Transcript Carom 1-5: Kites and Darts

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Activity 1-5: Kites and Darts
A tessellation is called periodic if you can lift it up and shift it
so that it sits exactly on top of itself again.
Sometimes a periodic tiling can be tweaked so that it
becomes non-periodic.
Exactly the same tiles, but non-periodic this time.
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’.
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.
The matching rule is this:
the tiles can only be placed
with the Hs together and the Ts together.
The only ways that tiles can legally meet at a point are as follows:
Star
Sun
Ace
King
Jack
Queen
Deuce
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.
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.
Sheet of Tiles
pdf
http://www.s253053503.websitehome.co.uk/
carom/carom-files/carom-1-5.pdf
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.
Note too that every point in the tiling is in a cartwheel shape.
Sometimes
Kite and Dart tilings
demonstrate
striking 5-fold and
10-fold symmetry.
The red shape
at the centre here
is called ‘Batman’.
There are many remarkable facts about Kite and Dart tilings.
There are an infinite
number of them,
and they are
always non-periodic.
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?
In any infinite Kite and Dart tiling,
the ratio of Kites to Darts is  to 1,
where  is the Golden Ratio.
Notice how the Darts ‘hold hands’
in this tiling (and every tiling) to form rings.
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!
Deflate
Deflate
Deflate
Inflate
How do we inflate a
tiling?
Cut every dart in half,
and then glue together all
the short edges of the
original pieces.
Inflate or deflate twice, and you get back
to the tiling you started with (scaled differently).
One consequence of the inflation/deflation property
is that any finite Kite and Dart tiling must appear
in any infinite Kite and Dart tiling.
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.
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.
There are other pairs of
shapes that always give
non-periodic tilings too.
This picture shows
Roger Penrose
on a tiled floor
at Texas A&M university,
showing a
non-periodic tessellation
employing two rhombuses
that he discovered
after the Kite and Dart.
One last question;
is there a single tile
that only tiles
non-periodically?
If you can find one, it will be a passport to immortality...
With thanks to:
Roger 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, [email protected]