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volume derivative

  • To: mathgroup at
  • Subject: [mg8034] volume derivative
  • From: Russell Towle <rustybel at>
  • Date: Sat, 2 Aug 1997 22:32:42 -0400
  • Sender: owner-wri-mathgroup at

Hi all,

I wish to evaluate the derivative of a function and discover its zeros
in order to identify maxima and minima of the function.  I am at a loss
as to how to proceed.

To begin with, the volume of a parallelepiped is equal to the absolute
value of the determinant of the 3 by 3 matrix of the three vectors
which meet at any vertex.  A convex zonohedron is bounded by
parallel-sided centrally symmetrical convex polygons, which may be
rhombs, parallelograms, hexagons, octagons, decagons, etc.  It is said
to be determined by a set of n vectors in 3-space.  Such a zonohedron
may be dissected into Binomial [n, 3] parallelepipeds, and by summing
the absolute values of the determinants for each parallelepiped, we
obtain the volume of the zonohedron.  A neat way to do this in
Mathematica is as follows, where "vectors" denotes the set of n
vectors, and "q" is the flattened list of determinants:

q = Flatten[Abs[Minors[vectors, 3]]];

Volume = Sum[q[[i]],{i,Length[q]}]

So, in the first place I wish to find the derivative for this function,
which is a sum of absolute values of determinants of vector triples in
3-space.  For my purposes, it is entirely sufficient and even desirable
to consider only unit vectors.

Then, hopefully, I wish to find the zeros of the derivative.  I am
especially interested in those zeros which lead to zonohedra of maximal
volume.  If by some method this process could actually return a list of
n vectors, then we would, for various values of n, discover the
vertices, mid-edge points, and face centers of the five Platonic
solids.  For instance, when n=3, we would discover the n=3 zonohedron
of maximal volume, a cube.

The next step would be to apply constraints to the function(s) so as to
admit more than one solution to the n-zonohedron of maximal volume. 
For instance, we would discover, for n=6, and in order of decreasing
volume, Kepler's rhombic triacontahedron, an n=6 polar zonohedron of
particular proportions, and the truncated octahedron.  By this means we
would also discover, from scratch and without any foreknowledge as it
were, the icosahedral, dihedral, and octahedral symmetry groups.

I will be most grateful for help on any part of this problem.

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