From: rusin@cs.niu.edu (David Rusin)
Newsgroups: sci.math
Subject: Re: Torus shaped polyhedra???
Date: 15 May 1997 00:16:50 -0500
In article <19970508194000.PAA27352@ladder01.news.aol.com>,
Cartesys wrote:
>A polyhedron is a solid whose faces are regular polygons. As such and
>forgetting for the time being the stellated varieties a polyhedron
>approximates a sphere.
>Is there anything comparable with a torus?
>Can you assemble several regular polygons (not necessarily of the same
>kind) and "approximate" a torus?
The faces of a polyhedron do not have to be _regular_ polygons.
Certainly if you don't insist on any kind of regularity, then yes, all
the compact orientable surfaces admit (fairly simple) realizations as
polyhedra. The question of whether or not there is a polyhedron
homeomorphic to a given topological space is the study of "triangularization",
and it's definitely nontrivial in higher dimensions -- especially if you
ask for polyhedra with particular properties, such as being embedded
in the same Euclidean space, being convex, containing a minimal number of
faces, and so on.
If you _do_ want to insist on regularity, you'll have to decide what that
means. Presumably that means there are a lot of isometries (i.e.,
functions which preserve lengths and angles) floating around. But
how exactly will you set the requirement? The Platonic solids are
particularly nice because given any two faces of the solid, and any
choice of vertices on the faces, there is an isometry f : R^3 -> R^3
which takes the solid to the solid, the one face to the other, and the
one vertex to the other. You can't expect so much for the torus. I can't
imagine there's _any_ surface in R^3 homeomorphic to the torus which
admits more isometries than the "standard" torus (made by swinging a
circle around an axis); that torus has a circle's worth of isometries,
and in particular there are no isometries carrying points near the
"inside rim" to points near the "outside rim". Thus I seriously doubt
that there is a polyhedron in R^3 homeomorphic to the torus which admits
a group of ambient isometries large enough to be transitive on the vertices.
In plainspeak: there is no polyhedral torus which is so regular that
it looks the same at every vertex.
But there are still plenty of questions to ask. For example, are there
polyhedral tori which at least have all faces congruent and regular?
That's easy if you can cheat a little. Take nine children's blocks, and
arrange them in a 3 x 3 cluster on the floor. Throw away the middle block and
glue each of the others to the ones next to it (clockwise and counterclockwise
neighbors). There are still 8 x (6-2) = 32 congruent square faces, whose
union is clearly homeomorphic to the torus. Of course it's not quite
what you want, I suppose, since some of the edges are "degenerate" (the
two adjoining faces are coplanar). You can fix (?) this by glueing an
additional block onto each of the 12 faces that's in the middle of a
coplanar set-of-three.
I can do a little better by trying to build a torus out of congruent
regular triangles. (This is getting fun.) Take those nine blocks again,
but arrange them into a triangle (three blocks on each side, with the
sides joined only along an edge). Reduce the inner triangular hole to
a regular hexagon by making three blocks to fit into the inside corners.
(You'll need these new ones to be equilateral triangular prisms.)
So far you have 39 square faces and 6 equilateral triangular ones.
Now take 39 square-based pyramids whose triangular sides are
equilateral, and glue them to the square faces. (The whole point to
having a hexagonal hole in the middle is allow these tetrahedra to
fit inside without intersecting. A square hole wouldn't work.)
The outer skin of your lovely creation is now a polyhedron with, um,
162 faces, all congruent equilateral triangles. It's homeomorphic to
the one-holed torus.
You can get the surfaces of higher genus, too. With 6 more blocks you
can make two more links of three and so make another big triangle sharing
a side with the original big triangle. (You can do this with many sets
of 6 additional blocks as long as you try to keep building out in
more or less the same direction, so as not to self-intersect eventually.)
Then repeat with the pyramids as before.
There's probably a way to glue pairs of icosahedra (say) along faces
so as to get them to form a big loop (looking like icosahedral pearls
on a necklace) but it's too much work for me to figure out how to make
the last pair of faces exactly meet as you try to join the ends of the
necklace.
As you may already know, if you try to build a polyhedron using only
regular n-gons, then the number m of them that meet at a vertex is
limited; indeed the only combinations are (n,m)=
(3, 3), (3, 4), (3, 5), (3,6)*
(4, 3), (4, 4)*
(5, 3),
(6, 3)*
( * = excluded if you wish to avoid degenerate vertices ). The combination
(3,3) is pretty useless, since any such set of exterior faces could be
replaced by the convex hull of its boundary, i.e., by the equilateral
triangle forming the base of this tetrahedron. We have already used
the (4,3) combination in our first model, the (3,4) combination in the
second, and the (3,5) and (5,3) combinations would arise if stringing
together necklaces of icosahedra and tetrahedra, respectively.
The Platonic solids have the additional virtue of being convex, which is
not possible for the surfaces of higher genus (due to total curvature).
I don't know what a suitable replacement condition would be, nor whether
it could be attained.
dave
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: Torus shaped polyhedra???
Date: 15 May 1997 17:35:32 GMT
In article <5le682$c1m@mp.cs.niu.edu>, David Rusin wrote...
What rubbish! Who is this guy? Why do they let him post?
>You can fix (?) this by glueing an
>additional block onto each of the 12 faces that's in the middle of a
>coplanar set-of-three.
Nice try, but then the central "hole" is lined with 4 sets of three coplanar
faces. As an alternative, glue an additional block onto each of the 4
faces of each of the four corner blocks.
>As you may already know, if you try to build a polyhedron using only
>regular n-gons, then the number m of them that meet at a vertex is
>limited; indeed the only combinations are (n,m)=
The listed combinations are the only ones which can occur in _convex_
polyhedra. Of course this does not apply with positive genus.
Those who like this sort of thing will like this book:
AUTHOR: Stewart, Bonnie Madison.
TITLE: Adventures among the toroids; a study of quasi-convex,
aplanar, tunneled orientable polyhedra of positive genus
having regular faces with disjoint interiors ... written,
illustrated and hand-lettered by B. M. Stewart.
^^^^^^^^^^^^^ !
PUBL.: (Okemos, Mich., : B. M. Stewart,
FORMAT: 206 p. illus. 34 x 13 cm.
^^^^^^^!
DATE: 1970
dave
==============================================================================
Date: Wed, 19 Aug 1998 02:33:57 +0000
From: "C.E.Peck"
To: rusin@cs.niu.edu
Subject: ADVENTURES AMONG THE TOROIDS
David,
Just saw your post concerning Bonnie Stewarts
ADVENTURES AMONG THE TOROIDS. I learned of this
fascinating book many years ago, finally located
the author and got a copy. Quite beyond my
abilities, but a lovely piece of work.
Prof. Stewart died a few years ago, and I have
been selling his book from my website www.winkworks.com.
This site may be closing soon, but I am sitting on
many copies of the book and will continue to sell
them. The website should remain visible, but may
not have much content. I will maintain my email
account at wink@southwind.net, if nothing else,
and sell copies to interested parties.
And, I will continue to sell my little booklet,
A TAXONOMY OF FUNDAMENTAL POLYHEDRA AND TESSELLATIONS.
Regards,
Charles Peck