Re: Geometry in Art - Help with magazine article?

From: James Buddenhagen (foo_bar_at_texas.net)
Date: 02/28/05 

Date: Mon, 28 Feb 2005 18:57:51 GMT

"The Last Danish Pastry" <clivet@gmail.com> wrote in message
news:388m3lF5jjdd3U1@individual.net...
> "The Last Danish Pastry" <clivet@gmail.com> wrote in message
> news:388fr3F5kueudU1@individual.net...
> > "Scott Brown" <scott@finebooksmagazine.com> wrote in message
> > news:200502242249.j1OMngV30700@proapp.mathforum.org...
> >
> > > sorry about the dividers-compass switch. The article text says dividers.
> > >
> > > Your snub disphenoid idea seems promising. It does bear some resemblance
> > > to one. However, my Oxford dictionary cites the origin of disphenoid to
> > > 1895. The painting dates from the early 1700s. Do you have any idea when
> the
> > > dispenoid was worked out mathematically (perhaps before it got that
> name?)
> >
> > The disphenoids are a class of irregular tetrahedra, probably known since
> > antiquity, under what name I do not know. Their nets are particularly
> simple
> > to make.
> >
> > The snub disphenoid is a member of the class of deltahedra. The term
> > "deltahedron" itself was apparently coined by H Martyn Cundy in 1952. As
> far
> > as I know Cundy did not use the term "snub disphenoid". In fact, he
> remarked
> > that if a name was required for that solid it would presumably have to be
> > the dodecadeltahedron.
> >
> > I suspect that the snub disphenoid has been known for centuries. Its net
> is
> > just a set of equilateral triangles (as are the nets of all the
> deltahedra)
> > so it is easy to make one. Anybody trying to find polyhedra with faces all
> > of which are equilateral triangles is likely to stumble across it.
> >
> > The snub disphenoid itself is of little mathematical interest, although
> the
> > shape is somewhat intriguing, being symmetrical about its center and also
> > having two perpendicular planes of symmetry.
>
> Here is a perspective view of a snub disphenoid, together with the relevant
> part of the painting:
> http://www.pisquaredoversix.force9.co.uk/SnubDisphenoid.png

Nice picture. It makes a good case for the snub-disphenoid, but I still think
it might just as well not even be a 3d object. Still, you got me interested in
the snub-disphenoid, and I worked out some 'vital statistics' that may be of
interest. One realization of it in R^3, with center at the origin puts the
eight vertices at the points:

[a,a,-1], [-a,-a,-1], [-a,a,1], [a,-a,1], [-b,b,-c], [b,-b,-c], [b,b,c],
[-b,-b,c]

where the approximate values of a,b,c are: a= .4510006936, b=.5814159087,
and c=.2622189781. The number c is a root of the polynomial 1-5*c+4*c^2+2*c^3
and a,b may be computed from c via: a^2 = (-c^2+1-2*c)/2 and b^2 = 1-2*c-2*c^2.

The 8 by 8 Gram matrix whose i,j entry is the inner product of vertex i with
vertex j (regarded as vectors from the origin) may easily be calculated. It
completely determines the object in a coordinate free way. The eigenvalues of
the Gram matrix are [0, 0, 0, 0, 0, 4+4*c^2, 6-12*c-10*c^2, 6-12*c-10*c^2].
The matrix is positive-semi-definite rank 3, as it must be. The positive
eigenvalues are approximately equal to 4.2750351699, 2.1657843380, 2.1657843380.

Note that two of the positive eigenvalues are exactly equal. Probably this
is related to the symmetry of the object.

I have general question about Gram matrices of a set of points (in say R^3):
what is the connection between equality of 2 or 3 (or none) of the
eigenvalues and symmetry (or lack of it) of the arrangement of points?

Jim Buddenhagen

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> Clive Tooth
> http://www.clivetooth.dk
>