Ci,i,i,2 B12, 0,0,0,0 /5 r A 1 4 \ \ 5 r \ 5 7 2 4/ /> A 5 A 2 4\ 2\\ \5 V 1 2\2 1/1 3\3 2/V 4/ ^\i^ ^1/7 Av 3, 3 5 s 4 7 2 l\ /s°*\ / 4 \ l^3^\ V\4// N 5/ W3/7 \l 3/3 2^ .2 4. 5 A 2 1/ //4 N -^3V\ V\ 5 ^ \3>V /l b\ 4 3 '3 4 \ 5 2 A l\ Z-^is^X "'2^ Z^4^A v\ 3 ^/ v.2^ Y\5/7 \i 4 A 4 v5 4 '5 A 2 1/ l^^> ^l\^ Fig. 20. EQUILATERAL TRIANGLE PASTIMES 19 13. The remaining contact system Ci,2,2, viz. 1 to 1, 2 to 3, 4 to 5, yields the sixteen types: Colours 1 12 10 3__4|_5_ I o ! o 1 I o o 2 | o o 1 ! 1 3 i o 2 ! 2 ; I I 2 3 4 4 4 4 4 3 3 1 4 2 2 2 2 5 s 2 4 4 1 1 2 3 4 2 3 3 2 6 6 5 S 1 4 4 2 3 3 3 with varieties of each type except the first. Examples, which include two varieties of ^0,3,3,3,3, are given in fig. 21. Cl,2,2 Bl2,0,0,0,0 ^\lx \<^^? /l A 5 2 '3 A 5 7 s i\ /^3^ ^"XX r^^ K2/7 \3/Y \\2/7 \i 4 /5 4 v5 2 A A 3 1/ r^ /^4\\ V\3^ K8/7 /1 A 5 4 '5 2\ 3 a /3 A 3 '1 1 a 5 A A \\ 2 //' **J>s K4/J \ 5 V* 1 .1 3 '2 3\2 3/ /^a^ '"'i^A \\2-" a^ A A 3 1 2 .3 4 AJ 3\ v\ 2 -^ \5^ vvS^/ \ 5 5 ,4 1, 1 A 2 4/ Ci,2,2 Bo,3,3,3,3 Ci,2,2 Bo,3,3,3,3 Fig. 2r. This set is prolific of types, the three systems of contact yielding altogether 49. 20 NE W MA THEM A TIC A L PAS TIMES Experiments with these will be found by many to be very interesting. Eum odi sapientem qui sibi non sapit: hee is an ill cooke that cannot licke his own fingers. Withals' Diet., Ed. 1634. 14. We can divide the complete set into two portions: (i) those which involve a particular colour, twelve in number ; (ii) the remainder, eight in number, which involve only four colours 1, 2, 3, 4. The first set of 12 is symmetrical in the four colours 1, 2, 3, 4. There are 36 compartments which occur 6, 6, 6,6, 12 times for the five colours 1, 2, 3, 4, 5 respectively. The members of the set may be assembled in either of the forms of fig. 22, each of which has 10 boundary compartments. Fig. 22. For the contact system 0,1,1,1,1 each colour must appear an even number of times upon the boundary, so that the types are fifteen in number: Colours I 2 3 4 5 10 2 8 4 6 2 2 6 6 4 1 2 3 4 4:2 2, 2 2 6 2 4 4 4 2 2 1 2 3 4 2 2 2 2 6 4 6 2 2 4 4 2 4 2 2 2 Examples are given in fig. 23. EQUILATERAL TRIANGLE PASTIMES 21 Certain of these types exist for only one of the two boundary shapes, as it is seen at once that the first of the types cannot exist for the parallelogram, while it does exist for the double hexagon, Ci,i,i,i,i Bio,o,o,o,o Ci,i,i,i,i 64,4,2,0,0 C2,i,i,i Bo,o,2,o,8 Vn 5 -^ /5 3 A l5\ K2/7 \-^<^y \5 7 3 2 \ 5/ V-vA/-/ 7 1 5\ \\2/7 K4// V V 1 3 V 5/ \ 2 J/7 5/5 2 W iN\/ /2 \ 5 4 /ll\ \l 7 5 \ 4 2/ V\5/7 /53V 71 5\ //l\\ /^4\\ K2/7 V^/V \ 5 Vaj 5\ 5 3/ //4\\ Y\4/y /5 i\ 2 5 A 3\ r^\ r"*"^ VsJ/7 \\4/V \5[3/ 3 2\1 5/ ^o"A Fig. 23. Fig. 24. 15. For the contact system £2,1,1,1 we find that the colours 1, 2 must occur equally often and the colours 3, 4, 5 each an even number of times. The types are 30 in number: I 2 3 4 5 5 S O O i O 4 4 2 O O 4 4 O 2 3 3 4 O 3 3 O O 4 3 3 2 2 1 3 3 O 2 2 2 2 6 O 1 O 2 2 6 2 2 4 I 2 I I I I I I I I 3 4 4 O 2 2 2 O 2 O 4 O 2 2 6 O 4 2 6 2 I 2 3 4 5 I I 4 4 O O O 10 O O 2 O 8 O O 4 6 O O 2 2 6 O O 6 O 4 O O 4 2 4 O O 6 2 2 O O 4 4 2 O O 6 4 An example is given in fig. 24. Some of these are probably non-existent for one or both forms of boundary. We have further the contact systems Cl,I,I,2, C2,2,I, C2,I,2, which may be left to the reader to examine. How this geare will cotton I know not. True Tragedie of Ric. Ill, 1594. 22 NEW MATHEMATICAL PASTIMES 16. The second set of eight pieces puts up into the parallelo- gram of fig. 25 with eight boundary compartments. The set is symmetrical in four colours. Each colour occurs on six compartments. The types for the different contact systems are Ci,i,i,i 2 3 2 4 O 2 2 2 2 C2 ,1,1 c 2,2 4 I 2 3 4 1 2 |3 4 O 4 4 4400 O 3 3 2 3 3 I I O 2 2 4 2 2 O O 2 2 2 2 2 1 1 6 1 1 4 2 6 2 4 4 Colours 1 "6 4 4 2 Examples are given in fig. 25. Cl,I,I,I B4,2,2,0 C2,2 B4,4,o,o C2,i,i B4,4,o, Fig. 25. SQUARE PASTIMES 23 SQUARE PASTIMES As a stream descending From his fair heads to sea, becomes in trending More puissant. G. Tooke'S Belides. 17. Taking next the square, divide it into four equal and similar compartments by drawing the two diagonals and, as usual, regard each compartment as the location of a colour or number. Fixing upon three colours and allowing repetitions of colour in the compartments of the same square we obtain twenty-four different pieces as in fig. 26 which can obviously be arranged in a rectangle 6x4. The 24 squares involve 4 x 24 or 96 compartments and, since each of the three colours is involved symmetrically, each must appear upon one-third of 96 or 32 compartments. 32 is an even Fig. 26. number and the rectangle has 20 boundary compartments so that for the contact system Ci,i,i, viz. 1 to 1, 2 to 2, 3 to 3, each colour must appear an even number of times upon the boundary. We have therefore the 14 types of boundary: Colours 123 123 123 20 14 4 10 8 2 18 2 12 8 O 10 6 4 16 4 12 6 2 8 8 4 16 1 n 12 4 4 8 6 6 14 6 10 10 with varieties of each type except the first. 2 4 NE W MA THEM A TIC A L PAS TIMES Most of these types have been actually arranged. Some of them are by no means easy but all are believed to be possible. Very little is known about the varieties of the different types. The reader will not have much difficulty in setting up the first type B20,o,o, with one colour everywhere upon the boundary, but he will find that thought and ingenuity are both required. For as the precious stone diacletes, though it have many rare and excellent sovraignties in it, yet loseth them all, if it be put into a dead man's mouth. Braith, Engl. Gent. p. 273. An arrangement of type i?lo,lO,0 is given in fig. 27. Ci,i,i Bio,io,o m IX 1 / 3 \ 1Y2 / 3 \ 2Y3 /l\ 3Y3 / 3 \ / 2 \ 1/2 \ 3 / 2V3 \s/ 3Y3 /• 3 \ 3Y3 \3/ 3 A 1 \2/ IX 2 /' 2 \ VX 3 / 2\ 3Y3 /2\ \ 3 / 3V2 / 3 \ \l / 2Y2 / l\ / 3 \ \ 2 / 3V2 / 2 \ \ 2 / iX 3 / 2\ V 2 / 3 /\ 3 / 2\ \ 3 / 3 /\ 2 / 2\ / 2 \ \3/ lY2 /2\ \2 / 2X 2 / 2 \ Fig. 27. 18. For the contact system Ci,2, viz. 1 to 1, 2 to 3, the colour 1 must occur an even number of times upon the boundary, whilst colours 2 and 3 must occur equally often. We have the eleven types : Colours 123 123 20 O 8 6 6 18 I 1 6 7 7 16 2 2 4 8 8 14 3 3 2 9 9 12 4 4 10 10 IO S 5 with the usual varieties. Every type is believed to be a possible arrangement and two examples are given in fig. 28. Come on, sir frier, picke the locke, This gere doth cotton hansome. Tronb. Reign of King John, p. 1. SQUARE PASTIMES 25 Cl,2 B2O.0.O 1Y1 ly <2 3 X 3 / 3\ 2 X 3 \iX 2X2 / 3 \ \l/ 3 X J / 2 \ \i / iX 3 V V \ 2 / 3 X a / 3 \ \ 3 / 3 X 3 / a\ \ 2 / 2 X 3 /2\ \ 3 X 2V1 / 2\ 1X2 /K 3) <3 \ 2 / 2X2 / 2 \ \ 3 X 3 X 3 /3\ \ 3 X 2 X 2 \ 3 / 3Yl X 2 \ I 3 / <2 3 X 2 3Y3 X \ 3 / 3 X 1 / 1\ 3Y1 O O 1X 3 / 2\ 2X 1 \ 1 / 1X 1 /2\ \l / 1 X 2 / 3 \ 3 X 3 / 3\ V <2 \ 3 / 3 X 3 / 3 \ \ 3 X 2 X 2 X 3 \ \ 3 X 3 X' / 1\ \ 2 X 1X 2 / 2\ 3 X 2 /3\ 3/ \ 2 3 X 3 /3\ \ 2 / 2 X 2 / 2 \ 3 X 3 / 2 \ \ 3 / 2 X 2 / 2 \ \ 2 / 3X2 V V \ 2 / 2 X 3 / 1 \ \3/ 2 X 3 / 1\ \ 3 / 2 X 2 /l\ \ 3 / 3 X 2 / x \ Ci,2 Bi2,4,4 Fig. 28. Pleased with the same success, vast numbers press'd Around the shrine and made the same request. The Temple of Fame. S 1 4 2 1X1 Fig. 29. 19. We may select from the complete set of 24 squares a lesser number upon definite principles. If we are restricted to two colours we have only six squares as in fig. 29 which fit into a rectangle three squares by two. The case is trivial from the present point of view, but since transformations of these diversions are effected in Part II of the book it is well not to leave it unexamined. 26 NE W MA THEM A TIC A L PAS TIMES Briefly the results are that for the contact system Ci,i only one boundary type exists, viz. £6,4., and for the contact system C2 only one, viz. ^5,5. Reflection, reason, still the ties improve, At once extend the interest and the love. Essay on Man. 20. To make a more useful selection we may in the first place discard all squares which remain unaltered when the colours 2, 3 are interchanged. We thus discard the four squares of fig. 30 and are left with a set of 20 squares which can be assembled into a rectangle 5x4. 1X1 2X3 1X1 Fig. 30. The set is not symmetrical in three colours, but only in the colours 2, 3. There are 18 boundary compartments and the colours 1, 2, 3 occur in the set 26, 27, 27 times respectively. We see that on the boundary the colours must occur for the system Ci,i,i 1 an even number of times. uneven uneven There are 25 boundary types: Colours 123 1 16 1 1 6 7 5 2 11 5 14 3 1 6 9 3 2 13 3 12 3 3 6 1 1 1 2 is 1 12 5 1 4 7 7 9 9 10 5 3 4 9 5 1 1 7 10 7 1 4. 1 1 3 13 5 8 5 5 4 13 1 15 3 8 7 3 2 9 7 17 1 8 9 1 and the usual varieties. SQUARE PASTIMES An example of the type Bo,g,g is given in fig. 31. Ci,i,i Bo. 9, 9 \sX 2 X 2 / 2 \ \ 2 X 2 X 3 X 2 \ \ 2 X 3 /\ 2 \ 2 X 2X 2 X 1 \ \ 2 X 2 X 3 X 1 \ 2X1 \ 2 X 1X 3 X J \ V\ 3 \iX 3 X 2 X 3 \ \iX 2 X 3 X 3 \ \iX X 3 \ \iX 1 X 2 X 1 \ vK 2 \sX 2 X 2 X 1 \ \sX 2 X 3 X 3 \ \sX 2 X J X 3 \ \iX 1X 1 X 3 \ 1V3 X 3 \ \iX 3 X 3 X 3 \ \sX 3 X 3 X 3 \ 27 Fig- 31- 21. For the contact system £2,1, viz. 1 to 2 and 3 to 3, we find that colour 2 must occur once oftener than 1 upon the boundary and 3 an uneven number of times; this leads to nine types: Colours 123 123 8 9 1 3 4 11 7 8 3 2 3 13 6 7 5 1 2 IS 5 6 7 1 17 4 5 9 These prove to be difficult to arrange, and some varieties of types appear not to be possible. Hard or difficile be those thynges that be goodly or honest. Taverner's Adagies, D. 5. And when he had taryed there a long time for a convenable wind, at length it came about even as he desired. Holinshed'S Chron. 