Activity 2-3: Pearl Tilings

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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. 13117

360

Here we build a pearl tiling for a regular pentagon with isosceles triangle oyster tiles.

180 – 360/n + 2a + p(180 - 2a) = 360

Generalising this...

So a = 90 – . )1(

180

pn

.

Any isosceles triangle with a base angle a like thiswill 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 .

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(counting reflections as the same...)

Does Qi .o. Qj for all i and j?

BUT...

we have a problem!

(Big) task: For how many i and j does Pi .o. Pj ?

There are 12 pentominoes (counting reflections as the same...) Task: find them all...

Sometimes...

but not always...

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’!

Footnote:(with thanks to Luke Haddow).

Consider the following two similar triangles:

Show that T1 .o. T2

With thanks to:Tarquin, for publishing

my original Pearl Tilings article in Infinity.

Carom is written by Jonny Griffiths, hello@jonny-griffiths.net