Activity 2-3: Pearl Tilings www.carom-maths.co.uk Activity 2-3: Pearl Tilings

Consider the following tessellation:

What happens if we throw a single regular hexagon into its midst? We might get this... The original tiles ‘manage to rearrange themselves’ around the new tile. Call this tessellation a pearl tiling. The starting shapes are the oyster tiles, while the single added tile we might call the iritile.

What questions occur to you? How about: Can any n-sided regular polygon be a successful iritile? What are the best shapes for oyster tiles? Can the same oyster tiles surround several different iritiles?

Here we can see a ‘thinner’ rhombus acting as an oyster tile . If we choose the acute angle carefully, we can create a rhombus that will surround several regular polygons. Suppose we want an oyster tile that will surround a 7-agon, an 11-agon, and a 13-agon. Choose the acute angle of the rhombus to be degrees.

Here we build a pearl tiling for a regular pentagon . Here we build a pearl tiling for a regular pentagon with isosceles triangle oyster tiles. Generalising this... 180 – 360/n + 2a + p(180 - 2a) = 360 So a = 90 – .

Any isosceles triangle with a base angle a like this will always tile the rest of the plane, since 4a + 2(180 - 2a) = 360 whatever the value of a may be.

This tile turns out to be an excellent oyster tile, since 2b + a = 360. One of these tiles in action:

Let’s make up some notation. If S1 is an iritile for the oyster tile S2, then we will say S1 .o S2 . Given any tile T that tessellates, then T .o T, clearly. If S1 .o S2, does S2 .o S1? Not necessarily. TRUE UNTRUE

Is it possible for S1 .o S2 and S2 .o S1 to be true together? We could say in this case that S1 .o. S2 .

There are only two triominoes, T1 and T2. What about polyominoes? A polyomino is a number of squares joined together so that edges match. There are only two triominoes, T1 and T2. We can see that T1 .o. T2 .

Task: do the quadrominoes relate to each other in the same way? There are five quadrominoes, And Qi .o. Qj for all i and j.

Task: what about the pentominoes? There are 12 pentominoes. Conjecture: Pi .o. Pj for all i and j.

Are there two triangles Tr1 and Tr2 One last question: Are there two triangles Tr1 and Tr2 so that Tr1 .o. Tr2? A pair of isosceles triangles would seem to be the best bet. The most famous such pair are...

So the answer is ‘Yes’!

Carom is written by Jonny Griffiths, mail@jonny-griffiths.net With thanks to: Tarquin, for publishing my original Pearl Tilings article in Infinity. Carom is written by Jonny Griffiths, mail@jonny-griffiths.net