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Prism-Expanded Dissected Cuboctahedron
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It seems Stewart was unaware of this model however, which Alex Doskey discovered more recently. It satisfies all of his criteria perfectly, but its convex hull has some rhombic faces. It has a genus of 11. See also my genus-41 toroid.

It's also interesting to note that aside from 8 equilateral triangles, the other 78 faces are all squares. It's somehow surprising that squares alone can form intricate surfaces like this. In fact, an infinite lattice can be made based on this shape, by building on the triangles, leaving nothing but square faces.

Start by putting together the inner section. Then glue the outer parts around it. | |

Close-up view of finished model. |

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Prism-Expanded Dissected Cuboctahedron
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