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www.carom-maths.co.uk. Activity 2-20: 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 t hemselves’ around the new tile.

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www.carom-maths.co.uk

Activity 2-20: Pearl Tilings



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 apearl tiling.

The starting shapes are theoyster tiles,

while the single added tile we might call theiritile.


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

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 S1is an iritile for the oyster tile S2, then we will say S1 .o S2.

Given any tile T that tessellates, then T .oT, clearly.

If S1.o S2, does S2 .o S1?

Not necessarily.

TRUE

UNTRUE


Is it possible for S1.o S2and S2 .o S1to 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 thatT1.o. T2 .


Task do the quadrominoes relate to each other in the same way
Task: do the quadrominoesrelate to each other in the same way?

There are five quadrominoes,

And Qi .o. Qjfor all i and j.


Task what about the pentominoes
Task: what about the pentominoes?

There are 12 pentominoes.

  • Conjecture:Pi.o. Pjfor all i and j.


One last question
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...



With thanks to tarquin for publishing my original pearl tilings article in infinity
With thanks to:Tarquin, for publishing my original Pearl Tilingsarticle in Infinity.

Carom is written by Jonny Griffiths, [email protected]


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