1577. In view of Part II of this book it is desirable to obtain symmetrical boundaries for choice and one such example is given in fig. 32 for the type ^2,3,13. 22. For the contact system C\,2, viz. 1 to 1, 2 to 3, which exists by reason of the want of symmetry in three colours, we 28 NEW MATHEMATICAL PASTIMES find colour i must occur an even number of times upon the boundary and colour 2 just as often as 3. C2,i 62,3,13 3Y1 / 3 \ \2 / 2 /\ 1 X 1 \ \2 / 2V2 /2\ \ 2 / lX 1 / l\ 2X3 X 2 \ \ 3 / 3 /\ 3 /3\ \ 2 / 3 /V / 1\ 2Y2 / J \ \ 2 / lX3 /]\ 3 X 3 3Y3 \ 2 / 3 X 2 / 2\ \ 2 / lX 2 / 3 \ 1X 2 X 2 \ 1X 3 / 2 \ 3 X 3 / 3\ \l / 3Y2 / 3 \ \ 3 / lY2 / 3 \ 1 X 2 / 3\ 1Y3 /• 3 \ Fig. 32. There are ten types : Colours 1 2 3 1 2 3 18 8 S ■5 16 1 1 6 6 6 14 2 2 4 7 7 12 3 3 2 8 8 10 4 4 9 9 with varieties possibly of all except the first. As an example a setting of i?i8,o,o is given in fig. 33. Ci,2 Bi8,o,o iX 3 / 1\ \iX 2 X 2 \iX 3X 2 / X \iX 3 X 3 X 2 \ \iX 2X 1 X 3 \ Xi/ 1x2 X 3 \ 3 X 3 \ 3 X 2Y2 X 2 \ \ 3 / 3 X 2 / 3 \ \ 2 / 3X 1 / 2 \ \ 2 X 1X 3 / 3 X \iX 2 X 2 / 2 \ \ 3 X 3 X 3 X 3 \ \ 2 / 2Y2 X 2 \ \ 3 X 3 X X / 3 \ \ 2 / 1X 2 / 1 \ \ 3 X 3 X 2 \ 2 / 3 X 2 \ 3 / 3Y1 / 1\ \ 2 / 1Y1 X 1 \ Fig. 33- 23. To reduce further the number of squares and, in the first place, with the object of obtaining a set of 16 we adopt as a discarding principle the rejection of all squares which involve a particular colour exactly twice and upon opposite compartments SQUARE PASTIMES 29 of the square. The reasons which prompt this rejection will appear in Part II, as it will be shewn therein that such pieces are inconvenient and even impracticable in the further develop- ments. We thus reject the four squares of fig. 34^ and we are left with the 16 pieces of fig. 34 £. The set is symmetrical in the two colours 2, 3 but not in the three colours. They form up naturally into a square 4x4, which has 16 boundary compartments. The colours 1, 2, 3 occur upon 20, 22, 22 compartments respectively. Fig. 34- For the contact system Ci,i,i each colour must appear an even number of times upon the boundary. We are led to 25 types of boundary: Colours 123 123 123 16 6 10 • 2 10 4 14 2 6 8 2 2 8 6 12 4 6 6 4 16 12 2 2 4 12 14 2 IO 6 4 10 2 12 4 IO 4 2 4 8 4 10 6 8 8 4 6 6 8 8 8 6 2 2 14 8 4 4 2 12 2 with the usual varieties. 30 NE W MA THEM A TIC A L PAS TIMES It is not known how many of these exist. He hath a person, and a smooth dispose To be suspected. Othello, I. iii. As examples we have the three arrangements of fig. 35 in the first of which, C\,\,l Bi6,o,o, symmetry is apparent*. Ci,i,i Bi6,o,o \iX 1 X 1 X 3 \ \iX iX 2 X 3 \ \iX 2 X 3 X 2 \ \iX 3X 1 X 3 \ \ 3 / iX 3 X 3 \ \ 3 X 3 X 3 X 3 \ \ 2 X 3 X 2 X 2 \ \ 3 X 2X 1 X 2 \ \ 3 X iX 3 X 2 \ \ 3 X 3 X 2 X 3 \ \ 2 X 2 X 2 X 2 \ \ 2 X 2 X X X 2 X \ 2 X iX 2 X 1 \ \ 3 X 2Y3 X 1 \ \ 2 X 3Y1 X 1 \ \ 2 X 1Y1 X 1 \ \ 3 X 3 X 3 X 3 \ \ 3 X \ 3 X 3X1 X 2 \ \ 3 X 1X 3 \ 3 X 3 X 2 X 3 \ \iX 2X 1 X 3 \ 1V 2 \iX 2 X 3 X 2 \ \ 3 X 2X 1 X 2 \ \ 3 X 1X 1 /\ \ \iX iX 2 X 3 \ \2/ 2 X 2 X 2 \ \aX 2X 1 X 2 \ \iX 1X 1 X 2 \ \ 3 X 1 X 3 \ 2 X 3X2 /2\ \ 2 X 2 X 2 X 2 \ \ 2 X 2 X 3 X 2 \ \ 2 X 3 X x X 1 \ \ 2 X 1X 2 / l\ X2X 2X 1 X 3 \ X2X 1X 1 X 1 \ \iX 1X 1 X 3 \ \iX X X 2 X 3 \ \sX 2 X X X 2 \ \iX 1X 3 X 3 x \ 3 / 3 X 3 X 3 \ vx 3 X 2 X 3 \ \ 2 X 2X 2 x\ X 3 X 2 X 3 Xj\ \ 3 X 3 X 3 X J \ \ 3 X 3 X 2 X 1 \ Ci,i,i Bo,8,8 Ci,i,i B4,i2,o Fig- 35- 24. For the contact system £2,1, viz. 1 to 2, 3 to 3, the colour 2 must occur upon the boundary a number of times greater by two than the number of times 1 occurs, while colour 3 must occur an even number of times. The possible types are Colours 123 123 7 6 5 4 o 2 4 6 3 2 1 o 10 12 14 and varieties. * If any compartment be reflected through the centre of the whole square (the point- image) its colour if i is unchanged, but, if 2 or 3, is changed to 3 or 2 respectively. SQUARE PASTIMES Examples are given in fig. 36. C2,i Bo, 2, 14 C2,i B2,4,io 31 \ 3 X 3X3 /3\ \2/ 3X2 / l\ IX 2 / 2\ \ 3 / 1Y3 / i\ \ 3 / 3Yl / 2\ \ 2 X 2Vl / 1 \ \ 1 / 2X1 X 2 / 2Y3 / 1\ \l/ 3Yl / l\ \ 2 X 2Y2 / 2 X \2/ 1Y3 X 1 X \ 2 / 3X 3 X 3 X \ 2 / 3Yl / 3\ \l/ 2Vl / 3\ \ 2 X 2V2 / 3 \ \sX lY3 / 3 \ \ 3 X 3Y2 X 3 \ \i/ 1Y2 Xi\ \l/ iY2 X 3 \ \ 3 / iX 3 / 3 \ X 3 X 3Yl / 1 \ \ 2 / 2Y1 X a X \ 3 / 2X3 / 1\ \ 3 X 3Y3 / 3 \ \ 2 / 3V2 / 2 \ \i/ 1Y1 X 3 \ \ 2 X 2Y2 / 2 \ \3/ 1Y3 X 2 \ X J /\ 3 / 3X2 1 X 2 \l/ lY2 / 2 \ \l/ lY3 X 2 \ Fig. 36. 25. For the contact system Ci,2, viz. 1 to 1, 2 to 3, we must have colour 1 an even number of times upon the boundary and colour 2 as often as 3. The types are nine in number: Colours 1 r6 O O H 1 1 12 2 2 10 3 3 8 4 4 3 5 5 Examples are given in fig. 37. Ci,2 Bi6,o,o 6 460 277 088 Ci,2 Bo,8,8 \lx 1X2 X 3 \ \i/ 3 X 3 / 3\ \l/ 2 Xl X 3 \ 1Y1 X 3 \ \ 2 / iX 3 X 3 \ \ 2 / 2Y2 X 2 \ \ 2 X 3 X 2 / 2 \ \ 2 X 3 X J X 3 \ \ 2 / IX 2 / 3\ \3y/ 3 X 2 / 3\ \ 3 X 3 X 3 / 3 \ \ 2 X 2Y1 X 2 \ \flX \ 2 X 1Y2 / l\ \a/ 3Y3 / i\ \ 3 X 3X 1 X 1 \ \ 2 X 2 X 3 / 2 \ \ 2 X 2Y2 X 2 \ \aX 3Y1 X 3 X \ 2 X 1Y2 Xi\ \ 3 X 2 X 3 X 1 \ X 3 / 2Y3 / 3 \ \ 2 X 2Y1 X 2 \ ^K \i/ 3 X 2 X 2 \ X 2 / 3 X J Xi\ \ 3 X 1X 3 Xi\ \iX 2 X 3 X 2 \ \ 3 X 3 X J X 3 \ \iX 1Y1 X 3 \ \iX •X 2 / 3 \ \3/ 3V3 X 3 \ Fig. 37- 32 NEW MA THEM A TIC A L PAS TIMES This set of pieces has been dealt with at some length because it possesses a large number of boundaries which are symmetrical in colour. Also because it lends itself particularly to the develop- ments of Part II. Yes, Sir, I study here the mathematics And distillation. B. JONSON, Alch. IV. i. The numbers i, 2, 3, ..., which have been used in connexion with the various sets, have been merely symbols for colours. No use has been made of their arithmetical properties. A single example will shew that it is possible to denote the colours by numbers and to call in the aid of arithmetic, in certain cases, to define the contact system that is before us. In the last contact system dealt with above, viz. 1 to 1, 2 x substitute for the numbers 1, 2, 3, the numbers 1,0, 2, so that the contacts in question are 1, 1 and o, 2 as in fig. 38a and we may define the conditions of contact to be that the sum of the numbers in adjacent compartments is 2. Clearly we .1/ \2/ \2/ \3. b Fig. 38- could increase each of the numbers 1, o, 2 by the same number with the same result. Thence the simplest way of carrying out the idea is merely to interchange the numbers 1, 2 so that the contacts are 2, 2 and 1, 3 as in fig. 38 & The sum of the numbers in adjacent compartments is then 4. In the case of five colours 1, 2, 3, 4, 5 and the contact system 1 to 1, 2 „ 3. 4 » 5. we similarly change the numbers I, 2, 3, 4, S to 3- 1. 5, 2,4 SQUARE PASTIMES 33 respectively, and define the contact system to be such that the sum of the numbers in adjacent compartments is 6. 26. There is an interesting set involving 15 pieces which is derived from the set of 16 pieces, dealt with above, by omitting the pieces in fig. 39a and adding the piece in fig. 39^. This is a 15 -piece set which may be defined as involving the pieces of the original set of 24 : (i) which have not more than three compartments of any piece of the same colour ; (ii) which do not involve any piece which has exactly two compartments of the same colour and those opposite com- partments. The pieces will form up into a rectangle 5 x 3 as shewn by the figured rectangle in fig. igc. Ci,i,i Bi6,o,o 3K ^ a \2/' 2X 3 / 3\ \iX 1X 1 / 2 \ \iX 1 X 1 / 3 \ 1Y3 / 2 \ 3 X 2 / 3 \ \iX 2 X X X 2 \ \ 2 / 1X 2 / 2\ 2 X 3 / 2 \ 3 X 2 X 2 \ \ 3 X 2 X 3 / 3 X \ 2 / 3 X J X 3 \ \ 2 X 1X 3 /l\ \ 2 X 3 X 2 \ 2 X 2 X 3 \ 3 X 3 X 3 X b c Fig- 39- To analyse the set we note that it is symmetrical in the three colours and that the compartments coloured 1, 2, 3 occur each 20 times. There are 16 boundary compartments. For the contact system Ci,t,i each colour must occur an even number of times upon the boundary. The types are therefore Colours 1 2 3 16 14 2 12 4 12 2 2 10 6 ten in number. I 2 3 O42 8 18 o 8 6 2 8 4 4 6 6 4 M. P. 34 NE W MA THEM A TIC A L PAS TIMES 27. For the contact system C\,2, viz. i to I, 2 to 3, the colour 1 must occur an even number of times upon the boundary and the colours 2, 3 equally often. The types are Colours 1 16 H 12 10 o 1 2 3 4 o 1 2 3 4 I 2 3 6 5 5 4 6 6 2 7 7 8 8 nine in number. All the types for both contact systems can probably be arranged and in many varieties. Above, fig. 39 £, is given an example of the type Bi6,o,o for the system Ci,i,i. In fig. 40 is given one of the type .#14,1,1 for the system Ci,2. Ci,2 Bi4,i,i 1X 2 / 2\ \ 1 / 3Y2 / 2\ \i/ 3 /\ 3 / 3 \ \l/ 2X.1 X 3 \ \i/ 1X1 X 2 \ \ 3 / 2 X 3 / 3\ \ 3 / 2 X 3 / 2 \ \ 2 X 2 X 2 / 3 \ \ 2 X 3 X X / 3 \ \ 3 X 1 X 3 / l\ >K \ 3 / 2 X 3 / 1\ \ 2 / 2 /\ 2 / 1\ \ 2 / 3X 2 / 1\ \iX 3Y1 / 1\ Fig. 40. 28. A 9- piece set can be formed of the six pieces each of which involves three compartments of the same colour and also the three pieces each of which has two colours twice represented on either side of a diagonal. The set is given in fig. 41 assembling into a square 3x3. 4 1 4' 2X2 2 | 2 M 1 X 3 / 3 \ X Fig. 41 SQUARE PASTIMES 35 The set is symmetrical in three colours, each colour occurring in 12 compartments. There are 12 boundary compartments. For the contact system c~i,i,i, the types are Colours 123 840 822 660 642 4 4 4 five in number. 