The Mathematical Pastimes of Major Percy Alexander MacMahon. Part 2. Triangles and BeyondAuthor(s): Paul GarciaSource: Mathematics in School, Vol. 34, No. 4 (Sep., 2005), pp. 20-22Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30216608 .

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Percy Alexander MacMahon (1854-1929), is one of mathematics best kept secrets. He began his career not as a mathematician, but as a soldier in the Royal Artillery, serving in India. After a serious illness, he returned to England and joined the Advanced Class in mathematics at the Royal Military Academy in Woolwich. It was here that he discovered he had a talent for computation, and ended up computing ballistics tables. This led him to study invariant theory, where he made an important discovery connecting invariants, symmetric functions and number partitions.

But while MacMahon was working on this important algebra and number theory (as a result of which he was elected a fellow of the Royal Society in 1890, and later awarded three important medals and four honorary doctorates) he also became very interested in mathematical puzzles, and took out three patents for puzzles in the early 1890s. The first patent I have described in part 1 of this series of three articles. In this article I want to introduce you to the kind of puzzle covered by the second patent.

MacMahon wrote a book, New Mathematical Pastimes, to help others create their own challenging puzzles. The book was first published in 1921, and reprinted in 1930, the year after his death. As part of the celebrations to mark MacMahon's 150th birthday in September 2004, the book has been reprinted a third time. In the original edition of the book, MacMahon expressed his disappointment that it had been too expensive to produce the book in colour. Things have not changed in the intervening 83 years, and a coloured print version of the book was still too expensive. But now we have technology that was unavailable to MacMahon; to accompany the print version, there is a CD-ROM containing a coloured version of the book, as well as some extra material. In particular, there are copies of all three patents for puzzles taken out by MacMahon.

In 1892, MacMahon took out his second patent, jointly with his friend, Major Julian R. Jocelyn. The puzzle described in this patent, is one of MacMahon's most lasting contributions to the world of recreational mathematics. It describes how a set of equilateral triangles may be constructed by dividing each triangle into three compartments, like this:

Each of the compartments is to be coloured in (or marked with a number or dots, as in dominoes). If you use four colours, and colour the triangles so that there is one triangle of each possible different colouring, you get a set of 24 triangles like this:

20

Fig. 1 24 four-colour triangle set (equivalent to 'treble threes')

If you use numbers in a domino-like manner then, for example, a set using the numbers 0, 1, 2, 3, 4 would need 45 triangles (a 'treble 4' set, analogous to dominoes, where a complete set using the numbers 0, 1, 2, 3, 4, 5, 6 is called 'doubles sixes'). Notice that each triangle would correspond to a partition of each of the integers up to 12 into three parts, no part greater than 4, allowing 0 as a part.

In the patent application, MacMahon described a game to be played on a hexagonal board marked out in equilateral triangles. You might like to read the patent, make a set of 'treble fours' and play the game.

In his book, MacMahon takes the idea of the triangles further. He suggests using a complete set of 24 four-colour triangles to construct a hexagon so that the border is all one colour, and internal colours match. Here is one solution:

Fig. 2

This is coded as B12,0,0,0 C1,1,1,1. The 'B' stands for Border, and the numbers mean that all twelve edge compartments are the same colour; the 'C' stands for Contact, and the numbers describe the way the colours must match internally; 1,1,1,1 means each colour must match itself. Here is another solution, with the same contact system, C1,1,1,1, but the border is B6,6,0,0.

Mathematics in School, September 2005 The MA web site www.m-a.org.uk

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Fig. 3

One of the problems MacMahon considers in the book is how many different border systems are possible.

Different internal contact systems are also possible:

C2,2 B6,6,0,0

Fig. 4

The C2,2 means that blue must always match red and green must always match yellow. C2,1,1 would mean that blue had to match red, green must match green and yellow must match yellow.

How many contact systems are possible ? Is it possible to find a solution for each possible combination of contact and border systems ?

MacMahon extended these ideas to right-angled triangles, squares and hexagons, as you can see in the book.

It is in the second part of the book where MacMahon created a whole new class of puzzle by forcing a particular contact system using not colour matching but shaped edges. This work produced some very exotic jigsaw puzzles, and designs that anticipated Escher's work by 15 years.

For example, we can replace the four colours in the 24 triangles shown above by adjusting the edges to force the matching. Where a compartment is red, we leave the edge straight, as in line 1 below. Where it is blue we replace the edge with line 2, and so on.

4 --0

Fig. 5 Edge replacements (from New Mathematical Pastimes, page 55)

You end up with a set of pieces that can only be assembled according to contact system C1,1,1,1. The result is the splendid looking jigsaw puzzle in Figure 6.

You can invent your own edge shapes to force different contact systems, to create an infinite variety of jigsaws.

Cr ,ta r B2,,o0o0

Fig. 6 The puzzle from Figure 2 using edge shape to force the contact system

(from New Mathematical Pastimes, page 56)

The next step, where MacMahon anticipated the work of Escher, is to apply the edge transformation system to tilings of the plane. Start with an equilateral triangle, and replace a straight edge of the triangle with a zig-zag, as in Figure 7 below.

Fig. 7

Fig. 8

Figure 8 shows how replacing all three edges by the zig- zag produces a shape that will tessellate.

MacMahon gives another example, on page 88, of a shape derived from a triangle that can be assembled to tile the plane in two different ways, one with rotational symmetry of order 2, and another of order 6 (Figures 9 and 10).

Two aspects Six aspects

Fig. 9 Fig. 10 The Helmet put together in 'Helmet' two different ways

With a square as a starting point, even more spectacular effects can be achieved, as the illustration below shows. Work out for yourself where the original square must have been, and what shape replaced the straight edges (Figure 11).

There is another particularly pleasing design derived from a square. The square is coloured as shown in Figure 12, and we want to force the contact system C2,2 - that is, red to green and blue to yellow. We can do this by altering edges,

Mathematics in School, September 2005 The MA web site www.m-a.org.uk 21

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Q 117

Fig. 11 Two tilings with different rotational symmetries derived from a square

(New Mathematical Pastimes, page 94)

moving a chunk of the yellow compartment to the blue, and a chunk of green to red:

3

Fig. 12 A square tiling with contact system C2,2

The resulting pentagon will tile the plane, as in Figure 13. (I've left the colours in, and the underlying square so you can see how it works.)

This is a very interesting tiling; you can find two sets of hexagons, running at right angles to each other. Figure 14 gives you a hint!

Escher also found this tiling, but not until 1937. (It is shown in Doris Schattschneider's book about Escher, Visions of Symmetry, on pages 28 and 106.)

There is a lot of scope for creativity here! M

Fig. 13 A pentagonal tiling derived from the square

Fig. 14 A hexagon hidden in the pentagonal tiling derived from a square

Reference

MacMahon, P. A. 2004 New Mathematical Pastimes & CD-ROM, Tarquin. Schattschneider, D. 1999 Visions ofSymmetry, Freeman.

Keywords: Partition; Tessellation; Escher.

Author Paul Garcia e-mail: paul@marybj.cix.co.uk

22 Mathematics in School, September 2005 The MA web site www.m-a.org.uk

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