A Grünbaum Polyhedron in Antiprism

25 Aug 2018

Tags: geometry , antiprism

A paper from 2012, The Regular Grünbaum Polyhedron of Genus 5, includes a polyhedron of genus 5, earlier discovered by Grünbaum, that is an embedding of the {3,8} Fricke-Klein regular map. The polyhedron (Figure 2 and 3) can be recreated using the wythoff program in Antiprism, although the exact proportions are a little different in the paper.

wythoff -p [3VE2F]0EV,0fe0EV0fevfV,0F,0V0F0evfe oct

The polyhedron in Figure 4 can be created with a small change:

wythoff -p [3VE2F]0EV,0fe0EV0fevfV,0F,0V0E0F oct

These can also be used to generate the genus 11 polyhedra in Figure 5: just replace oct with ico.

Here’s the titular Grünbaum polyhedron of genus 5:

The Grünbaum polyhedron of genus 5

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