29. For the contact system Ci,2, viz. 1 to 1, 2 to 3, the types are Colours 123 8 2 2 6 3 3 4 4 4 2 5 5 066 five in number. An arrangement of type .58,2,2 for the system Ci,2 is given in fig. 42a, the boundary colours exhibiting symmetry about the diagonal BD. Cl,2 B8.2.2 Cl,I,I B8,2,2 D A D B 1X 1 X 3 \J \ 3 X iX 3 X 3 \ \ 2 / 2 X 2 / 3 \ \ 2 / l X 2 X 2 \ \ 2 / 3 X 3 / 3 \ \2 / 2X3 / 3\ \3/ ! X 3 \2/ 2X1 / l\ \2/ lX 1 /l\ \l/ ! X 2 / 1\ \i/ 2 X 2 / 2 \ \l/ 2Y1 / 2\ iX 3 /3\ \ 2 X 3 X 2 / 3\ \ 2 X 2 X 2 / 3\ \ 3 / 1X 1 /l\ \ 3 / 1X 3 / 3 \ \ 3 X 3 X 2 / 3 \ Fig. 42 One of type B8,2,2 for the system Ci,i,i is given in fig. 42^ in which considering the colour symmetry about the diagonal AC, it will be noticed that colour 2 appears along DC in the same way as colour 3 appears along BC. It is this kind of symmetry that leads, for this contact system, to the greatest symmetry in the transformations of Part II. 36 NE W MA THEM A TIC A L PAS TIMES 30. In order to design a set based upon the square and involving five colours'we adopt as a principle that every square is to involve four of the colours ; there are to be no repetitions of colour in the compartments of the same square. Four different colours give rise to six squares because four different objects have just six permutations in circular proces- sion. We can select four colours out of five in five different ways so that we can obtain 5x6 or 30 different squares in- volving the five colours. There is little doubt that this would be an interesting set, but in this book we have limited ourselves to sets containing not more than twenty-four pieces and we can reduce the set of 30 as required by importing the condition that every piece is to involve one specified colour, say the colour 1. We thus get the set of twenty-four given in fig. 43 for which the rectangle 6 x 4 is available. Fig. 43- The colours 1, 2, 3, 4, 5 are involved upon 24, 18, 18, 18, 18 compartments respectively. These are all even numbers and as the number of boundary compartments is 20, we have as a condition that each colour must appear, in the contact system £"1,1,1,1,1, an even number of times upon the boundary. More- over each angular point of the rectangle must exhibit two different colours, so that a particular colour cannot appear upon the boundary more than 16 times. The number of boundary SQUARE PASTIMES 37 types is very large ; it is not necessary to write them all down. They proceed for the colours 12 3 4 5 from 16 4 o o o to 06644 Two examples are given in fig. 44. Ci,i,i,i,i Bi4,o,o,4,2 \lx 4 X 3 / 5\ \lX 3 X 4 / 2\ \*X 4 X 2 X 3 \ \*X 2 X 3 /5\ \iX 3 X 2 X 4 \ \iX 2 X 5 X 3 \ \ 5 X 4 X ! /3\ \ 2 / 'X 5 / 4 \ \ 3 X 5X4 X 1\ \ 5 / 4Y1 X 2 \ \ 4 X 1X2 / 5 \ \ 3 / 2Y1 X 5 \ \ 3 X 4X 1 / 5 \ \ 4 X 1X 5 X 2 \ \lx 5X4 X 3 \ \ 2 / 4X1 / 5 \ \ 5 X 1X 2 X 4 \ \sX 2X1 X3\ \5/ 4X3 \ 2 / 3X4 \ 3 X 4X2 /\ \ \ 5 X 2X3 X 1 \ \ 4 / 3X2 / 1\ \3/ 2 X 5 X 1 \ \iX 2X5 X 3 \ \iX 5X4 f 3 \ \ 1 / 4V 3 X 2 \ \ 5 X 3X4 /\ \ \ 5 X 4Y2 X 1 \ \ 5 X 2X3 Xi\ \ 3 X 2 X 4 Xi\ \ 3 / 4X1 X 5 \ \ 2 X 1X4 X 5 x \ \/ 4Y5 X 2 \ \iX 5X3 X 2 \ \iX 3X2 X 4 \ \iX 2X4 X 3 \ \ 5 X 4 X 1 X 3 \ \&x X 2 \ \ 2 X 4 X 5 X 1 \ \ 2 X 5 X 3 X ! \ \ 4 X 3 X 2 X ! \ \ 3 X 2 X 5 X 1 \ \ 3 X 5 X 4 X x \ \ 2 X 4 X 3 X 1 \ \1/ 3 X 4 X 5 \ \l / 4Y2 X 5 \ \iX 2Y3 X 5 \ Cl,I,I,I,I B6,6,2,o,6 Fig. 44. Since there is only symmetry in four colours, four other contact systems are available : C2,I,I,I Cl,2,I,I C2,2,I Cl,2,2 I to 2 I to I I to 2 i to I 3 » 3 2 „ 3 3 .. 4 . 2 „ 3 4 >, 4 4 >, 4 5 ,. 5 4 „ 5 5 ., 5 5 „ 5 The reader will know how to set forth the large numbers of boundary types that arise from these systems of contacts. 38 NE W MA THEM A TICAL PA S TIMES 3 1 . Allusion has been made to varieties of the different types. These should be chosen, for trial, so as to be symmetrical or semi-symmetrical about some axis passing through the centre of the figure. Thus in the case of the rectangle in fig. 45 a the chosen axis may be AB, or CD or any other axis EF passing through the centre of the figure. If just two equally numerous colours are upon the boundary we may, for the axis AB, take just half of the compartments of each colour on each side of AB and arrange them symmetrically. Or we may make the compartments to the left of AB any we please and then take a compartment, to the right, of the colour 1 when the corresponding compartment to the left has the colour 2. Such arrangements for the axes AB, EF might be as in fig. 45 b and c. E A C 1 1 \1 2 2 2 h 2 2 1 1 *2 1 1 1 2\ 2 2 Fig. 45- These are extreme cases of what should be attempted. The simplest case would be to place all the compartments of one of the colours on the same side of the axis chosen ; and the simplest should always be chosen in the first place. When symmetrical boundaries are not practicable the interest is greatly diminished. It will be found that symmetrical boundaries lend themselves particularly to the transformations of Part II. RIGHT-ANGLED TRIANGLE PASTIMES 39 RIGHT-ANGLED TRIANGLE PASTIMES It cottons well, it cannot choose but beare A pretty napp. Family of Love, D. 3 b. 32. Take a right-angled triangle, which is half of a square, as in fig. 46^. Find its centre O by taking CO equal to two-thirds of the perpendicular drawn from C upon A B. Then the three straight lines CO, AO, BO divide the triangle into three compartments of equal areas. If we have three colours and each triangle is to have three different colours in its compartments we obtain six different triangles which can be assembled, with the contact system Cs,i,\, in the form of fig. 46^ with any chosen colour monopolising the boundary. /1 3 s * -^ \^ '2 ^\ r^B S 3 2 / l\ ^^ f/ / £^—"2^ ■^3~~^~^ \ ^vJL' •~~3l^"7 \ / \V 3, ^3 2 S .2 1/ c '^l~~~ I I I ) Fig. 46. If we are given four colours we triangles which can be arranged in are led to 24 different the hexagonal form of fig. 46 c, a figure consisting of four triangles and ten squares. Two triangles may fit into any one of these squares with the long sides either vertical or horizontal. This is a new feature which adds interest to the study of the pieces. 40 NE W MA THEM A TICAL PASTIMES This is exhibited by two arrangements for the contact system Ci,i,i,i which are given m fig. 47. Cl,I,I,I Bi2,o,o,o Ci,i,i,i B6,6,o,o Fig. 47. The first is symmetrical in colour about the axis AB. The second diners from the first both in type of boundary and in internal structure. In the first the triangles have two aspects, and in the second four aspects. In order to deal with the internal structure by type we may suppose the type to depend upon the numbers of triangles which have their long sides horizontal and vertical respectively. We might then describe the first arrangement by B\2, ,0,0,0 ^24,0, and the second by £6,6,0,0 A 16,8. Altogether there are 72 compartments, 24 long (L for long) and 48 short (S for short). Since there is symmetry in four colours, each colour appears upon 6L and 12S compartments. RIGHT-ANGLED TRIANGLE PASTIMES 41 Upon the boundary there are 4Z and 8S. Hence for the contact system Ci, 1,1,1, each colour must have an even number of L and an even number of 5 compartments upon the boundary- So that as regards L we have for Colours 1234 4000 2200 two types ; and as regards 6" we have for Colours 1234 8000 6200 4400 4220 2222 five types ; with numerous varieties both in internal structure and in boundary. Cl,I,2 Bl2,0,0,0 A24,o C2,2 B6,6,o,o A24,o Fig. 48. 42 NE W MA THEM A TICAL PA S TIMES We must presume that each of the L types may occur with each of the 5 so that there may be ten types in all. Also of internal structure we may have the eleven types A24.fi, A22,2, A20,4, Ai8,6, ^16,8, A 14,10, Ai2,\2, Aio,i4, A8,i6, A6,i8, ^4,20, and of every type of boundary except one and of every type of internal structure except one there may be varieties. In regard to other contact systems, there is no reason why the L compartments should have the same system as the 6", because they do not clash at all. 33. Other systems available for both L and 5 are Ci,i,2, viz. 1 to 1, 2 to 2, 3 to 4, and C2,2, viz. 1 to 2, 3 to 4. Assemblages for the systems (for both L and S) Ci,i,2, 62,2 are given in fig. 48. THE CUBE PASTIME 34. A cube has six faces, twelve edges and eight summits. If we are allowed six different colours in order to colour the faces each with a different colour, we find that we can make 30 differently coloured cubes. It is a well-known rule, applicable to any regular solid, that in order to ascertain the number of different cubes or other solids that can be made by colouring the faces with different colours it is merely necessary to divide the factorial of the number of faces by twice the number of edges. Thus in the case of the cube we have 6x15x4x3x2x1 2x12 J So also in the case of the tetrahedron, composed of four equilateral triangles, which has four faces and six edges we have 4x3x2x1 - — = •> 2x6 and so on for any regular solid. We now construct these 30 cubes and, denoting the colours by numbers, we represent any such cube in a diagram as in THE CUBE PASTIME 43 fig. 49 a, the cube being supposed resting upon a table and viewed from above. A 3 A' 5 B 3 B ' 4 C 4 C' 5 2 1 4 2 1^4 2| 1 I5 2 l|5 2 13 2 1 3 5 3 4 3 5 4 D 3 D' 5 E 3 E' 4 F 4 F' 5 12 4 12 4 1 2 5 12 5 1 2 3 12 5 3 4 3 5 4 1 G 1 G' 5 H 1 H' 4 1 4 1' 5 4 2 3 4 2 3 4 2 3 5 2 3 5 2 3 1 2 3 R 1 a. 1 R Fig. 61. Cl,I,2 Fig. 62. Sic ludus animo debet aliquando dari, Ad cogitandum melior ut redeat tibi. Phaedr. /"a*, in. 14. 41. For the third contact system 62,2 which is 1 to 2, 3 to 4 we select for the colours as in fig. 63 ; one piece again vanishes for the transformation and we are left with the twenty- three pieces shewn in fig. 64. THE TRANSFORMATION OF PART I 59 6o NE W MA THEM A TICAL PA S TIMES In fig. 65 the pieces are assembled according to the diagram (72,2 B6,6,o,o of Part 1. C2.2 Fig. 65. Behold, said Pas, a whole dicker of wit. Pembroke, Arcadia, p. 393. 