# Splatts

And this is a do-dodecahedron.
— ‘The Man Who Mistook His Wife for a Hat’

## The splatt

The term ‘splatt’ was coined in 1986 by Matthew Richards and Mark Owen to refer to the truncated octahedron, an Archimedean solid which tessellates to fill space. We observed that three splatts could be joined together in six different ways consistent with this packing, and on that basis we designed a puzzle akin to the Soma Cube.

## The original puzzle

We wrote an article for Eureka, the journal of the Archimedeans, the Cambridge University Mathematical Society. It appeared in issue 47, published in February 1987. You can read a PDF version of the article ‘A Song of Six Splatts’: the PDF incorporates scans of the elegant original illustrations prepared by Matthew Richards. Sadly Matthew died in 1993, and the article is reproduced here with the kind permission of his family.

This page summarises the main aspects of the puzzle we invented. Interested readers are urged to consult the original article, which includes much additional material.

## The six 3-splatts

The six ‘3-splatts’, as we called them, are shown below: they are given names for convenience.

 Simon John Michael David Mark Matthew

The original article invited the reader to make a set of 3-splatts and attempt to use them build the following structures.

 Bridge Dragster Drum Eggboxes Giraffe Lozenge New Court Christ’s Prism Spinode Tortoise Tower Triangles University Library Ziggurat

Note that one of these structures is impossible to make: you can read the original article to find out which!

## The forty-four 4-splatts

Four splatts can be joined together in forty-four different ways consistent with their space-filling packing, as illustrated below: names have yet to be assigned to them. The set shown includes both of each of the nine pairs of 4-splatts that are mirror images of one another.

Although the number of pieces in this set is too large to make a satisfying puzzle, it is a pleasing fact that they can be packed into a rectangular cuboid whose sides measure 3 units, 5 units and 8 units. Here the ‘unit’ is the side of the circumcube of the splatt that includes its square faces. Now 3, 5 and 8 are successive terms of the Fibonacci sequence, and so we might call this cuboid a ‘Fibonacci box’. Because the ratio of two consecutive terms in the Fibonacci sequence approaches the ‘golden ratio’, four of the faces of the box approximate ‘golden rectangles’, which accounts for its handsome appearance. The box as a whole approximates the ‘golden cuboid’, apparently the favoured sitting-room shape of audiophiles.

The illustration below shows this cuboid, the colours hinting at one solution to the packing problem.

## How many n-splatts are there?

The number of distinct n-splatts is given in the table below. For n≥4 there are two different counts, depending on whether two splatts that are mirror images of one another are counted as distinct. Entries for n≤8 are confirmed by The On-Line Encyclopedia of Integer Sequences (sequences A038180 and A038181); I would welcome independent confirmation (or contestation) of the later entries as well as any results that extend the table.

 n Number of distinct n-splatts,counting mirror image pairsas distinct Number of distinct n-splatts,counting mirror image pairsas the same 1 1 1 2 2 2 3 6 6 4 44 35 5 394 251 6 4680 2602 7 59 361 30 900 8 789 303 400 818 9 10 742 595 5 401 599 10 148 921 162 74 617 105 11 2 093 400 002 1 047 497 078 12 29 769 338 104 14 888 851 869

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