PENTAGONAL TILES
Hope Roberts
Pentagonal tiling is a periodic tile method that covers the plane in such a way that there are no gaps. Every individual piece of the tiling is part of a pentagon.
In order to cover the plane with no gaps with a pentagon there are certain symmetries, angles and side length requirements that must be fulfilled.
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Measurements of my constructed tile.
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Brief
History
Reinhardt discovered the first five of the convex shapes in 1918,
Kershner discovered the next 3
in 1968 who incorrectly stated he had completed the list. The tenth type is
credited to James in 1975 and shortly after in 1977 a stay at home housewife
with a fascination for math named Rice found four more. There was a lull with
the fourteenth not being found till 1985. A fifteenth was found in October of
2015 by Mann/McLoud/Von Derau (2015) from computer algorithm.
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The reflections the tile type 5.
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The Tiles in General
The tiles are restricted by the
internal angles and requirements of the edges having to be certain lengths. The tiles also have certain wallpaper group symmetries
and some of the symmetries may repeat in different tiles. Because the tile
tessellates there are no spaces, which means that the internal angles must be
designed in a way that all the pieces of the pentagon fit together. The
pentagon has internal angles that add up to 540 degrees so when finding a new
type of pentagon tiling the angels must be arranged so that they are not
greater than 540 degrees which means that some of the edges are dependent upon
this fact. Certain sides may also have different requirements of length.
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My tiles in 6-tile primitive unit.
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The tile I chose
is type Monohedral convex pentagonal tiling #5. It
was discovered by Reinhardt
and was amongst the first of several tiles to be discovered. This tile differs
from the other tiles such as p6(*632) in that it does not have glide
reflections. This is one example of how the reflections for this tile are particular
to this tile making it different from the others. The tiles in general differ in the
requirements of the angles and side lengths that they are restricted to. In the
wallpaper group symmetry my tile is P6 (632) symmetry. This symmetry has one
rotation center of order six differing every 60 degrees. There are two centers
of rotations of order three differing by 120 degrees and three centers of
rotations of order two differing by 180 degrees.
The angle and sides requirements of this tile are
given by the equation;
a=b d=e A=60 D=120 (B+C+E=360)
My
angles and sides for my tile are;
Angles:
60, 136, 120, 72, 152
Sides: a=b=6.14, d=e=5.26, c=6.53
(see
picture)
I used a site Math
is Fun to construct my tile. I insured first the required angles of 60 and
120 were and the required sides that needed to be equal. After doing this the remaining sides and
angles were given so that I did not need to further manipulate.
This pattern does not have reflections or glide
reflections.
The tiling is periodic which means that there is
translational symmetry and the pattern will repeat.
SOURCES;
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