42. For the transformation of the 10-piece set on the contact system Ci,i,i we may take the colours as in fig. 66 a Fig. 66. and we find for the first figured assemblage Ci,i,i Bi,i,6, the fig. 66 b shewing the shapes of the transformed pieces and of the boundary for the particular case. THE TRANSFORMATION OF PART I 61 43. Skurffe by his nine-bones swears, and well he may, All know a fellon eate the tenth away. HERRICK. Similarly for the contact system £2,1, taking the colours as in fig. 67 a for the first figured assemblage £2,1 B\&,o we have fig. 67 b shewing the shapes of the transformed pieces and of the particular boundary. One piece vanishes. Fig. 67. 44. For the remaining contact system Ci,2 we may take the colours as in fig. 68 a, and for the first figured assemblage C\,2 i>S,3,o we obtain fig. 68 £ giving the shapes of the pieces which differ from those in the preceding case. Moreover there are ten of them instead of nine. Fig. 68. 45. For the transformation of the 13-piece set we may for the contact system Ci, 1,1,1 select the pieces from those trans- 62 NE W MA THEM A TIC A L PAS TIMES formed from the complete set of 24 for the same contact system. The second figured arrangement is then as in fig. 69. Ci, 1,1,1 63,3,3,0 Fig. 69. 46. For the contact system Ci,2,i, viz. 1 to 1, 2 to 3,4 to 4, we may take the colours as in fig. 70 a ; we have then fig. 70 b. Ci,2,i Bi,4,4,o Fig. 70. Affirm'd the trigons, chopp'd and changed. Hudib. 11. iii. 47. Coming next to the five-colour triangle we choose, for the case Ci, 1,1, 1,1, the colours as in fig. 71. The twenty pieces THE TRANSFORMATION OF PART I 63 are as in fig. 72 and the assemblage for Ci, 1,1, 1,1 B 12,0,0,0,0 is as in fig. 73. I- 2- Fig. 71. 6 4 NE W MA THEM A TICAL PA S TIMES Cl,I,I.I,I Bi2,o,o.o,o Fig. 73- 48. For the system Cl,i,i,2, taking the colours as in fig. 75 a, the pieces are as in fig. 75 &, the assemblage as in fig. 74. Cl,I,I,2 Bl2,0,0,0,0 Fig- 74- THE TRANSFORMATION OF PART I 65 be E r-> h cq go ^1 m THE TRANSFORMATION OF PART I Cl,2,2 Bl2,0,0,0,0 67 Fig. 77- or Fig. 78. (See page 68.) 5-2 68 NE W MA THEM A TICAL PA S TIMES Has matter more than motion? Has it thought, Judgment and genius? is it deeply learn'd In mathematics? The Consolation. 50. In the case of pieces based upon the square the com- partment boundaries must be modified so as to bear a convenient relation to the angles of the parallelogram of the compartments. The parallelogram is here a square and an angle of 45 " should be often in evidence. For the contact system Ci, 1,1 suitable boundaries for the 24-piece set may be as in fig. 78 a or b, p. 67. The system of pieces for the first of these is given in fig. 78. For any given boundary defined by colours as in Part I, the figure into which the pieces may be assembled can be drawn. Thus, as an example, if we take a type and variety such that the colours 2, 3 occur alternately we find the figure 79 b and so on. In each case, we make the boundary transformation. Ci,i,i Bio, 10,0 Fig. 79- THE TRANSFORMATION OF PART I 6 9 A great many symmetrical figures can usually be drawn, with the exercise of a little ingenuity, for any contact system. The assemblage for Ci,i,i B 10,10,0 is given in fig. ?ga. After mutch counsayle and great tyme contrived in their several examinations. Pal. of Pleas. D. d. 2. bfl E co)*(co 70 NE W MA THEM A TIC A L PAS TIMES The power and corrigible authority of this lies in our wills. Othello, I. iii. 51. With the contact system C\,2, viz. i to i, 2 to 3, we have a good choice for the compartment boundaries ; we may take, for example, the colours as in fig. 80 a, b, c. In the last straight line system of boundaries the piece, which has every compartment coloured 3, vanishes so that the set is one of 23 pieces. A reference to the boundary types of Part I shews that the set will fit into a large number of symmetrical boundaries. It should be noted that this 23-piece set is really not suitable because the pieces in fig. %od,e,f transform into those in fig. 80^, h, k, each of which, consisting of two pieces meeting at a point, cannot be handled when constructed in cardboard or wood, but only in diagrams. These pieces must always be avoided. The twenty-four pieces are as in fig. 80/ and for the assemblage Ci,2 .#20,0,0 we have fig. 80 m. 52. In the case of the 20-piece set of Pastime no. 20 and the contact system Ci,i,i we have only to discard the pieces in fig. 81 and then to transform the coloured boundaries in the Fig. 8 1 usual manner to obtain a set of figures into which the pieces may be assembled. 53. For the contact system Cl,2, viz. 1 to 1, 2 to 3, we discard from the corresponding set of 24 pieces those shewn in fig. 82 and proceed in the same way. CiatJ Fig. 82. THE TRANSFORMATION OF PART T 7i 54. For the contact system £2,1, viz. 1 to 2, 3 to 3, we may discard from the set of 24 pieces corresponding to Ci,2 the four shewn in fig. 83 and take the correspondence between colour and boundary to be that shewn in the same figure. CO □ □ : Fig. 83. The reader will find the 20-piece set very interesting but the pieces appear to be difficult to assemble, compared with some of the other sets. Nihil tam difficile est quin quaerendo investigari possiet. Terence. 55. The 16-piece set transformed for the contact system Ci,i,i is shewn in fig. 84 made up into a square of boundary type 5i6,o,o. Ci,i,i Bi6,o,o Fig. 84. These same pieces may be fitted into a number of symmetrical boundary lines as will be evident to the reader on comparison with the colour schemes of Part I. 56. The reader will probably have no difficulty in dealing with the contact system C2,i, viz, 1 to 2, 3 to 3. The author has not particularly examined it, but he recommends it with confidence. Sempre avviene Che dove men si sa, piii si sospetta. Machiavelli. 72 NE W MA THEM A TIC A L PA S TIMES 57. He has however put the contact system Ci,2, viz. i to i, 2 to 3, through much experimental work, and will deal with it in some detail. He transforms through the correspondence given in fig. 85 a. It is shewn in the form of 15 pieces since the 16th piece derived from fig. 85 b vanishes, but it still must be regarded as a 16-piece set forming up into a square 4x4. The assemblage for C\,2 i?i6,o,o is given in fig. 85 c. Cl,2 B16.O.O Fifteen symmetrical boundaries, out of a large number that may be constructed, are shewn in figs. 86, 87 and 88, inside each of which the same pieces may be assembled. Much design Is seen in all their motions, all their modes : Design implies intelligence and art. The Consolation. And soo they thre departed thens and rode forth as faste as ever they my3t tyl that they cam to the forbond of that mount. Morte cV Arthur, 1. 139. For the transformation which has the correspondence of fig. 85 d the pieces are shewn assembled in a square and three diagrams of boundaries are also given in fig. 89. Mark how the labyrinthian turns they take, The circles intricate and mystic maze. The Consolation. THE TRANSFORMATION OF PART I 75 Fig. 86. 74 NE W MA THEM A TIC A L PA S TIMES Fig. 87. THE TRANSFORMATION OF PART I 75 Fig. 88. The 1 5 -piece set for the contact system Ci,i,i is shewn transformed and put up into the rectangle of fig. 90 and other boundaries for the same pieces are shewn in fig. 91. For the contact system C\,2 we have similar results as in fig. 92. For the 9-piece set we may, for the system Ci,i,i, transpose to the set in the two upper rows of fig. 93 and for the system Ci,2 to the set in the two lower rows of the same figure. For the system Ci ,i,i a symmetrical boundary B8,2,2 is shewn in fig. 94. For the system Ci,2 symmetrical boundaries are shewn in %• 95- 76 NE W MA THEM A TIC A L PAS TIMES Ci,2 Bi6,o,o Fig. 89. C 1,1,1 B 16, 0,0 Fig. 90. THE TRANSFORMATION OF PART I 77 Fig. 91. Ci,2 Bi6,o,o Fig. 92 78 NEW MA THEM A TICAL PA S TIMES Ci,i,i Cl,2 Fig- 93- Fig. 94. Ci,i,i B8,2,2 Fig- 95- THE TRANSFORMATION OF PART I 79 58. In fig. 96a the set of 24 squares involving five colours unrepeated, are shewn assembled for the contact system 1 to 1 , 2 to 3, 4 to 5 and a transformation according to fig. 96 b is shewn in fig. 96 c. The symmetry about the central horizontal line is to be remarked. Cl,2,2 B20.0.0 \l / 2X4 /5\ \ 1 / 5X 3 / 4\ \ 1 / 2 X 5 / 4 \ 4 X 3 X 2 \ \iX 2 X 5 X 3 X Vx 4 X 2 X 3 \ \ 4 X 3 X 2 \5/ 3 X 2 / 1 \ \ 5 / 3 X 4 / 1\ \sX 5 X* X 1 \ \ 2 X 5X 4 / 1\ \ 2 X 5 X 3 X 1 \ 3V2 X \ 3 X 2 / 5 \ 3 X 4 X 5 \ \iX 5X 4 X 3 \ \iX 5 X 4 X 2 \ \iX 5 X 3 X 2 \ \ 5 X 2X4 X\ \ 4 / 5X3 / 1 \ \ 4 / 2X5 /1\ \ 2 X 4 X 3 X 1 \ \ 3 X 2 X 5 X 1 \ \ 3 X 4 X 2 X 1 \ a Cl,2,2 Bl2,4,4 Fig. 96. It is obvious that there is ample room for experiments with this interesting set. PART III THE DESIGN OF 'REPEATING PATTERNS' FOR DECORATIVE WORK The story without an end that angels throng to hear. M. F. TUPPER. 