Newsgroups:sci.mathFrom:tea@netcom.com (Tom Ace)Subject:Szilassi polyhedronKeywords:toroidal polyhedronOrganization:NETCOM On-line Communication Services (408 261-4700 guest)Date:Sat, 4 May 1996 23:17:55 GMT

In 1977, the Hungarian mathematician Lajos Szilassi found a way to construct a seven sided toroidal polyhedron. Each face of his polyhedron is a hexagon (none of them is a regular hexagon). The polyhedron shares with the tetrahedron the property that each of its faces touches all the other faces. A tetrahedron demonstrates that four colors are necessary for a map on a surface topologically equivalent to a sphere; the Szilassi polyhedron demonstrates that seven colors are necessary for a map on a surface topologically equivalent to a torus. (Neither polyhedron demonstrates anything about how many colors are sufficient.) tetrahedron: 4 faces 4 vertices 6 edges 0 holes Szilassi polyhedron: 7 faces 14 vertices 21 edges 1 hole What other kinds of polyhedron could have the property that each pair of faces shares an edge? Probably none. Euler's formula f + v - e = 2 is easily generalized for polyhedra with one or more holes; if h is the number of holes, f + v - e = 2 - 2*h. In a polyhedron every pair of whose faces shares an edge, f and e are related by the equation e = f * (f - 1) / 2. In a polyhedron every pair of whose faces shares an edge, each vertex must be at the intersection of three edges. (If there were more than three, some of the faces couldn't share an edge.) Thus, as each edge connects two vertices, v = e * 2 / 3. Combining the above equations and simplifying gives h = (f*f - 7*f + 12) / 12 which can be factored as h = (f - 4) * (f - 3) / 12 .. Only certain values of f yield whole numbers of holes. f=4 applies to the tetrahedron; f=7 applies to the Szilassi polyhedron. The next value of f that yields a whole value for h is 12, which would apply to a polyhedron with 12 faces, 66 edges, 44 vertices, and 6 holes. I find it inconceivable that such a polyhedron could exist, and higher values of f and h seem even less likely. Obviously I'm a bit short of a rigorous proof, but you get the picture. Szilassi gave a set of equations that describe a particular instance of 7-sided toroidal polyhedron; Martin Gardner reproduced patterns for the faces in his Mathematical Games column in Scientific American (Nov. 1978). If any of this interests you, and in particular if you would like patterns for making models of a Szilassi polyhedron, let me know. I have software I'd be glad to share which lets you play around with the angles and proportions of the polyhedron. Tom Ace tea@netcom.com

From:Laura Helen <email address removed by request of LH>Newsgroups:sci.mathSubject:Re: Szilassi polyhedronDate:Sun, 5 May 1996 09:29:04 -0700Organization:University of California, Los Angeles

On Sat, 4 May 1996, Tom Ace wrote: > In 1977, the Hungarian mathematician Lajos Szilassi found a way to > construct a seven sided toroidal polyhedron. If you exchange faces and vertices, this is the inverse of a torus with 7 vertices, 21 edges and 14 faces. To find polyhedra with each face sharing an edge one can invert polyhedra with each vertex sharing an edge. Looking for polyhedra with each vertex sharing an edge, this is combinatorially satisfied by a polyhedron with 9 vertices, 36 edges and 24 faces with Euler characteristic -3 -- it is non-orientable. This looks like: 1 2 3 1 2 4 1 3 5 1 4 6 1 5 7 1 6 8 1 7 9 1 8 9 2 3 6 2 4 5 2 5 7 2 6 9 2 7 8 2 8 9 3 5 8 3 6 7 3 7 9 3 4 8 3 4 9 4 5 9 4 6 7 4 7 8 5 6 8 5 6 9 V=9, EC=-3,F=24,E=36 There is an orientable such polyhedron with 12 vertices, 66 edges and 44 faces: 1 2 3 1 2 4 1 3 5 1 4 6 1 5 7 1 6 8 1 7 9 1 8 10 1 9 11 1 10 12 1 11 12 2 3 6 2 4 5 2 5 8 2 6 7 2 7 9 2 8 11 2 9 12 2 10 11 2 10 12 3 5 8 3 6 11 3 8 9 3 9 10 3 4 10 3 4 12 3 7 11 3 7 12 4 5 9 4 6 10 4 9 11 4 7 11 4 7 8 4 8 12 5 7 12 5 9 10 5 10 11 5 6 11 5 6 12 6 7 10 6 8 9 6 9 12 7 8 10 8 11 12 V=12, F=44, E=66, EC=-10, orientable This may even be embeddable in 3-space. If so, one can try to make the faces of the inverse flat. There is an orientable example for each #vertices > some minimal number where #vertices choose 2 is divisible by 3 and V-E+F is even -- reference: MR- 82b:57012 | AU- Jungerman,-M.; Ringel,-G. | TI- Minimal triangulations on orientable surfaces. | PY- 1980 | JN- Acta-Math. [Acta-Mathematica] 145 (1980), no. 1-2, 121--154. | LA- English | AB- A polyhedron on S, a compact 2-manifold, is called a triangulation if each face is a triangle with 3 distinct vertices and the intersection of any two distinct triangles is either empty, a single vertex, or a single edge. Let delta (S) denote the minimum number of triangles in any triangulation of S. The second author determined delta (S) for nonorientable S (Math. Ann. 130 (1955), 317 - 326; MR 17, 774). A routine application of Euler's formula shows that if S is an orientable surface of genus p, then delta (S)>4(p - 1)+2((7+{radlin}1+48p))/2). It is the authors' considerable achievement to show that equality holds except when p=2 (here delta (S)=24). They assert, ''The proof of the formula is a problem similar in nature and at least equivalent in complexity to the problem of determining the genus of the complete graph K{sub}n. In both problems one must exhibit triangular embeddings of graphs which are complete or nearly complete (where some edges ar

