Talk:Pentakis dodecahedron

Is this the result of adding a pentagonal pyramid to each face of a dodecahedron? Judging by the picture, it appears to be. --Ravi12346 02:21, 30 April 2006 (UTC)

Yes, if you mean isosceles triangles at least (Johnson solid version has equilateral triangles). It's more considered the dual polyhedron of the vertex uniform Truncated icosahedron. -- Tom Ruen 06:22, 30 April 2006 (UTC)

convex or not

There seems to be a bit of inconsistency between:

• the deltahedron article claims that the "equilateral pentakis dodecahedron" is "nonconvex"
• this pentakis dodecahedron article claims "pentakis dodecahedron is a Catalan solid.", and the Catalan solid claims "The Catalan solids are all convex."

I suppose they could both be true, if they are talking about two slightly different -- but topologically identical -- shapes:

• the "equilateral pentakis dodecahedron", formed by adding equilateral pentagonal pyramid to each face of a dodecahedron, results in 6 equilateral triangles around some vertices -- necessarily resulting in 6 equilateral triangles that make a flat hexagon that is technically not strictly convex, although it's not concave either.

• the kind of "pentakis dodecahedron" (is there a more specific name?), formed by "pushing" those 6-triangle vertices out slightly so the hexagon is slightly convex, but not pushing them quite far enough that the pentagons become flat or convex, resulting in isosceles triangles that are not exactly equilateral. Any arbitrary amount of push between those two extremes -- flat hexagon vs flat pentagon -- gives a convex shape. In particular, there is one shape in that range where those vertices have been pushed out exactly enough so they and all other vertices exactly touch the bounding sphere.

Is there a good way to clear up this apparent inconsistency? --68.0.124.33 (talk) 04:58, 7 November 2009 (UTC)

I think you've got it. The catalan solid doesn't have equilateral triangle faces. They can be made equilateral, but then its no longer convex (There's no image of this case). This pentakis dodecahedron should express these varied geometric forms. Tom Ruen (talk) 12:11, 7 November 2009 (UTC)

I have attempted to clarify/disambiguate these concepts by editing the page to reflect the typical usage of "pentakis dodecahedron" to refer specifically to the Catalan solid as (geometric) dual to the truncated icosahedron, introducing the more general term "elevated dodecahedron" that is typically used for any polyhedron produced by augmenting every face of a dodecahedron with some pentagonal pyramid. Although this distinction and these usages are not universal, in my fairly extensive experience they are the most common usages of these contrasting terms. Are citations needed to back up this disambiguated terminology? If so, what sort of references are appropriate? I could find a number of original sources that use the two terms in these ways, but as there exist references that use the term "pentakis dodecahedron" for any "elevated dodecahedron", no list of references for the more specific meaning would really prove that the restricted meaning is more common. Thanks for any guidance. GlenWhitney (talk) 01:51, 15 July 2023 (UTC)

a range of geometrically distinct forms of pentakis dodecahedron

Enlarging on Tom Ruen's comment above, there are (by my count) at least six geometrically distinct stages the pentakis dodecahedron (PD) passes through as the pentagonal pyramids are progressively flattened, beginning with pyramids of 5 equilateral triangles, and proceeding through flatter pyramids, made of isosceles triangles with progressively wider bases:

• The equilateral or deltahedron version of the pentakis dodecahedron (PD) consisting of 12 pyramids made of precisely equilateral triangles. This is the PD depicted in the red figure [1] in the Kleetope section. This version of the PD is non-convex, and it has two kinds of vertex, with the twelve 5-edge vertices at the apexes of the pyramids lying farther from the center of the polyhedron than the twenty 6-edge vertices.
• A family of nonconvex polyhedra with progressively flatter pyramids between the deltahedron PD and the following degenerate case.
• The rhombic triacontahedron where triangles on adjacent pyramids become coplanar, forming rhombuses, and reducing the number of faces from 60 to 30. Unlike the preceding stages, this polyhedron, and all the following ones, are convex. The rhombic triacontahedron's vertices are of two kinds, 3-edge and 5-edge, lying at different radii.
• A family of polyhedra, again with 60 isosceles triangular faces, intermediate between the rhombic triacontahedron and the standard Catalan PD below.
• The standard Catalan PD depicted in the green figure [2] at the beginning of the article. Being a Catalan solid, this polyhedron, dual to the truncated icosahedron, has all its faces equidistant from the center, but not all its vertices. The vertices are of two kinds: twelve 5-edge pyramidal vertices and twenty 6-edge vertices which seem to lie closer to the center than the 5-edge vertices, but I am not sure.
• A family of PDs with 60 isosceles triangular faces forming pyramids shallower than the Catalan PD above. For some particular value of the pyramid height, I think it must happen that all 32 vertices of the PD become equidistant from the center.
• When the pyramids become completely flat, the PD reduces to a regular dodecahedron.

Can some expert in polyhedron geometry confirm and/or correct this?CharlesHBennett (talk) 04:43, 16 April 2015 (UTC)

I agree about the existence of all of the above cases, but as I've edited the page to reflect, I think typically the term "pentakis dodecahedron" is used to denote the "standard Catalan PD" as you term it. The others I would call various "elevated dodecahedra", the final one being trivially elevated. There are additional forms to the ones you list when the augmenting pyramids have greater altitude than the equilateral pentagonal pyramid; one of particular note is the exterior surface of the Small stellated dodecahedron considered as a non-convex polyhedron with 60 isosceles triangular faces, rather than a self-intersecting polyhedron with 12 regular pentagram faces. But I don't particularly feel that this page needs to include this entire taxonomy of elevated dodecahedra... GlenWhitney (talk) 02:00, 15 July 2023 (UTC)

Add reference

Should add reference to class 2 icosahedron geodesic domes. Hugh Kenner's book, Geodesic Math and How to Use It (1976), gives the algorithm for generating this type of dome. — Preceding unsigned comment added by 67.184.118.144 (talk) 21:56, 11 August 2011 (UTC)