From: "John Weidner"
Subject: Re: SOCCER BALLS AND SPHERICAL GEOMETRY
Date: Mon, 27 Dec 1999 06:26:00 -0500
Newsgroups: alt.math,alt.math.undergrad,sci.math
Keywords: regular, semiregular polyhedra
A soccer ball shows the projection of one of the possible "semi-regular"
polyhedron onto a sphere.
A regular polyhedron is a polyhedron in which all faces are the same
regular polygon, and for which the configurations at each vertex are the
same. (I must confess that I have not looked at whether the regularity
at each vertex is a necessary constraint.) A couple of quick counting
formula and Euler's formula for the surface of a sphere show that there
are only 5 regular polyhedral.
A "semi-regular" polyhedron allows two different regular polygons as
faces, still with the condition that there be only one vertex
configurations. (I apologize in advance is "semi-regular" is the wrong
name. I think the reason I forget so much is that I am growing old, but
I forgot whether that is really the reason.)
Example of a semi-regular polyhedron: Put an equilateral triangle on
the floor. Form a square vertically up on each edge of the triangle.
This creates a triangular box, top and bottom equilateral triangles, and
each side a square. Clearly this can be done with any regular
polyhedron as the base, thus giving an infinite class of semi-regular
polyhedra.
Example 2: Rotate the top of the box by half the central angle of the
base polygon, and fill in the sides with equilateral triangles.
Of course, neither of these examples is used for making soccer balls.
There are other, to my taste-more beautiful, semi-regular polyhedra. A
soccer ball shows one such. Take a bunch of regular polygons and figure
out how you wish to put them together at a vertex. Do the calculations;
Euler's formula again. I think there are only a finite number other
than the two families given above.
There are books of regular polyhedra. I don't have the references.
Most of them feature the stellated polyhedra; those that allow faces to
cross, giving beautiful star patterns. Try a search on "Constructing
polyhedra".
Note: Euler's formula: For a connected graph drawn on a plane or
sphere,
number of vertices + number of faces - number of edges = 2
For a connected graph drawn on any other surface, the same calculation
gives a constant, which depends on the surface. I think the value is 0
for a torus and 1 for a projective plane.
Spherical Geometry is not what you need here; graph theory is.
> Have you played any Soccer lately? My sons have gotten me into the
> game and I enjoy it quite a bit. One day I looked a little
> more intently at the Surface Geometric Pattern of an ordinary Soccer
> Ball and have been thinking about it since:
>
> I'm fascinated by the Tessellation achieved by the 5 "Regular
> Hexagons" surrounding a "Regular Pentagon" repetitive pattern on
> an ordinary Soccer Ball.
>
> For a more general understanding, Some months back I asked people for
> an explanation or mathematical formula or description
> of allowable "Tessellations" on a Sphere.
>
> I was directed to Coxeter's classic Book "Regular Polytopes," which is
> a wonderful discussion in its own right, but it does not
> deal with Spheres except for one very small, irrelevant section
> (Coxeter himself calls it a "digression.") Additionally, some
> respondents opined that the Soccer Ball must be a Projection of a
> Dodecahedron or Icosadodecahedron on a Sphere, which is
> false. Recall that these 2 polyhedra consist of Pentagons or
> Pentagons/Triangles Only - not Hexagons.
>
> [For reasons known for centuries, Hexagons are not allowable Polygons
> in these Regular or QuasiRegular Polyhedra. They do
> not satisfy 1/p + 1/q >1/2]
>
> That still leaves me with an interesting puzzle:
>
> Regular Hexagons/Pentagons are not a viable Tessellation in Planar
> Geometry. There are 17 such combinations, but none uses
> both Regular Hexagons and Regular Pentagons.
>
> Yet on a Sphere, with each Edge/Angle deformed by Spherical Geometry,
> the combination of Curved Hexagons/Pentagons works
> like a charm.
>
> Why? And What are the precise mathematical Rules for Tessellating a
> Spherical Surface? They should be expressable in terms
> of Spherical Angle Measure or derivable as such. Concepts of Regular
> Spherical Shapes Analogous to Regular Planar Polygons
> would help complete the picture.
>
> I studied spherical Geometry a bit but that was 25 years ago and
> forgot just about all of it - so I need some help here.
[HTML deleted --djr]
==============================================================================
From: Douglas Zare
Subject: Re: SOCCER BALLS AND SPHERICAL GEOMETRY
Date: Mon, 27 Dec 1999 13:47:31 -0500
Newsgroups: alt.math,alt.math.undergrad,sci.math
Keywords: disphenoid
John Weidner wrote:
> A soccer ball shows the projection of one of the possible
> "semi-regular" polyhedron onto a sphere. A regular polyhedron is a
> polyhedron in which all faces are the same regular polygon, and for
> which the configurations at each vertex are the same. (I must confess
> that I have not looked at whether the regularity at each vertex is a
> necessary constraint.) A couple of quick counting formula and Euler's
> formula for the surface of a sphere show that there are only 5 regular
> polyhedral. A "semi-regular" polyhedron allows two different regular
> polygons as faces, still with the condition that there be only one
> vertex configurations. (I apologize in advance is "semi-regular" is
> the wrong name. I think the reason I forget so much is that I am
> growing old, but I forgot whether that is really the reason.)
If you only assume that the faces look the same and the vertices look
the same, then you get the regular polyhedra plus the disphenoid, which
is the result of folding an acute triangle along the lines connecting
the midpoints of its sides (see
http://www.math.columbia.edu/~zare/disphenoid.html ). On the other hand,
the usual assumption for semi-regular polyhedra (called Archimedean
solids rather than Platonic solids) is that the vertices look the same
and the faces are regular. One can have, for example, a triangle, a
square, a pentagon, and another square at each vertex.
I'd comment more, but your post showed up as extremely awkwardly
formatted in my newsreader.
Douglas Zare