59. The ideas which have been prominent in Parts I and II lead to a most interesting Pastime — the design of repeating patterns for various kinds of decoration. In Part I the notion was to connect colours with the compartments of various polygons, usually triangles and squares, in such wise as to realise every possible combination of colours on the compartments. The set thus formed has the property that no two pieces are alike. There are no duplicates. Smaller sets are chosen from these according to definite principles and laws but the pieces are in every case differently coloured. In the transformations of Part II the passage is made from pieces of the same shape but differently coloured to pieces of the same colour but differently shaped. The transformations do not affect the cardinal property that no two pieces of any set are alike. It was shewn that the pieces of a set can be assembled so as to fit inside a boundary which can be specified as to colour or shape and that, for a given set of pieces, the different boundaries that can be predicted exist, for various contact systems, in very large numbers. We now examine what can be done with pieces which, far from being all different, are all exactly similar in shape and size. Certain pieces of the same size and shape can, everybody knows, be fitted together so as to completely cover any floor, wall, ceiling or other flat surface, and no attention need be paid to the boundary. The pieces, when assembled, can be cut along any desired boundary. For practical purposes the boundary may be ignored. The simplest repeating patterns are shewn in fig. 97 and are met with everywhere. THE DESIGN OF REPEATING PATTERNS 81 We make a systematic search for pieces of other shapes which possess the same property of completely covering any flat surface by simple repetition. These ' repeating patterns ' are everywhere visible in con- siderable variety. Public buildings and private homes exhibit examples on floors, walls and ceilings. Street pavements, palings, furniture, wearing apparel, woven fabrics and artistic designs of all sorts bring them constantly before our eves. We feel that the ideas brought before the reader in this book will enable him to take quite a fresh interest in these matters. Without making any pretence of exhausting the subject we base our study upon the equilateral triangle, the square, and the regular hexagon. These are probably the most important bases. I saw it budding in beauty ; I felt the magic of its smile. M. F. Tupper. 60. It is a very trivial fact that the equilateral triangle is a repeating pattern for any flat surface ; but one useful remark may be made, viz. that in such an assemblage, a portion of which is represented above, the triangle has always just two aspects or orientations, because the fitting is invariably so that of two adjacent pieces one is always like the other, only upside down, as may be readily seen. This is not the case in either the square or hexagonal pave- ments, which present only one aspect. The equilateral triangle, which as a repeating pattern has two aspects, gives immediate rise to other repeating patterns upon very simple principles. If we can in any manner separate it into three or more parts of the same size and shape we obtain again a repeating pattern. m. p. 6 82 NE W MA THEM A TICAL PA S TIMES For example we can as in fig. 98 from the centre draw perpendiculars upon the sides and obtain the equal and similar quadrilaterals of which one is ODBF, a repeating pattern which in an assemblage has 3 x 2 or 6 aspects.. A B D' D Fig. 98. Also we may draw the lines OD ', OE' , OF' making equal angles with OD, OE, OF and obtain the repeating pattern OD'BF'. This system of quadrilaterals has two opposite angles 60° and 120 respectively. 61. Similarly from the square we can at once derive an unlimited number of repeating patterns by drawing lines, straight or not, through the centre as in fig. 99. Fig. 99. The first of these figures shews a separation into two or four parts, the second into two parts, the third into two parts. There is no limit to the number of such separations. Any number of squares may be placed in contact to form a pattern, and a little consideration shews that it follows that every rectangle is a repeating pattern. 'Who made the spider parallels design, Sure as de Moivre, without rule or line ? Essay on Man. If we draw a series of equidistant parallel lines and cut them with any other series of equidistant parallels, we see at once that every parallelogram is a repeating pattern with one aspect, and since every parallelogram can be separated into two equal and similar triangles ; and vice versa every triangle is the half of a THE DESIGN OF REPEATING PATTERNS 83 parallelogram ; it follows that every triangle is a repeating pattern, with two aspects. As a general principle, if any repeating pattern can be divided into two or more equal and similar parts, a part thus formed is a repeating pattern*. 62. When we assemble a number of patterns so as to cover a flat space we can always observe combinations of patterns which are themselves patterns. If the pattern has a certain number of aspects in the assemblage and we. draw a boundary which includes one pattern of each aspect, the figure enclosed by the boundary is a pattern with one aspect. For example the equilateral triangle has two aspects, and we can draw a regular hexagon which encloses six triangles, three of each aspect. Thence we conclude at once that the regular hexagon is a pattern having one aspect. 63. The regular hexagon may be derived as a repeating pattern from the equilateral triangle in another interesting manner. We can colour an assemblage of triangles as in fig. 100 a with two colours so that each triangle is adjacent to three triangles of a different colour. Fig. 100. It is by others termed a fesse between two gemels. And that is as farr from the marke as the other ; for a gemel ever goeth by paires, or couples, and not to be separated. R. Holme, Academy of Armory, I. iii. 77. We can now transform from colour to shape by the method of Part II on the contact system 1 to 2, taking for the colours * Triangles may be taken to be equal and similar in plane geometry when they can be made to exactly coincide by movements in the plane; a movement such as ' turning over ' is not in view. 6—2 84 NE W MA THEM A TIC A L PAS TIMES the boundaries as in fig. 100 b, when we find that the triangles coloured 2 entirely disappear and the triangles coloured 1 become regular hexagons. We have transformed the tessellation from being triangular to being hexagonal. 64. The mathematical reader will recall that, in space, the cube and the rhombic dodecahedron are similarly associated. In that case we start with a number of cubes and give them two colours as in a three-dimensional chessboard. The cubes coloured 2 are divided into six portions by lines joining the centre to the eight summits ; each portion being a pyramid whose base is a face of the cube and whose height is equal to half of the cube's edge. Each cube coloured 1 is adjacent to six cubes coloured 2, and transformation upon the contact system 1 to 2 with pyramidal boundaries causes the cubes coloured 2 to disappear and those coloured 1 to become rhombic dodecahedra. Space is thus exhibited as partitioned into rhombic dodeca- hedra, a fact of great importance in crystallography and other parts of applied mathematics and mathematical physics. The foregoing considerations bring out the importance of designing patterns which possess symmetry about one or more axes ; from these other patterns can be at once derived. 65. In Part II we adopted certain forms of compartment boundaries in association with the contact systems, and the design of such boundaries was subject to the transformed pieces being readily handled. This operated as a considerable restric- tion which can now be removed because our pieces are merely delineated ; they are not handled at all. The restriction took two forms. b Fig. 101. A(JBO in fig. 101 a being the compartment parallelogram, in drawing the new boundary between A and B and within the THE TRIANGLE BASE 85 parallelogram we decided that no perpendicular to AB should cut the new boundary in more than one point ; because if it did so the piece would be inconvenient to handle. Again it was tacitly agreed that the portion of the piece bounded by the new boundary line and AOB should have no holes in it. Both of these restrictions can now be removed and we can deal with shapes such as in fig. 101 b and 101 c where in the second case there is an inner as well as an outer boundary, the triangular piece AEB being cut out. THE TRIANGLE BASE 66. Consider the triangle to be divided into compartments numbered i, 2, 3 so that, travelling clockwise round the figure, the numbers are in ascending order of magnitude as in fig. 102 a. Fig. 102. It is not quite a trivial remark that any number of triangles identical with this one can be assembled so that the compart- ments numbered 1, 2, 3 are adjacent to others numbered 1, 2, 3 respectively. The annexed diagram, fig. 102 b, establishes this and shews that the number of aspects of the triangle is not increased by the numbering. We knew before that it could not be less than two. We know now that it remains precisely two. Also the internal structure is always the same. The reader who has become acquainted with Part II will observe that the straight line boundaries of the compartments of the triangles can be altered in many ways so that the pieces remain of the same shape and size. In other words the numbered triangular piece can have its shape altered to an indefinite extent and still preserve the property of being a repeating pattern. The assemblage under examination has the contact system 1 to 1, 2 to 2, 3 to 3, and we may employ any of the boundaries 86 NEW MATHEMATICAL PASTIMES which belong to the system. The numbers I, 2, 3 need not differ from one another. The particular cases 1, 1, 1; 1, 1, 2 are included in the discussion. 