# Questions tagged [polyhedra]

Questions related to polyhedra and their properties.

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### Trying to create a 3D model of Kelvin’s Tetradecahedron / Tetrakaidecahedron polyhedra

How can I go about creating a 3D model / 3D image of a Kelvin’s Tetrakaidecahedron Cell / Tetrakaidecahedron. I planned on using Octave to 3D model an image it mathematically then convert that into a ...
0answers
15 views

### The point of intersection of the four space diagonals of a general parallelepiped, the centroid

I know that the centroid of a parallelogram is the intersection of the diagonals, but is it true that the centroid of a parallelepiped is the intersection of the space diagonals? I'm talking about the ...
0answers
20 views

### Embedding a polyhedron given a relative interior point

How can I embed a polyhedron given in an H-representation (hyperplane inequalities) which is not full-dimensional, given a relative interior point?
1answer
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### Platonic solids - topological and geometrical conditions

With V, E, F as the numbers of vertices, edges and faces of a given polyhedron and based on Euler‘s polyhedron formula $$V - E + F = 2$$ it is quite simple to derive a necessary topological ...
0answers
28 views

### Algorithm for variant of Alexandrov's uniqueness theorem

Alexandrov's uniqueness theorem states for each development, there is a unique convex polyhedra with the metric induced by the development. For our purposes, we can view developments as finite planar (...
1answer
66 views

0answers
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### What is the largest volume of a polyhedron whose skeleton has total length 1? Is it the regular triangular prism?

Say that the perimeter of a polyhedron is the sum of its edge lengths. What is the maximum volume of a polyhedron with unit perimeter? A reasonable first guess would be the regular tetrahedron of side ...
1answer
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1answer
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1answer
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### Given isosceles triangles $\triangle ABC$ and $\triangle DBF$ (all spherical chords), identify the chord $\overline {DF}$ so that $|AD| = |DF| = |FC|$

tl;dr: As shown in the image below, find the chord $\overline {DF}$ so that $|\overline {AD}| = |\overline {DF}|$, and have the answer be in the form of the ratio between $|\overline {AC}|$ and \$|\...
0answers
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### farkas-minkowski theorem specific exmaple

I was reading about farkas-minkowski theorem, which basically said a convex cone is polyhedral iff it's finitely generated. The theorem make sense, but when I played with some example I met the ...