# Tagged Questions

Questions related to polyhedra and their properties.

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### What honeycomb has the highest volume to edge length ratio?

This question is analagous to the Kelvin Problem where the solution, the Weaire-Phelan Structure, has the highest volume to surface area ratio; however, the cell volume is compared to edge length ...
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### What does an Euler characteristic of a topological space greater than 2 topologically mean?

Recently I've found a polyhedron with Euler characteristic $\chi=9$. This is made from the Octahemioctahedron with adding the intersections of the hexagon faces as vertices. It has $V=13, E=36, F=32$....
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### Why is Octahemioctahedron topologically a torus?

I'm afraid, that I have a very bad space vision, because I don't see, that Octahemioctahedron is topologically a torus. Could somebody explain it for me, why is it? Scene 2. @aes: Finally, ...
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### Largest of the smallest angles of incidence from arbitrary point to tetrahedron vertex/centroid line

Picture a regular tetrahedron where each vertex has a line through the centroid and a plane normal to it. I need to show that the range of the smallest angles of incidence from an arbitrary point to ...
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### The skeleton of Eulerian polyhedra

There is (at least) two kind of validity domain of Euler's $v−e+f=2$ polyhedron formula. One is the "Eulerian" polyhedra, i.e. simply connected polyhedra with simply connected faces (see here). The ...
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### How to find out the circumscribed radius of a snub dodecahedron?

A snub dodecahedron is produced by partially rotating all 12 regular pentagonal faces of a small rhombocosidodecahedron. A snub dodecahedron has 80 congruent equilateral triangular faces, 12 congruent ...
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### Is there any simply connected polyhedron with a not simply connected face?

According to Wikipedia, For a convex polyhedron or more generally for any simply connected polyhedron whose faces are also simply connected, χ = 2. Is it really necessary to specify here, that ...
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### How does the existence of Platonic graphs imply the existence of Platonic solids?

I will use the following definitions Platonic graph: A 3-connected planar graph with faces bounded by the same number of edges and vertices having the same number of incident edges. (remark: ...
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### computing Lefschetz number

We have a fixed point theorem which says that : Let $X$ be a compact polyhedron, $f:X\rightarrow X$ be a continuous map. If $L(f)\neq 0$ then $f$ must have a fixed point. (Lefschetz number is ...
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### Is the adjective 'regular' necessary in the definition of Platonic solids?

The definition I mean can be found in the tag Wiki of Platonic solid tag and also in Wikipedia: Definition 1: A Platonic solid is a regular, convex polyhedron with congruent faces of ...
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### Convex polyhedra with edges of equal length

This is again a natural category of polyhedra without having an own name. Is it possible, that their graphs are the same as the graphs of polyhedra with faces of regular polygons? My question is ...
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### Dihedral angles of a pentakis dodecahedron

I'm new to the world of mathematical descriptions of polyhedra, and I'm wondering if, for a Pentakis Dodecahedron, the dihedral angles are uniform at each vertex. The visualization of the P.D. on the ...
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### What is the name of convex polyhedra with congruent faces of regular polygons?

The definition of platonic solids is the following (see Wikipedia): In Euclidean geometry, a Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same ...
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### Is there a term for an “unbounded simplex”?

Is there a general term for regions like $\{(x,y):x>y\}$ and $\{(x,y,z): x>y>z\}$, i.e., regions which are simplexes with one open?
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### Conway polyhedra notation calculator?

I recently read about Conway polyhedra notation, and I want to experiment with it. Are there any programs that take the notation, and output a representation of the shape?
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### 3D Dodecahedron model: Construction question.

The image below is a 2D construction that, when cut-out and folded appropriately (hopefully it is intuitively clear how to cut and fold), forms a 3D dodecahedron. It works great: I've successfully ...
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### possible polyhedra from euler's formula

I'm not very clear with the euler's formula, and I couldn't find it anywhere. I'm sorry if it is a double post. F + V - E = 2 Is the euler's formula. If the equation balances, is it polyhedra all ...
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### surface area of 'cylinder' with the top cut at an angle

I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ...
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### Analytic-geometrical properties of dodecahedron

Consider the following projection of a dodecahedron: An equilateral triangle can be projected to make points $A, B, C, D, E, F$ intersect with it's edges. What would be the mathematical proof (if ...