Pentagonal Dipyramid

The pentagonal dipyramid is one of the convex Deltahedra, and Johnson Solid $J_{13}$. It is also the Dual Polyhedron of the Pentagonal Prism. The distance between two adjacent Vertices on the base of the Pentagon is

$${d_{12}}^2 = [1-\cos({\textstyle{2\over 5}}\pi)]^2+\sin^2({\textstyle{2\over 5}}\pi)$$

$$= [1-{\textstyle{1\over 4}}(\sqrt{5}-1)]^2+\left[{(1+\sqrt{5})\sqrt{5-\sqrt{5}}\over 4\sqrt{2}\,}\right]^2$$

$$= {\textstyle{1\over 2}}(5-\sqrt{5}\,),$$ (1)

and the distance between the apex and one of the base points is

$${d_{1h}}^2 = (0-1)^2+(0-0)^2+(h-0)^2=1+h^2.$$ (2)

But

$${d_{12}}^2={d_{12}}^2$$ (3)

$${\textstyle{1\over 2}}(5-\sqrt{5})=1+h^2$$ (4)

$$h^2={\textstyle{1\over 2}}(3-\sqrt{5}),$$ (5)

and

$$h=\sqrt{3-\sqrt{5}\over 2}.$$ (6)

This root is of the form $\sqrt{a+b\sqrt{c}}$, so applying Square Root simplification gives

$$h = {\textstyle{1\over 2}}(\sqrt{5}-1)\equiv \phi-1,$$ (7)

where $\phi$ is the Golden Mean.

See also Deltahedron, Dipyramid, Golden Mean, Icosahedron, Johnson Solid, Triangular Dipyramid