A second eight-faced polyhedron in which every two faces share an edge

Source abstract

We report a novel polyhedral surface of genus~3 embedded in $\mathbb{R}^3$ with eight planar, simple, non-convex nonagonal faces, 24 vertices and 36 edges, in which every two faces share at least one edge: 20 pairs of faces share one edge and 8 pairs share two collinear edges. Its face planes are $3x-4y-2z=5$, $-2x+5y-5z=3$ and their images under the half-turns about the three coordinate axes, and all vertices are rational. The polyhedron has the same face vector, face sizes and number of edge multiplicities as the polyhedron described by Mizhaev, but it is not combinatorially equivalent to it. Our realisation has the symmetry group $D_2$ of order~4, whereas Mizhaev's polyhedron has a rotoreflection symmetry. The example was found by a computational geometric search, and all its properties were verified in exact rational arithmetic.