From:mathwft@math.canterbury.ac.nz (Bill Taylor)Newsgroups:sci.mathSubject:Re: Szilassi polyhedronDate:8 May 1996 04:58:12 GMTOrganization:Department of Mathematics and Statistics, University of Canterbury, Christchurch, NZ.Keywords:toroidal polyhedron

tea@netcom.com (Tom Ace) writes: |> In 1977, the Hungarian mathematician Lajos Szilassi found a way to |> construct a seven sided toroidal polyhedron. Each face of his |> polyhedron is a hexagon (none of them is a regular hexagon). |> ... ... Martin Gardner reproduced patterns for the |> faces in his Mathematical Games column in Scientific American (Nov. 1978). Yes; nice one! I have one here in my hand, in fact, this very second. It was made and and given to me by my late colleague Derrick Breach, who alas died suddenly the other day. He also constructed what's been said to be one of the world's most complete sets of (cardboard) solids - regular, Archimedean, Johnson, stellated, compound, and many many others. Anyone who likes can see them on display here at Canterbury NZ - perhaps on your way through while collecting your chocolate fish credits! Derrick will be sorely missed. But getting back to the Szilassi polyhedron - it's very nice. The faces come in three pairs of congruents, and another; so the whole thing has a neat skew-symmetry. The six paired faces are all nonconvex hexagons, and the other one is convex and almost regular - it's a rectangle with two opposite truncated corners on my model, though I might guess it could be made regular by distorting the others a bit more. Cool. |> I have software I'd be glad to share which lets you play around |> with the angles and proportions of the polyhedron. Oh great. Give it a whirl, Tom, and tell us if that one can be made regular. |> h = (f - 4) * (f - 3) / 12 |> |> .. Only certain values of f yield whole numbers of holes. Yes, the whole Heawood formula thing is fascinating, both mathematically and historically. When Kempe produced his false proof of the 4-color theorem, (ca 1886), it took a year for someone to notice it was wrong. That someone was Heawood, and ironically, the very paper where he announced the error, contained his formula, which was given as an equation, ALSO in error! (He'd in fact proved an inequality, but didn't notice his logical lacuna.) It took all of TEN (!) years for someone to notice THIS error. It was Tietche, who proved the equation correctly, produced examples for up to (I think) 6 holes, and extended it to non-orientable surfaces. As far as I know there was no actual error in *his* paper - but: The Heawood formula, though then known to be correct, was still only known to be applicable as a sufficient but NOT necessary bound on the number of colors - there could not be MORE than the Heawood number of colors required, but there may be less. The Tietche paper with examples, tended (by silence) to imply it was nec & suff. Finally, in 1930 odd, Franklin proved that the Klein bottle, though having Heawood number 7, in fact only ever required 6 colors. This paper is very easy to follow, and a tour-de-force in simple chromatic combinatorics. There were NO errors, even implied, in the Franklin paper. But oddly, it led to an error in mathemnatical folklore! Thuswise. Because the Klein exception was the only one known for so long, it became "well-known" that the Heawood formula had been proved nec-&-suff for ALL other cases. This was mentioned to me by my own lecturer as fact, in fact; I recall it very well. That was in 1965, when in fact it was NOT YET KNOWN to be so! (My old lecturer is still here on the staff with me - I never fail to remind him of this gaff when appropriate! :) ) It wasn't completely proved till Ringel & Young tidied it all up in a series of papers in the late 60's. Klein is the sole oddity! Phew!! |> The next |> value of f that yields a whole value for h is 12, which would apply to |> a polyhedron with 12 faces, 66 edges, 44 vertices, and 6 holes. Yes; I took years of odd moments of time, constructing my own dodecahedral holocontiguous hexatoroidal map. Great fun. Later I found a fun way to construct new maps from olds, using man-machine interaction. As always, of course, the real fun was in the programming rather than the results. One other thing I *did* do by hand, though, was to find a fully symmetric map of this type. It's neatly modellable on either a dodecahedron with 6 (bent) tubes going between opposite faces, or a cube with six tubes going between opposite edges. Way cool! I have models in my office if anyone wants to see. |> I find it inconceivable that such a polyhedron could exist, (For those who've forgotten what I'm waffling about, Tom means a polyhedron with 12 (possibly nonconvex) faces, all touching all, with Euler characteristic -10.) I would have said so too, once. However, the existence & shape of the Szilassi polyhedron, combined with my own fully symmetric simple-tube model, now suggests to me that it may well be possible! Maybe I'll try and construct it one day, and earn undying fame... ------------------------------------------------------------------------------- Bill Taylor wft@math.canterbury.ac.nz ------------------------------------------------------------------------------- Klein bottle for sale. Apply within. -------------------------------------------------------------------------------

