A Pentagonal Tiling of the Euclidean Plane We’ ve chosen one of the tile families from the Wikipedia article on pentagonal tilings (shown below) and we aim to see, using Mathematica, how this tiles the plane. In this tile we have: a = b, d = c+e angle(A) = angle(C) = angle(D) =120° We arbitrarily set the point “B” at the origin: pointB = 80, 0<; p1 = Graphics@8PointSize@LargeD, Red, Point@pointBD<, Axes fi TrueD -1.0 -0.5 0.5 1.0 -1.0 -0.5 0.5 1.0 We arbitrarily place the point “A” on the horizontal axis at (-2,0) and join point “A" and point”B” with a line:
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A Pentagonal Tiling of the Euclidean Plane
We’ve chosen one of the tile families from the Wikipedia article on pentagonal tilings (shown below) and
we aim to see, using Mathematica, how this tiles the plane.
In this tile we have:
a = b, d = c+e
angle(A) = angle(C) = angle(D) =120°
We arbitrarily set the point “B” at the origin:
pointB = 80, 0
-
pointA = 8-2, 0
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pointE = 8-3, Sqrt@3D
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-3.0 -2.5 -2.0 -1.5 -1.0 -0.5
0.5
1.0
1.5
2.0
Having located the coordinates of the vertices of the pentagon we can use Mathematica’s Polygon
function to draw a filled polygon with these vertices:
vertexlist = 8pointA, pointB, pointC, pointD, pointE
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vertexlist2 = [email protected]@@kDD, 8k, 1, Length@vertexlistD, 90, -2 3 =>F
We draw the new pentagon, in a different color, along with the basic pentagon:
a2 = Graphics@8Yellow, pentagon2
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a2 = Graphics@8Yellow,Polygon@Table@vertexlist2@@kDD + 8-3, Sqrt@3D
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a3 = Graphics@8Red, Polygon@Table@vertexlist3@@kDD + 8-3, -Sqrt@3D
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