67. We are about to study the design of repeating patterns and the reader will readily realise that a pretty pattern must always be an object, and a principal object, in the designer's mind. This can best be accomplished by paying attention to symmetry of shape. We must learn how to select boundaries which will produce symmetrical patterns. This matter must be studied in the case of each contact system and each base. To assist us we must have a typical boundary before us, say A in fig. 103 a, and associate with it another boundary which we will call the inverse of A and denote by iA. b N^ c L Fig. 103. iA is the reflexion of A in a mirror placed to its right, or it is A rotated about its right-hand extremity, in a plane perpen- dicular to the plane of the paper, until it occupies the position of iA. It will be noticed that if A be the original single straight line, inversion does not alter it. In the first place it is clear that if the numbers 1, 2, 3 are associated with the same boundary A, however we choose A, the piece will possess trilateral symmetry. For example, for the typical boundary we get fig. 103 b. The only other kind of symmetry is that about an axis which bisects one of the angles of the triangle ; say about the line chain-dotted in fig. 103 c. As regards the sides which meet at the angle fixed upon, the only, possibility is to associate them with the boundaries A, iA respectively. THE TRIANGLE BASE 87 The reader is reminded that a compartment-boundary is to be viewed from the centre of the base, looking outwards. In regard to the third side it may remain associated with the single straight line. We call this L for short. Notice also that the shape is altered when we interchange A and iA. For the typical boundary we thus get the two patterns of fig. 104 a and b. b c a ) Fig. 104. Inspection of the numbered diagrams shews that we can pass from them to repeating patterns, fig. 104 c and d, of which the first is the regular hexagon again. In both cases the sides L have been made adjacent. Other patterns are obtained by making the sides A or the sides iA adjacent in each case. 68. The particular case 2 equal to 3 may be noticed. It is interesting from the circumstance that all pieces of this type may be assembled so as to have either two or six aspects. This is made clear by the diagrams in fig. 105 in which the diagram on the left exhibits two aspects and that on the right six. \\ 2 ^7 A A 1 a /a l\ Z^2\\ /^2X\ \\2^V Y\2/7 \l 7 2 lV 2/ //2\\ Two aspects Six aspects Fig. 105. 88 NE W MA THEM A TICAL PASTIMES In fig. 106 some patterns and assemblages belonging to this system are given and in particular the ' Helmet ' pattern is Two aspects Six aspects Fig- 1 06. THE TRIANGLE BASE 89 shewn assembled so as to exhibit two aspects, on the left, and six aspects, on the right. Besides the equilateral triangle, since every triangle is a repeating pattern, we may take as base the isosceles triangle in general as in fig. 107 and the isosceles right-angled triangle in particular. Fig. 107. The diagram shews that, for this contact system, the treat- ment is the same. The reader will find no difficulty with this general case. So that the art and practic part of life Must be the mistress to this theoric. Hen. V. I. i. 69. A second and entirely different system of patterns is obtainable from the contact system 1 to 1, 2 to 3 which we call C\,2. Inspection of the depicted assemblages of six triangles in the form of a hexagon as shewn in fig. 108 shews that the piece has six aspects in each of two constructions, which are at the disposition of the designer. Part 1 1 again shews us how to alter the shape of the piece so that it will remain a repeating pattern. Herm. Methinks I see these things with parted eye, When ev'rything seems double. Hel. So methinks : And I have found Demetrius like a gemel, Mine own, and not mine own. Mids. N. Dr. iv. i. The boundary of compartment 1 is altered in any manner which appertains to the first system already dealt with, whilst 90 NE W MA THEM A TICAL PASTIMES ^) Six aspects Fig. 1 08. THE SQUARE BASE 9i for compartments 2 and 3 we have the boundaries of Part II as in the numbered row of fig. 108 where the 3rd is the inverse of the 2nd, while the 1st and 4th are self-inverse. Some examples of repeating patterns, issuing from this system, and an assemblage are given in the same fig. (108). Symmetry can be secured by taking six identical pieces as in the annexed diagram, fig. 109 a, which is a well-known pattern composed of three hexagons. /2/ \i\ y/\ l\" / IJ2 \\ / 2 / \ \2 1 // /2/ \\ 2T1 / \1V b M rs Fig. 109. The isosceles right-angled triangle is also available for this contact system, as shewn by the diagram of fig. 109 £. The system is 1 to 2, 3 to 3, and the derived patterns have four aspects when assembled. THE SQUARE BASE For the vvals glistered with red marble and pargeting of divers colours, yea all the house was paved with checker and tesseled worke. Knolles' Hist, of Turks. But I of these will wrest an alphabet, And by still practice learn to know thy meaning. Tit. Androti. ill. ii. 70. We recall that the piece does not, of necessity, have more than one aspect. Numbering the compartments as in fig. 1 10 a we can always assemble according to the contact system 1 to 1, 2 to 2, 3 to 3, 4 to 4 which is denoted by Ci, 1,1,1. This is shewn in fig. uo£, a diagram which can be repeated indefinitely. 92 NE W MA THEM A TIC A L PAS TIMES The arrangement proves that it is possible in one way only and that the piece has two aspects, one being the other rotated through two right angles. iX 3 / * \ \*X 3Yl / 2\ 3X1 / 2 \ \ 2 / 1 X 3 /4\ X ^ \i/ W 1X1 ix 1 /2\ /2\ 1V1 iX 1 /l\ /l\ X / \ 1 / \2 / 1X2 2X1 /2\ / 1\ \ 2 / \l/ 2X1 iX 2 /l\ /2\ \iX iX 2 /2\ \iX 2V1 /2 \ \ 2 / iX 2 /1\ \ 2 / 2Yl / 1\ h \2/ iVl /2\ fc 1X 2 \l/ *X 2 / 3 \ \ 3 X 2V1 / 1 \ \sX 2Y1 /l\ \iX 1X 2 X 3 \ 2 X3 \ 2 X 1Y1 / 3 \ \ 2 X 1V1 /3\ \sX 1X1 X 2 \ \3/ 1X1 /I \ \2/ \3/ 1Y1 1V1 /3\ / 2 \ \ 3 X \v 1X1 00 /2\ /3\ m n Fig. no. In the particular cases: fig. hoc has one arrangement with one aspect of piece. Fig. nod has only one symmetrical arrangement which deserves consideration as shewn in fig. noe with two aspects of piece. The piece in fig. no/" has two arrangements shewn in fig. no^(two aspects), in fig. 110^ (four aspects). The piece in fig. 110 k has obviously one arrangement with one aspect of piece. The piece in fig. 1 10/ has one arrangement with two aspects as shewn in fig. 110m; and finally the piece in fig. non has the arrangements of fig. no, and/, each with two aspects. 71. To construct repeating patterns we take four shapes of boundary drawn from the first system of boundaries, taking care to choose those that are appropriate to the square base. THE SQUARE BASE 93 We are restricted to the square A 0B0' in fig. in and no circular arc employed must lie outside of it at any portion of its course between A and B. When the arc passes through A the centre from which it is struck should be taken upon either AO' 1 %J Cf4 V\y£ or AO produced if necessary so that it touches OA or O'A at the point A. Part II points out convenient forms and leads us to a variety of patterns of which samples are shewn in fig. ill and assemblages in fig. 1 1 2 where a and b exhibit two aspects and c four. 94 NE W MA THEM A TIC A L PAS TIMES Seizes the prompt occasion, makes the thought Start into instant action, and at once Plans and performs, resolves and executes ! Hannah More. ^-^v Fig. 112. Two other assemblages are shewn in fig. 1 1 3 where fig. a exhibits two aspects and b four. Fig. 113. 5 YMMETR Y OF PA TTERN 95 SYMMETRY OF PATTERN Veluti in speculum. Let no face be kept in mind, But the fair of Rosalind. As You Like It, ill. ii. 72. We now consider some more principles which lead to symmetry of pattern. We take four compartment boundaries as in fig. 114a; we have already defined iA ; we call the one beneath iA the com- plement of A and denote it by cA. It is called the ' complement ' because the compartment with the boundary cA will fit into (lie adjacent to) the compartment with the boundary A. So also the boundary beneath A is called the complement of the inverse of A. It will be noticed that ciA and cA are reflexions of A and iA in the horizontal chain-line delineated. Also that icA is the same as ciA and that ccA, UA both leave A unaltered. A^~^ „ ^~«_ iA X A b iA X iA iA A A A C iA d Fig. 114. cA and A on the one hand and ciA and iA on the other have to do with the second kind of contact between compartments. We are immediately concerned only with the first kind of contact and we see that we may have, as in fig. 114^, the same boundary to each compartment giving quadrilateral symmetry. Or we may have the designs with diagonal symmetry in fig. 114*: and d where the first has symmetry about both diagonals, because both bisect angles contained by A and iA ; and the second has only symmetry about the diagonal drawn because the other does not satisfy the A and iA condition. Examples are shewn in fig. 1 14 £ and/! 9 6 NEW MATHEMATICAL PASTIMES We will have, if this fadge not, an antick. I beseech you, follow. Love's L. L.v.i. 73. To obtain symmetry about a straight line through the centre parallel to a side the form that must be taken is that in fig. 1 1 5 a of which an example is fig. n$b. iA '<> < B iA B A > iB C iA d iB e L a b iB B iB B h k Fig. ii m With two boundary forms A, B we have four forms, fig. 1 1 5 c, d, e,f, each of which has symmetry about one diagonal. Examples for the boundaries in fig. 115^ are shewn in fig. 115/2, k, I, and m. 74. The next contact system to study is that in which we have 1 to 1, 3 to 3, 2 to 4 and inspection of the diagram in fig. 116 shews that there is one arrangement with two aspects of piece. \2 / \4/ 1X3 3X 1 \2/ \V 1Y3 3X 1 /4\ /2\ Fitf 116. This is also the case when 1 and 2 are identical. For the boundaries of compartments 2 and 4 we choose from those brought forward in Part II that are appropriate to the square base. Patterns of quite a new character emerge, as is evident from the samples in fig. 117. SYMMETRY OF PATTERN 97 /\ X \ /\ w, A/\ /\ s\/s *7sX vw Two aspects Two aspects Fig. 117. Two aspects 75. Here the only possible symmetry is that about an axis through the centre parallel to a side. The general forms are as in fig. 118 a and b, where B, A belong to the first and second kinds of contact respectively. In particular either A or B may have the form L. 9 8 NE W MA THEM A TIC A L PAS TIMES A must also be symmetrical about its extremities, so that A=iA. As an example, for the forms in fig. 118c we find fig. nSd, a piece with windows. iB iB iA a iA B B b A Fig. 1 1 8. 