Newsgroups:sci.mathFrom:tea@netcom.com (Tom Ace)Subject:Re: Szilassi polyhedronKeywords:toroidal polyhedronOrganization:NETCOM On-line Communication Services (408 261-4700 guest)Date:Fri, 10 May 1996 04:14:58 GMT

mathwft@math.canterbury.ac.nz (Bill Taylor) writes: >But getting back to the Szilassi polyhedron - it's very nice. The faces >come in three pairs of congruents, and another; so the whole thing has a >neat skew-symmetry. The six paired faces are all nonconvex hexagons, and >the other one is convex and almost regular - it's a rectangle with two >opposite truncated corners on my model, though I might guess it could be >made regular by distorting the others a bit more. Cool. > >Oh great. Give it a whirl, Tom, and tell us if that one can be made regular. I haven't had any luck making that one convex face a regular hexagon. In Szilassi's original, it has 180-degree rotational symmetry and looks about like this: _____________________ / | / | / | | / | / |_____________________/ whereas I find it aesthetically desirable to twist the polyhedron so the face starts to look more like ___________________ / \ / \ / \ \ / \ / \___________________/ but that's about as close to regular as I've been able to make it. > [...] >|> I find it inconceivable that such a polyhedron could exist, > >(For those who've forgotten what I'm waffling about, Tom means a polyhedron >with 12 (possibly nonconvex) faces, all touching all, with Euler >characteristic -10.) > >I would have said so too, once. > >However, the existence & shape of the Szilassi polyhedron, combined with my >own fully symmetric simple-tube model, now suggests to me that it may well be >possible! Maybe I'll try and construct it one day, and earn undying fame... If you succeed, I'd love to have your autograph. :) Tom Ace tea@netcom.com

From:Tom Ace <crux@qnet.com>To:eppstein@ics.uci.eduSubject:re: geometry junkyardDate:Fri, 26 Feb 1999 14:37:14 -0800

David, Hi there. Your page at http://www.ics.uci.edu/~eppstein/junkyard/szilassi.html (which I found from a link on your page at http://www.ics.uci.edu/~eppstein/junkyard/polytope.html ) has a copy of a sci.math posting I'd made several years ago about the Szilassi poyhedron, and several follow-up postings. Since then, I've changed my email address, and put up a web page about the Szilassi poyhedron at http://www.qnet.com/~crux/szilassi.html which is an updated version of the topic I'd originally brought up in sci.math, with illustrations. If you're willing to add to a link on your site to my Szilassi page, I'd be grateful. As it is now, anyone trying to contact me at the old address given on your page (e.g., to get a copy of my Szilassi polyhedron software) won't find me. In any case, thanks for your web site, I've enjoyed reading it. regards, Tom Ace crux@qnet.com