```Newsgroups:     sci.math
From:           tea@netcom.com (Tom Ace)
Subject:        Szilassi polyhedron
Keywords:       toroidal polyhedron
Organization:   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.math
Subject:        Re: Szilassi polyhedron
Date:           Sun, 5 May 1996 09:29:04 -0700
Organization:   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.math
Subject:        Re: Szilassi polyhedron
Date:           8 May 1996 04:58:12 GMT
Organization:   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.math
From:           tea@netcom.com (Tom Ace)
Subject:        Re: Szilassi polyhedron
Keywords:       toroidal polyhedron
Organization:   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
_____________________
/                     |
/                      |
/                       |
|                       /
|                      /
|_____________________/

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.edu
Subject:        re:  geometry junkyard
Date:           Fri, 26 Feb 1999 14:37:14 -0800
```

```David,

http://www.ics.uci.edu/~eppstein/junkyard/szilassi.html
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
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
```