 In 1977, Hungarian mathematicianLajos Szilassi found a way to construct a toroidal heptahedron. Each face of his polyhedron is a hexagon (although none of them is a regular hexagon).
To make your very own model of a Szilassi polyhedron, clickhereto download a pattern in PDF (52807 bytes, viewable with the freeAdobe® Acrobat® reader). For aesthetic reasons, I've altered the polyhedron's angles and proportions slightly from Szilassi's originals.Clickherefor Dr. George Hart's tips on constructing paper polyhedron models.(And if you make a polyhedron with this pattern, feel free to let me know how it came out. I'm happy to hear from people who've checkedout this web page.)
The Szilassi polyhedron shareswith 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 necessaryfor a map on a surface topologically equivalent to a torus. (Neither demonstrates anything about how many colors are sufficient.)
The vital statistics of the two polyhedra are as follows:
| tetrahedron | 4 faces | 4 vertices | 6 edges | 0 holes |
|---|
| Szilassi polyhedron | 7 faces | 14 vertices | 21 edges | 1 hole |
|---|
What other kinds of polyhedra could have the property that each pairof faces shares an edge? Maybe none.
Euler's formula f + v - e = 2is easily generalized for polyhedra with holes;
ifh is thenumber of holes, f + v - e = 2 - 2*h.
In a polyhedron every pair of whose faces shares an edge,
f ande arerelated by the equatione = f * (f - 1) / 2.
Each vertexmust be at the intersection of three edges. (If there were more thanthree, some pairs of faces couldn't share an edge.) Thus, as each edgeconnects 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 off yield whole numbers of holes.f=4 appliesto the tetrahedron;f=7 applies to the Szilassi polyhedron. The nextvalue off that yields a whole value forh is 12, which would apply toa polyhedron with 12 faces, 66 edges, 44 vertices, and 6 holes. Thatconfiguration doesn't sound workable to me; higher values off andh seem even less likely. Obviously I'm a bit short of a proof,but you get the picture. Clickhereto see a model of the configuration of the dual of such an unlikelypolyhedron.
Szilassi gave a set of equations that describe a particular instance of a7-sided toroidal polyhedron; Martin Gardner reproduced patterns for thefaces (in the context of an entertaining discussion about math and minimalist art) in his Mathematical Games column in Scientific American (Nov. 1978). Szilassi's seven faces, in their original shapes, are also illustrated below.
There is a dual to the Szilassi polyhedron. It was discovered in thelate 1940s by Ákos Császár and has 14 triangular faces, 7 vertices, 21 edges, and one hole. Patterns for a model of the Császár polyhedron can be found in Martin Gardner's column in the May 1975 Scientific American. (The math above was adapted from Gardner's description of Donald W. Crowe's analysis of the relationships amongf,v,e, andh in the Császár polyhedron.)
There's an applet athttp://mathworld.wolfram.com/SzilassiPolyhedron.htmlthat shows how the Szilassi polyhedron looks at various rotations that you can control with your cursor. (Last I checked, the Java code in thatapplet exhibited a bug in its hidden-surface removal methods; the polyhedronisn't rendered properly at certain rotations. The bug isn't a big deal;I found it more amusing than annoying.)
Images on this page were generated by a program I wrote to play withaltering the angles and proportions of the polyhedron. To download C++ source and documentation for the program (gzipped tarball, 44379 bytes; requires X Window graphics), clickhere.If you'd like further information, send me email.
Tom Ace
return to Tom's home page
Clickherefor a Hungarian language version of this page (translated by Noémi Vanderstein).
The polyhedron's seven faces:
Other views of the polyhedron in perspective:
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