The missing fact, I think, is that if you augment two adjacent pentagons of a dodecahedron, the edge between the two pyramids will be a valley and not a ridge, so the resulting polyhedron won't be convex. The Wikipedia article on "Johnson solid" lists the four possible ways to augment a dodecahedron: on one face, two different ways to do it to two faces, and on three faces. You can't pick four faces of a dodecahedron without two being adjacent, so that's all there is. This is one of those enumeration problems where the Frobenius-Burnside-Polya-Uncle-Tom-Cobbley-And-All counting formula is just overkill. On Mon, Mar 11, 2019 at 4:45 PM Bill Gosper <billgosper@gmail.com> wrote:
"Peraugmented Dodecahedron"? And what about (Mono)Augmented? Ortho-, Meta, & Para Biaugmented?, <Help-me-Polya> Triaugmented? . . . Undecaaugmented?? What am I missing? —rwg
On Mon, Mar 11, 2019 at 11:44 AM Bill Gosper <billgosper@gmail.com> wrote:
Johnson 93? < https://www.wolframcloud.com/objects/810fbe92-6ab3-480b-9f86-e4869990cf46> —rwg And why can't we tumble that figure?
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