Activity 1-5: Kites and Darts

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www.carom-maths.co.uk
Activity 1-5 Kites and Darts
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A tessellation is called periodic if you can lift
it up and shift it so that it sits exactly on
top of itself again.
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Sometimes a periodic tiling can be tweaked so
that it becomes non-periodic.
Exactly the same tiles, but non-periodic this
time.
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The question arises is there a tile or a set of
tiles so that EVERY infinite tiling of the plane
they make is non-periodic?
Roger Penrose came up with two tiles that fit
this criteria in 1974. He called them the Kite
and the Dart.
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Of course, kites and darts can be used on their
own and together to generate periodic tilings.
But the matching rules for the tiles stop these
tilings from counting.
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The matching rule is thisthe tiles can only be
placed with the Hs together and the Ts together.
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The only ways that tiles can legally meet at a
point are as follows
Star
Sun
Ace
King
Queen
Deuce
Jack
The H-T rule can be enforced using red and green
lines as above, or by using bumps and dents on
the tiles. Some of these configurations force
other tiles around them.
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Task have a play with some Kites and Darts, and
get a feel for how they tile together.
A sheet of tiles to cut up can be found below.
http//www.s253053503.websitehome.co.uk/carom/caro
m-files/carom-1-5.pdf
Sheet of Tiles pdf
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With these matching rules, it turns out
that every infinite tiling that these tiles make
is non-periodic.
Task find each of the seven ways that tiles
can meet at a point in this tiling.
Note every point in the diagram is in an ace.
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Note too that every point in the tiling is in a
cartwheel shape.
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Sometimes Kite and Dart tilings demonstrate
striking 5-fold and 10-fold symmetry.
The red shape at the centre here is called
Batman.
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There are many remarkable facts about Kite and
Dart tilings.
There are an infinite number of them, and they
are always non-periodic.
In any infinite Kite and Dart tiling, the ratio
of Kites to Darts is ? to 1, where ? is the
Golden Ratio.
You notice in this tiling it has been possible
to colour the tiles with only three colours so
that no two tiles of the same colour share an
edge. Is this possible in any Kite and Dart
tiling?
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Notice how the Darts hold hands in this tiling
(and every tiling) to form rings.
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You can inflate or deflate any Kite and Dart
tiling to give another Kite and Dart tiling with
bigger or smaller tiles.
This shows that the Penrose tiling has a scaling
self-similarity, and so can be thought of as a
fractal. Wikipedia
To deflate, add these lines on the left to every
Kite and Dart in your tiling. Your tiles will
get smaller, but they will all remain Kites and
Darts!
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Deflate
Deflate
Deflate
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How do we inflate a tiling?
Inflate
Cut every dart in half, and then glue together
all the short edges of the original pieces.
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One consequence of the inflation/deflation
propertyis that any finite Kite and Dart tiling
must appear in any infinite Kite and Dart
tiling.
Inflate or deflate twice, and you get back to the
tiling you started with (scaled differently).
We can prove now that every Kite and Dart tiling
is non-periodic. Suppose we have a periodic such
infinite tiling, with translation vector s.
All inflations and deflations of the tiling must
also be periodic period s.
Now simply inflate the diagram until s lies
within a single tile. Now clearly periodicity is
impossible.
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If we start with either the Star (left) or the
Sun (right) and insist on perfect five-fold
symmetry, then every tile is forced as above...
If we inflate or deflate one of these tilings,
we get the other.
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There are other pairs of shapes that always give
non-periodic tilings too. This picture shows
Roger Penrose on a tiled floor at Texas AM
university, showing a non-periodic
tessellation employing two rhombuses that he
discovered after the Kite and Dart.
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One last questionis there a single tilethat
only tiles non-periodically?
If you can find one, it will be a passport to
immortality...
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With thanks toRoger Penrose. Wikipedia, for a
brilliant article on Penrose tilings.John Conway
for his talk on Kites and Darts back in 1979.
Carom is written by Jonny Griffiths,
hello_at_jonny-griffiths.net