activity 2-3: pearl tilings
TRANSCRIPT
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Activity 2-3: Pearl Tilings
www.carom-maths.co.uk
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Consider the following tessellation:
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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.
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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:
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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
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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
.
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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.
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This tile turns out to be an excellent oyster tile,
since 2b + a = 360.
One of these tiles in action:
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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
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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 .
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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 .
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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?
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BUT...
we have a problem!
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(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...
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Sometimes...
but not always...
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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...
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So the answer is ‘Yes’!
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Footnote:(with thanks to Luke Haddow).
Consider the following two similar triangles:
Show that T1 .o. T2
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With thanks to:Tarquin, for publishing
my original Pearl Tilings article in Infinity.
Carom is written by Jonny Griffiths, [email protected]