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

pointE = 8-3, Sqrt@3D

-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

vertexlist2 = Table@M.vertexlist@@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

a2 = Graphics@8Yellow,Polygon@Table@vertexlist2@@kDD + 8-3, Sqrt@3D

a3 = Graphics@8Red, Polygon@Table@vertexlist3@@kDD + 8-3, -Sqrt@3D