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 can 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? 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? How about:

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 with isosceles triangle oyster tiles. 180 – 360/n + 2a + p( a) = 360 Generalising this... So a = 90 –..

Any isosceles triangle with a base angle a like this will always tile the rest of the plane, since 4a + 2( a) = 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 S 1 is an iritile for the oyster tile S 2, then we will say S 1.o S 2. Given any tile T that tessellates, then T.o T, clearly. If S 1.o S 2, does S 2.o S 1 ? Not necessarily. TRUEUNTRUE

Is it possible for S 1.o S 2 and S 2.o S 1 to be true together? We could say in this case that S 1.o. S 2.

What about polyominoes? A polyomino is a number of squares joined together so that edges match. There are only two triominoes, T 1 and T 2. We can see that T 1.o. T 2.

Task: do the quadrominoes relate to each other in the same way? There are five quadrominoes (counting reflections as the same...) Does Q i.o. Q j for all i and j?

BUT... we have a problem!

(Big) task: For how many i and j does P i.o. P j ? There are 12 pentominoes (counting reflections as the same...) Task: find them all...

Sometimes... but not always...

Are there two triangles Tr 1 and Tr 2 so that Tr 1.o. Tr 2 ? A pair of isosceles triangles would seem to be the best bet. The most famous such pair are...

So the answer is ‘Yes’!

Footnote: (with thanks to Luke Haddow). Consider the following two similar triangles: Show that T 1.o. T 2

With thanks to: Tarquin, for publishing my original Pearl Tilings article in Infinity. Carom is written by Jonny Griffiths,