4 (Deh come e ver che) subito travato II bello piace a chi non e malato. Bracciolini. 76. A few minutes trial will convince the reader that the contact system i to i, 2 to 2, 3 to 4 is impossible unless 1 and 2 are identical. For the case specified we have the arrangement shewn in the diagram on the left of fig. 1 19 involving four aspects, and no other essentially different from it. \l / 1X3 /4\ \l/ 4X 1 /3\ 1X 4 / 1 N 3X 1 / 1\ Fig. 119. This system does not lend itself to interesting symmetrical patterns, for two reasons. In the first place 3 and 4 are not opposite compartments, and in the second the inverse principle cannot be applied to the two identical compartments 1. However, a few forms are given in fig. 1 19. 77. A more interesting set of patterns arises from the contact system 1 to 3, 2 to 4. Here the contacts are of oppositely situated compartments and the inverse principle is also available. The arrangement is shewn in fig. 120 a. It involves only one aspect of piece. It can be repeated to any extent on all sides. Exceptionally the piece in fig. 1 20 b with the contact system 1 to 3, can be assembled in three different ways having one, two and four aspects respectively. SYMMETRY OF PATTERN 99 This is shewn by the diagrams 1 20 e with one aspect, 1 20 d with two aspects and 1 20 e with four aspects. \2 / 1Y3 / 4 \ \2// IX 3 / 4 \ \ 2 / 1Y3 / 4 \ \ 2 / lY3 / 4 \ 4 3 1X3 1X3 / 3 \ \l/ 1X3 /3\ IX 3 /3\ \1/ \3/ lX3 IX 3 /3\ /l\ \ 1/ \3/ 1X 3 1X 3 /3\ /1\ \l/ 1X 3 / 3\ \ 3 X 1X 3 3\l X 3 \ 3X 1 /1 \ Fig. 120. d Some patterns and assemblages are shewn in fig. 121 where a has one aspect, b two aspects, c four aspects and d one aspect. ^Qoaa Fig. 121. 7—2 IOO NE W MA THEM A TICAL PA S TIMES 78. The three symmetrical forms are shewn in figs. 122 a, b, and c where for the first of these A must be identical with its inverse. Examples of the last two are figs. 122 d and e, and of the first fig. 122/. iA cA iA a A ciA b cA cA ciA d Fig. 122. 4 79. Lastly we have the contact system 1 to 4, 2 to 3. The arrangement is as in fig. 123. 1Y3 / 4 \ 2 X 4 \ 4 / 3V1 /2\ Four aspects Fig. 123. The arrangement can be, clearly, repeated to any extent. Some examples of the patterns that arise are given in fig. 124 where assemblage a has two aspects and assemblage b has four. The reader will observe that the tessellation (fig. 124^) is composed of two sets of hexagons at right angles to one another. Each hexagon contains four equilateral pentagons, one of each aspect. 80. For symmetry about an axis parallel to a side we have fig. 125a where A is a form which is identical with its inverse; and for symmetry about a diagonal the two forms in fig. 125 £ and c when A is as in fig. 125 d, the triangle numbered zero being cut out ; the pattern arising from the last form but one is curious, as shewn in fig. 125 e. It has four aspects and requires four colours when assembled. In the assemblage shewn in fig. 125/" the numbers denote different colours. Pieces having the same aspect are given the same colour. 5 YMMETR Y OF PA TTERN 101 codooQa Fig. 124. 102 NE W MA THEM A TICAL PASTIMES / V^ Fig. 125. Parfois dans un coin triste et noir pousse une fleur. FRAN901S COPPEE. See'st thou the gaze-hound ! how with glance severe From the close herd he marks the destin'd deer. Steele's Miscellanie. 81. The tessellation of pentagons that has been depicted is one of the most remarkable that can be met with in the subject. If the patterns that have been given above, for the contact system under examination 1 to 4, 2 to 3, be inspected they will be found to include three convex pentagons, out of an infinite number that it is possible to construct. In particular the first given of the three is equilateral. Taking the base square ABCD as in fig. 126 we have to find compartment boundaries for the contacts 1 to 4, 2 to 3 that will Fig. 126. lie wholly inside the compartment square AOBE and the three others similarly situated in regard to the other sides BC, CD, DA. In order that the resulting pattern may be a convex pentagon we choose any point P in EB and take a compartment boundary S YMME TR Y OF PA TTERN 103 APB for the compartment 1. This necessitates the choice of a similarly situated point Q on the line OD and the compartment boundary AQD for the compartment 4. Thus the projection APB on one pattern will fit into the indent AQD in another. We now select for the compartment 3, the boundary DQC (an indent). This selection necessitates the projecting boundary BRC, BRC being similar to DQC and the inverse of APB, for the compartment 2. The result is the convex repeating pentagonal pattern APRCQ. It will be noticed that PAQ and RCQ are necessarily both right angles. We can thus obtain an infinite number of convex pentagons as repeating patterns, by simply varying the situation of P upon the line EB. If P coincides with B the pentagon degenerates to the square ABCD. If P coincides with E the pentagon degenerates to the rect- angle A EEC. In order to make the pentagon equilateral we first notice that by construction it ha.s four sides equal, viz. AP, A Q, QC, CR. The outstanding side is PR in which PB is, by construction, equal to BR. We have therefore to make AP equal to twice PB to make aWfive sides equal. For this to be so we find that the angle PAB must be (to the nearest minute) 20° 42' and thence we find that the angles APR, PR Care each 114 18' and the angle AQC 131° 24'. Laying off the angle PAB with mathematical instruments the pentagon is constructed. This however is not necessary because we can readily make the construction with ruler and compasses only. On a larger scale but with corresponding letters at various points, let RBP in fig. 127 be a side of the required pentagon, B its middle point, BQ a perpendicular and BA drawn so as to make an angle of 45" with BP. With centre P and radius PR strike a circle to cut BA at A. Then PA is a second side of the pentagon. Similarly find the corresponding point C by the same con- struction to the left of B ; CR is a third side of the pentagon. io4 NE W MA THEM A TICAL PASTIMES With centre A and radius PR strike a circle cutting BQ in Q. The remaining two sides are A'Q, QC and the construction is complete. If the depicted pentagonal tessellation be studied it will- be found to consist of a series of oblong hexagons in direction / cut by a second similar series in direction \. This observation supplies the simplest mode of construction of the tessellation. The area of this hay-stack shaped pentagon is equal to the square upon AB. If we produce BQ to D so that QD = RB (half the length of the side of the pentagon) the figure ABCD is the square upon AB. This is very evident directly it is noticed that the triangles CRB, APB are equal and similar to the triangles CQD, AQD respectively. Of 4-sided figures, besides the square we may conveniently take as bases the various forms assumed, by the rhombus and rectangle. The reader should have no difficulty in dealing with these upon the principles set forth above. In the case of the rectangle he must observe that sides of unequal lengths must be associated with different colours when designing repeating patterns. We may adopt the equilateral pentagon as a base upon which to construct a system of repeating patterns. THE PENTAGON BASE 105 THE PENTAGON BASE Some indeed there have been, of a more heroical strain, who striving to gaincope these ambages by venturing on a new discovery, have made their voyage in half the time. JOH. Robotham to the Reader in Comenius's Janua Ling. Ed. 1659. To take out other works in a new sampler. MlDDLETON. When the pentagon is assembled (see fig. 128) the contact system, as the reader will see at once on trial, is 1 to 1, 2 to 3, 4 to 5. Fig. 128. 7—5 io6 NE W MA THEM A TICAL PA S TIMES The side being 20 millimetres, the altitude is 265 mm. and the point upon the central axis to which the angular points are joined is 1 1 mm. from the base. The five compartments thus formed have no angle less than 45 u . The labelling of the sides shews, for any appropriate boundary A, four distinct symmetrical patterns. The sides labelled A, iA in the first figure are inclined to the horizontal line at the same angle that the sides labelled cA, ciA are inclined to the vertical. The angle in question is approximately 24° 18'. This fact leads us to take for A in the first place the boundary as in fig. 128 a; easily constructed because the tangent of 24° 18' is 45. The base angles of the boundary are thus each 24 18'. The four resulting patterns degenerate to two, as in fig. 128 b and c, because A = iA in this special case. Next take for A the form fig.~ 128^ and we obtain the full number of patterns as shewn in the third row. So to their work they sit, and each doth chuse What story she will for her tapet take. Spenser, Muiopotmos. I went alone to take one of all the other fragrant flowers that diapred this valley. Greene's Quip for an Upstart Courtier, B. 2. THE REGULAR HEXAGON BASE 82. Numbering the compartments, as before, as in fig. 129 a, there are apparently only three possible arrangements. One is shewn in fig. 129^ in which compartments 1, 2, 4, 5 are adjacent 6 \ 1 5 / / 5 1 6 \ 4 2 3 3 ' 2 \ 2 /\ 6 * 1 5 / / 1 V/3 \ / 5 1 6 \ 6 /\ 4 / \ 4 9. 3 5 \ / 3 / 2 \ 6 4 2 / / b 1 3 \y - \ 4 2 6 3 '5 i\ 6 , 4 2 / / 5 1 3 \ 4 9. 