constructing the bisectors of angles in polyhedra

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joachim_weinhold
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Joined: 24 Jun 2022, 13:11

constructing the bisectors of angles in polyhedra

Postby joachim_weinhold » 09 Aug 2023, 12:04

Hi all,

again I am held back by a simple hurdle (… one from the large number of hurdles for complete greenhorns...) while beginning an attempt to evaluate my construction of a sculptural object (see attachment) that I constructed on the basis of the Tetartoid: How can I construct the perpendicular bisectors (which is hopefully the correct term) of all angles of the edges that meet in the single cornerpoints of a polyhedron with polymake ? (please excuse my stumbling explanation…)
Additionally I fail even by by the question of how to describe the Tetartoid (https://polytope.miraheze.org/wiki/Tetartoid) for polymake ?
(… with the aim in mind to use this description as well in the helpful algorithm you kindly and very helpful made available to me some months ago.)

Kind regards & thanks
Joachim Weinhold
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joswig
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Re: constructing the bisectors of angles in polyhedra

Postby joswig » 22 Aug 2023, 12:50

What is the "perpendicular bisector" of an edge? Do you mean the affine hyperplane which is perpendicular to a given edge, passing through the midpoint of that edge?

Your "tetartoid" is given without coordinates, and from the description given on that web page I cannot deduce how to obtain them. I guess they come from some (possibly oscure) variational principle. If that should be the case, then this is strictly outside of polymake's capabilities. This would require some numerical solver to produce the coordinates, which you could then pass on to polymake.

joachim_weinhold
Posts: 4
Joined: 24 Jun 2022, 13:11

Re: constructing the bisectors of angles in polyhedra

Postby joachim_weinhold » 22 Aug 2023, 22:48

Many thanks for your answer ! I was not aware if the term I was using describes correctly what I am looking for: A line that is orientated in the middle of the angles of all edges that meet in the vertices of a polyhedron - which I need to construct the orientation of the rings around the holes in the final object.

I found this solid first mentioned on the website of George Hart (https://www.georgehart.com/virtual-poly ... hedra.html) and constructed (better: modeled) the object that you saw in the attachment basing on this solid. I will list the coordinates of the vertices as accurate as possible and then try again.

kind regards & thanks !
Joachim Weinhold
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joswig
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Re: constructing the bisectors of angles in polyhedra

Postby joswig » 23 Aug 2023, 11:27

I still don't understand. Let's postpone this.

Concerning your tertatoid, I think I know a way, modulo some experimenting. Here are the steps.
(1) Construct the regular tetrahedron as the convex hull of every other vertex of the cube [-1,1]^3. Call this polytope T.
(2) From each edge of T construct the reflection at the (linear) hyperplane containing that edge and the midpoint of the opposite edge. Depending on whether you are manipulating the affine or the homogeneous coordinates these reflections will yield 3x3- or 4x4-matrices.
(3) The group generated from these six reflections is isomorphic to the symmetric group S4, which has 24 elements. Actually, three out of the six reflections suffice. If you don't want to figure out which three you need, use all six of them.
(4) From those 24 matrices, pick those which have positive determinant; there are exactly twelve of them, and they form the alternating subgroup A4.
(5) Now pick one facet of the tetrahedron T and perturb it slightly. This is where experimentation matters.
(6) Apply the twelve matrices from A4 to that one affine halfspace (which is oriented such that it contains the origin).
(7) If your perturbation is small and generic enough, the intersection of those 12 affine halfspaces will be combinatorially equivalent to the dodecahedron. That is a version of your tertatoid.
(8) Experiment with the perturbation until it looks like you want.


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