Pentagonal Tiling Type 9

Michael Tritle
MATH 401 3D printing
George Mason University
09/18/2019

Type 9 pentagons must satisfy the following criteria:

If A, B, C, D and E are the interior angles and a, b, c, d, e are the lengths of edges of the pentagon, then

2A + C = 360°,
D + 2E = 360°, and
b = c = d = e

I. FINDING THE INTERIOR ANGLES

Since we are working with a pentagon, we know that the sum of the interior angles is 540°, so

A + B + C + D + E = 540°.

We would like to find values for A, B, C, D and E that simultaneously the system of equations:

A + B + C + D + E = 540°
2A + C = 360°
D + 2E = 360°.

This system is equivalent to this system

A - B + E = 180°
2B + C - 2E = 0°
D + 2E = 360°

This system has an infinite number of solutions. Choose a B and E such that 0° < B < 180° and 0° < E < 180°, then we get:

A = 180° + B - E
C = 2E - 2B
D = 360° - 2E.

II. FINDING THE MISSING EDGE

For now, we will assume b = c = d = e = 1, This will simplify the calculation for