written by Rene K. Mueller, Copyright (c) 2007, last updated Fri, February 25, 2022

Geodesic Pentakis Dodecahedron

It's not a platonic or archimedean polyhedra, but the dual of the archimedean truncated icosahedron:

Pentakis Dodecahedron

Pentakis Dodecahedron

Dual: truncated icosahedron

s_{1}: 1/19 * (18*√5-9)

s_{2}: 3/2 * (√5-1)

V: s_{1}^{3} * 5/3 * √(1/2 * (421+63*√5))

A: s_{1}^{2} * 5/36 * (41+25*√5)

Vertices/Connectors: 32

12 x 5-way

20 x 6-way

Faces: 60 (3-sided)

Edges/Struts:

A x 60: 0.67765

B x 30: 0.76393

total 90 struts (2 kinds)

strut variance 12.7%

In its original version there is no straight hemisphere line, and already two kinds of edge lengths.

Pentakis Dodecahedron vs Geodesic Dodecahedron L1

A

B

Strut Variance

Pentakis Dodecahedron

1.000000

1.127322

12.7%

Geodesic Dodecahedron L1

1.000000

1.113583

11.4%

While I thought I knew this form, and going through previous forms I found it looks like the Geodesic Dodecahedron L1, and to my surprise the structure almost is the same,
yet the ratio between both strut length is off a bit.
The difference of B's is 1.21%, very small but still too significant - so the Pentakis Dodecahedron and the Geodesic Dodecahedron L1 (triangulated the pentagons and spherical projected)
are structure-wise the same, but the strut ratios are a bit off.

Geodesic Pentakis Dodecahedron L2

Geodesic Pentakis Dodecahedron L2

Vertices/Connectors: 122

12 x 5-way

110 x 6-way

Faces: 240 (3-sided)

Edges/Struts:

A x 60: 0.32036

B x 60: 0.32910

C x 120: 0.33867

D x 60: 0.36284

E x 60: 0.38161

total 360 struts (5 kinds)

strut variance 19.4%

Geodesic Pentakis Dodecahedron L2T

Geodesic Pentakis Dodecahedron L2T

Vertices/Connectors: 182

90 x 4-way

60 x 6-way

12 x 10-way

20 x 12-way

Faces: 360 (3-sided)

Edges/Struts:

A x 60: 0.18738

B x 120: 0.21357

C x 60: 0.32036

D x 60: 0.32910

E x 60: 0.36284

F x 60: 0.36388

G x 120: 0.40553

total 540 struts (7 kinds)

strut variance 117.1%

The triacon version provides hemisphere lines for a dome version.

Epcot Spaceship Earth

Epcot Spaceship Earth on opening day (1982)

The pentakis dodecahedron is quite famous by its Epcot (Experimental Prototype Community Of Tomorrow) "Spaceship Earth" version at Disney World in Tampa, Florida (USA):

As far I figured out from the photo, the geodesic sphere is 8V, whereas each 8V triangle is center point triangulated into 3 faces again,
60 x 8 x 8 x 3 = 11'520 faces finally. Yet, according Wikipedia: Spaceship Earth the sphere is composed by

pentakis dodecohedron 60 faces, which each is

subdivided 16 equilateral triangles, which each is

subdivided in 4 triangles, which each is

subdivided in 3 isoceles triangles

60 x 16 x 4 x 3 = 11'520 faces. It would be interesting to get more details to see why the leveled subdivisions were chosen the way they were.

Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (1 of 2)

Epcot Construction of Spaceship Earth (courtesy by Von Johnson and Associates, Inc., 1981) (2 of 2)

These construction photos confirm the description of the Wikipedia article.

Other Solids

From the Johnson Solid I tried

Gyroelongated Triangular Bicupola

Gyroelongated Square Bicupola

but didn't provide a straight hemisphere line at L1 or L2, and strut lengths were much higher than the platonic and archimedean solids.

As mentioned in the Geodesic Dome Notes I deepen some of the dome options for real-life applications.