6 3 5 i\ Fig. 129. THE REGULAR HEXAGON BASE 107 to 1, 2, 4, 5 respectively and 3 adjacent to 6 The piece has two aspects. A second is shewn in fig. 129 c according to the contact system 1 to 4, 2 to 5, 3 to 6. The piece has one aspect. In these systems numbers which are not linked with different numbers may be made identical to any extent ; also two numbers linked together may be made identical and numbered at pleasure without interfering with the assemblage. The particular cases are too numerous to be separately examined in this book. The patterns and assemblages which are shewn in fig. 130 belong to the first of these contact systems. The assemblages exhibit each two aspects. Nicophanes gave his mind wholly to antique pictures, partly to exemplify and take out their patterns. Holland's Pliny. The patterns and assemblages which are shewn in fig. 131 belong to the second of these contact systems. The assemblages have each one aspect. Take me this work out. Othello, III. iv. 83. When the contact system is 1 to 1, 2 to 2, 3 to 6, 4 to 4, S to 5 the symmetry may be either about an axis perpendicular or parallel to a side. In the former case, we have for one boundary form of the first kind the set shewn in fig. 132 a, b, c, d. Taking for A the boundary as in fig. 1322 and for B as- in fig- 1 32/" we find the patterns given in fig. \$2g, h, k, I. If the symmetry be about a diagonal we have the cases of fig. 132;;/, n, o,p. Employing two boundary forms of the first kind B, C we have fig. 132^ and seven others by replacing B and Cby B, iC\ iB, C\ iB, iC\ C, B ; C, iB ; iC, B ; iC, iB respectively. We also have fig. 1 32 r and three others by replacing B and C by B, iC ; iB, C ; iB, iC respectively. 84. When the contact system is 1 to 4, 2 to 5, 3 to 6 we have symmetrical arrangements when the axis of symmetry is perpendicular to a side and when parallel to a side. Thus we have fig. 133 a and three others obtained by replacing B by cB, iB, ciB. We also have fig. 1 33 b and c for the diagonal symmetry. 1 08 NE W MA THEM A TIC A L PAS TIMES YZaJ, 0^0 Two aspects P'ig. 130. Two aspects THE REGULAR HEXAGON BASE 109 One aspect Fig. 131. no NEW MATHEMATICAL PASTIMES m L n l o L V L Bl 2 1 3 W 4 \,B ] 5 L\ 6 i _JL_ Ac r L Fig. 132. Fig- 133- THE REGULAR HEXAGON BASE in L is suitable because it is the simplest form of the second kind which is both its own complement and its own inverse. Other hexagons may be taken as bases ; such for instance as arise from the square base and are shewn in fig. 133 d and e giving assemblages as in fig. 133/(1 to 4, 2 to 5, 3 to 6) and as in fig. 133^(1 to 1, 2 to 5, 3 to 3, 4 to 4, 6 to 6). As with the regular hexagon two contact systems are available. The third contact system 1 to 2, 3 to 4, 5 to 6 is established by the diagram shewn in fig. 134^, which exhibits three aspects. a' ii2 NEW MA THEM A TICAL PA S TIMES Symmetry of pattern can be obtained either about a line bisecting opposite sides or about a line bisecting opposite angles as in fig. 1 34 b and c respectively. In the first diagram A' is a self-inverse boundary. In the second diagram A is any boundary. L as usual is the unaltered straight line. The first of these does not require attention because it is derivable from the contact system I to 4, 2 to 5, 3 to 6 which has been already considered. But giving A various forms, as in fig. 134^ to k, in the second we find assemblages of which those shewn in fig. 134/, m, n are examples. Exceptionally the pentagon pattern, which appears, can be assembled so as to exhibit either three (fig. 134 m) or six aspects (fig. i34«). It will be noticed that the pentagon is a repeating pattern on the principle given early in this Part — that if a repeating pattern can be dissected into a number of parts of the same size and shape, the shape involved is also a repeating pattern. Here the regular hexagon is divisible into either 3 or 9 similar and equal pentagons, the pentagon being the one before us. 85. The equilateral triangle, the square, and the regular hexagon are satisfactory bases for the construction of repeating patterns because they are themselves repeating patterns. In fact any repeating pattern may be taken as a basis for the evolution of other repeating patterns. This general principle will oc- casionally and exceptionally be found of interest and importance. In order to throw some light upon this development we will consider a very well known repeating pattern for linoleum, pavements, etc., viz. the combination of the regular octagon and the square (fig. 135 a). How can we obtain this form upon the principles that have been set forth ? When, as in this case, we are given a repeating pattern and we wish to determine the base and contact system to which it appertains, we have a problem which sometimes requires clever- ness and ingenuity for its solution. Take a square base as in fig. 135 b and the contact system 1 to 3, 2 to 4 with the boundaries as in fig. 135 c. THE CONSTRUCTION OF THE PASTIMES 113 The result is the repeating pattern of fig. 135^. Numbering its compartments, take it as a new base as in fig. 135 e with the contact system 1 to 4, 2 to 5, 3 to 6 and with the boundaries as in fig. 135^ The result is the repeating pattern (fig. 135 «) of which we are in search. It has emerged by combining a square base and the contact system 1 to 3, 2 to 4 with an hexagonal base having the contact system 1 to 4, 2 to 5, 3 to 6. We now find that the pattern may be made to emerge at once from the square base as in fig. 1 35^ with the contact system 1 to 3, 2 to 4 by taking the boundaries as in fig. 135 h. In colouring assemblages of repeating patterns it is a useful rule to employ at least _as many colours as the pattern has aspects. Usually each aspect would be associated with a separate colour, but this need not be the invariable rule. THE CONSTRUCTION OF THE PASTIMES Thou art a three-pil'd piece, I'll warrant thee. Meas.for Meas. I. ii. 86. In Part I the triangles may be made at home out of stout cardboard and have a side two-and-a-half or three inches in length. The compartments should be coloured with good water-colours or oils. Four suitable colours are black, white, red and blue, as they are readily distinguishable at night. A more permanent set should be made of good dense wood. The writer's set has a three-inch side and the thickness is one-fifth of an inch. ii4 NEW MA THEM A TICAL PA S TIMES Oil colours are used and for the five-colour set the colours are black, white, red, olive green and dark orange. If made at home, . in cardboard, pains should be taken to make the shape as accurate as possible. The pleasure of handling the pieces is much in- creased when they fit accurately. The ideal pieces would be as heavy as ordinary dominoes. The heavier they are the better. The square pieces may be of one-and-a-half or two inches side and one-sixth or one-fifth of an inch thick. It is desirable to have boards upon which to assemble the pieces, boards with a rim against which the boundary pieces may be placed. For the equilateral triangular pieces, assuming a piece of three inches side, the board should be a regular hexagon of slightly over six inches side so that the pieces will just fit in comfortably. For the 20-set 5-colour triangular pieces the board should be the figure-of-eight shape as depicted. For the square pieces the only board required is the 6x4 board and for the right-angled triangles the shape of the hexagon used. The irregularly shaped pieces of Part II and the designing of Part III will be much facilitated by the use of squared millimetre paper. This is preferable to squared paper in sub- divisions of the inch because, at any rate in this country, it appears to be of more reliable accuracy. The pieces should be set up with some simple boundary and then drawn on the millimetre paper. This can then be pasted upon 3 -ply or other suitable wood and handed over to some one with a fret-saw to cut out. The pattern-maker should be called in if sets of great accuracy are required, and they should be as heavy as possible. BIBLIOGRAPHY Bachet de Mesiriac. Problemes plaisans et delectables qui se font par les nombres. 1612. A. Labosne. Paris, 1884. Leuvechon, Jean. Recreation mathematique, etc. 1624. Leake. Mathematical Recreations. London, 1653. Oughtred, William. Mathematical Recreations. London, 1674. Ozanam, Jacques. Recitations mathematiques, 1694. English translation by Charles Hutton. London, 18 14. Berckenhamp, J. A. Les amusements math. Paris, 1749. HOOPER, \V. Rational Recreations. London, 1774. Allizeau, M. A. Les metamorphoses ou amusements geometriques. Paris, 1 8 18. Jackson. Rational Amusements for winter evenings. London, 1824. Hugonlin. Premiere collection de recreations mathematiques. Paris, 1828. Robinson, N. H. Mathematical Recreations. Albany, 1851. 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Math. Unterhaltungen und Spiele. Leipzig, 1901. Ignatiev, E. J. Math. Spiele, Ratsel und Erholungen. Petersburg, 1903. TeyssonneaU, E. Cent recreations math. ; curiosites scientifiques. Paris, 1904. FOURREY, E. Curiosites geometriques. Paris, 1907. DUDENEY, H. E. The World's Best Puzzles. Strand Magazine. 1908. n6 BIBLIOGRAPHY Dudeney, H. E. The Canterbury Puzzles. 1908. Ahrens, W. Mathematische Unterhaltungen und Spiele. Leipzig. Aufl. 1, 1910; 11, 1918. Ernst, E. Math. Unterhaltungen. Ravensburg, 1911-12. Ghersi, Italo. Matematica dilettevole e curiosa. Milano, 1913. Genau, A. Math. t/berraschungen. Arnsberg, 1913. Dudeney, H. E. Amusements in Mathematics. London and New York, 19 17. Ahrens, W. Altes und Neues aus der Unterhaltungsmathematik. Berlin, 1918. And now he has pour'd out his ydle mind In dainty delices and lavish joys. Spenser, F. Q. ii. v. 28. PRINTED IX ENGLAND BY J. B. PEACE, M.A. AT THE CAMBRIDGE UNIVERSITY PRESS s •• \