Re: [math-fun] Can an equilateral toroidal polyhedron
----- have fewer than 32 faces? —rwg ----- If it's not embedded anywhere, it can have just two faces. If it's embedded in n-space for fixed n, this is a terrific question! (Likewise for other surfaces, like the Klein bottle.) —Dan
For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex. The problems for n in {3, 4, 5} are certainly interesting! -- APG.
Sent: Saturday, May 23, 2020 at 4:48 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Can an equilateral toroidal polyhedron
----- have fewer than 32 faces? —rwg -----
If it's not embedded anywhere, it can have just two faces.
If it's embedded in n-space for fixed n, this is a terrific question!
(Likewise for other surfaces, like the Klein bottle.)
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I still have B M Stewart's book, Adventures among the toroids, which is conspicuous for its size (13" X 5" X .75"), it stands out, or rather up, in any book case. The subtitle is A study of ORIENTABLE POLYHEDRA with REGULAR FACES. Best regards Neil Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com On Sat, May 23, 2020 at 1:08 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex.
The problems for n in {3, 4, 5} are certainly interesting!
-- APG.
Sent: Saturday, May 23, 2020 at 4:48 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Can an equilateral toroidal polyhedron
----- have fewer than 32 faces? —rwg -----
If it's not embedded anywhere, it can have just two faces.
If it's embedded in n-space for fixed n, this is a terrific question!
(Likewise for other surfaces, like the Klein bottle.)
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
I have Stewart's book too -- a delightful romp, in my opinion unrivalled until Winning Ways came out. I will die in my boots for the 30-faced toroid. It is quiet adequately proved valid in Stewart. The convex hull is a truncated octahedron; a tunnel composed of three rings connects two hexagonal faces of the hull. The two outer rings are triangular cupolae; the middle one is a regular octahedron. That the hexagon-based height of the truncated octahedron is equal to twice the height of a triangular cupola, plus the altitude of an octahedron, is an easy exercise. On Sat, May 23, 2020 at 2:24 PM Neil Sloane <njasloane@gmail.com> wrote:
I still have B M Stewart's book, Adventures among the toroids, which is conspicuous for its size (13" X 5" X .75"), it stands out, or rather up, in any book case. The subtitle is A study of ORIENTABLE POLYHEDRA with REGULAR FACES.
Best regards Neil
Neil J. A. Sloane, President, OEIS Foundation. 11 South Adelaide Avenue, Highland Park, NJ 08904, USA. Also Visiting Scientist, Math. Dept., Rutgers University, Piscataway, NJ. Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com
On Sat, May 23, 2020 at 1:08 PM Adam P. Goucher <apgoucher@gmx.com> wrote:
For n >= 6, you can take a Császár polyhedron sharing the vertex-set (and indeed edge-set) with a regular 7-vertex simplex.
The problems for n in {3, 4, 5} are certainly interesting!
-- APG.
Sent: Saturday, May 23, 2020 at 4:48 PM From: "Dan Asimov" <dasimov@earthlink.net> To: "math-fun" <math-fun@mailman.xmission.com> Subject: Re: [math-fun] Can an equilateral toroidal polyhedron
----- have fewer than 32 faces? —rwg -----
If it's not embedded anywhere, it can have just two faces.
If it's embedded in n-space for fixed n, this is a terrific question!
(Likewise for other surfaces, like the Klein bottle.)
—Dan
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (4)
-
Adam P. Goucher -
Allan Wechsler -
Dan Asimov -
Neil Sloane