Re: [math-fun] Pentagram snowflakes? In a square grid?
<< On Fri, Sep 11, 2009 at 2:18 AM, Fred lunnon <fred.lunnon@gmail.com> wrote:
This might make an interesting wall design for a public space ...
Any comments on how it's done? WFL
I'd guess it's something like the dodecahedra in pyrite crystals: http://math.ucr.edu/home/baez/dodecahedron/14.html
That dodecahedron is described as non-regular on its web page. But (as is well-known) a perfectly regular dodecahedron can be inscribed in a perfect cube with 3 pairs of opposite edge passing through the face centers of the 3 pairs of opposite faces, resp., of the cube. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
Quoting Dan Asimov <dasimov@earthlink.net>: <snip>
But (as is well-known) a perfectly regular dodecahedron can be inscribed in a perfect cube with 3 pairs of opposite edge passing through the face centers of the 3 pairs of opposite faces, resp., of the cube.
Is there something similar connecting the regular icosahedron and the regular octahedron? It will have to be a little different, since 6|12 but 8~|20. Rich
On Fri, Sep 11, 2009 at 3:34 PM, <rcs@xmission.com> wrote:
Quoting Dan Asimov <dasimov@earthlink.net>: <snip>
But (as is well-known) a perfectly regular dodecahedron can be inscribed in a perfect cube with 3 pairs of opposite edge passing through the face centers of the 3 pairs of opposite faces, resp., of the cube.
Is there something similar connecting the regular icosahedron and the regular octahedron? It will have to be a little different, since 6|12 but 8~|20.
Rich
You can realize an icosahedron as the convex hull of six line segments that sit inside an octahedron -- the segments lie under its six vertices and their endpoints are on the icosahedron's edges, cutting them up in some golden-ratio-ish sort of way. --Michael -- Forewarned is worth an octopus in the bush.
Quoting rcs@xmission.com:
Is there something similar connecting the regular icosahedron and the regular octahedron? It will have to be a little different, since 6|12 but 8~|20.
They are face duals. This may carry over to embedment. -hvm ------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
All this stuff is to be found in Coxeter's book "Regular Polytopes", or (less technically) in Cundy & Rollet "Mathematical Models". There are 5 ways to inscribe the 8 vertices of a cube among the 20 vertices of a dodecahedron; and dually, 5 ways to circumscribe the 8 faces of an octahedron around the 20 faces of an icosahedron. Googling "regular compound polyhedron" turns up among others http://en.wikipedia.org/wiki/Polyhedral_compound http://www.georgehart.com/virtual-polyhedra/compounds-info.html Fred Lunnon On 9/12/09, mcintosh@servidor.unam.mx <mcintosh@servidor.unam.mx> wrote:
Quoting rcs@xmission.com:
Is there something similar connecting the regular icosahedron and the regular octahedron? It will have to be a little different, since 6|12 but 8~|20.
They are face duals. This may carry over to embedment.
-hvm
------------------------------------------------- www.correo.unam.mx UNAMonos Comunicándonos
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participants (5)
-
Dan Asimov -
Fred lunnon -
mcintosh＠servidor.unam.mx -
Michael Kleber -
rcs＠xmission.com