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.
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
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?
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(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
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?
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 .
There are five quadrominoes,
And Qi .o. Qjfor all i and j.
There are 12 pentominoes.
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’!
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