Questions related to polyhedra and their properties.

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Group of rigid motion of tetrahedron

Let us take the group $S_4$. It has $24$ elements. Its group $A_4$ has $12$ elements. It is well known that the group of rigid motions of the tetrahedron has $12$ elements ($A_4$). Why? Because if we ...
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2answers
335 views

How to compute the volume of the polyhedron with vertices at centre of a cube?

The centers of the faces of a cube are also the vertices of polyhedron. How to Compute the ratio of the volume of the polyhedron to that of the cube containing it?
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3answers
175 views

Is it possible to inscribe a regular tetrahedron in every convex body?

Is it possible to inscribe at least one regular tetrahedron in every convex body?
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Orthocentric tetrahedron homothety [duplicate]

Possible Duplicate: Tetrahedron parallel planes feet of altitudes I was wondering if you could help me with the following problem: Let H be the orthocentre of a tetrahedron ABCS incribed ...
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Tetrahedron parallel planes feet of altitudes

Could somebody help me prove that the plane KLM, where K, L, M are feet of altitudes of an orthocentric tetrahedron ABCS inscribed in a sphere is parallel to the base ABC of this pyramid? I would ...
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1answer
362 views

Angles for a great dodecahedron

Could someone describe to me how to find the angle between two intersecting pentagonal faces on a great dodecahedron? Thanks
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1answer
162 views

Is the Euler characteristic $\chi =2$ for the prism with a hole?

I keep getting $\chi=2$ for the solid in the picture. It's a prism with a hole joining two opposite sides. I remember reading that $\chi=0$ for such solids. Help me find my error. I'd appreciate if ...
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1answer
534 views

How can I determine the radius of a dodecahedron?

I am making a dodecahedron that needs to fit inside of a sphere. The sphere has a diameter of 56mm. What is largest possible measurement of one segment of a pentagon side of a dodecahedron that would ...
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1answer
175 views

How to determine the intersection of 6 planes?

ABCD is a tetrahedron (not necessarly a regular one). A Monge's plane is a plane which is perpendicular to an edge and goes through the middle of the opposite edge. I want to prove that the 6 ...
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1answer
165 views

Are there formulae to determine close-packing polyhedra?

Is there a formula to determine which polyhedra will tessellate in 3D without any spaces?
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4answers
6k views

Calculating the probability of a coin falling on its side

A classical example that's given for probability exercises is coin flipping. Generally it is accepted that there are two possible outcomes which are heads or tails. However, it is possible in the real ...
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1answer
141 views

An unbounded convex polyhedron realizing the primes?

Does there exist an unbounded convex polyhedron with faces that have 3, 5, 7, 11, 13, ... edges, i.e., such that the number of edges of each face realize exactly the odd primes, with each prime ...
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3answers
121 views

Intuition about the faces in the connected planar graphs

In the Euler formula, for counting the number of faces, we count the regions bounded by edges, including the outer, infinitely-large region, so in the graph $K_1$ there is only one face which is outer ...
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2answers
483 views

Radius of a Sphere inscribed in a Convex Polyhedron

My teacher gave me this problem in class as a challenge. It has stumped me for days, yet he refuses to give me the answer! Let $PQRSTU$ and $PQR'S'T 'U'$ be two regular planar ...
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3answers
2k views

Angle between lines joining tetrahedron center to vertices

What are the angles formed at the center of a tetrahedron if you draw lines to the vertices? I'm trying to make these: I need to know what angles to bend the metal.
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1answer
398 views

Space-filling polyhedra (or honeycomb) survey?

Is there a survey anywhere of space-filling polyhedra? MathWorld's article, space-filling polyhedron, mentions about 400 being seen in pre-1981 books and papers. Wikipedia mentions 28 convex uniform ...
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2answers
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3D picture of the 38-sided Engel space-filling polyhedron

On page 220 of Peter Engel's Geometric Crystallography, he describes a 38-sided convex polyhedron that can fill space. I've seen this this accepted as the record in various places, but I've never ...
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2answers
216 views

Is there such a thing as the “edge-face dual” of a polyhedron, and is the “edge-face dual” of a cube a rhombic dodecahedron?

The dual of a polyhedron is a polyhedron where the vertices of one correspond to the faces of the other, and vice versa. Is there always a similar correspondence between a pair of polyhedra where the ...
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1answer
87 views

Does max { $w^Tx$ subject to $x$ is a point on a given polyhedron } optimize at an extreme point?

Is it necessary that the linear program max { $w^Tx$ subject to : $x$ is a point on a given polyhedron } attain its maximum at an extreme point of the polyhedron for any arbitrary w ? Let $c$ = ...
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221 views

How to find extreme bases of a polyhedron?

Suppose there is a submodular function $f$ over a ground set $V$ with cardinality $|V| = n$. Let $x \in \mathbb{R}^V$ is a function. Define $x(S) = \sum_{v \in S} x(v)$ I define a polyhedron in ...
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1answer
417 views

Name of this convex polyhedron?

Does anyone recognize / know the name of the convex polyhedron depicted below as the intersection of a Cuboctahedron and a Rhombicdodecahedron? Please note you have to interpret this picture and ...
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2answers
256 views

Number of distinct nets of dual polyhedra

There are 11 non-congruent nets of a cube as well as 11 distinct nets of an octahedron. Both a dodecahedron and an icosahedron have 43380 distinct nets. Is it true that any pair of dual convex ...
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1answer
272 views

Chebyshev center = center of mass?

I would like to know for which convex polyhedra $P$ in $\mathbb{R}^3$, is the center of the largest sphere enclosed in $P$ (a.k.a. the Chebyshev center, or the incenter) the same as the center of ...
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3answers
218 views

What shape is this?

im doing a question that involves a shape with 8 faces, 10 vertices and 16 edges. Can anyone enlighten me as to what this shape is called? Many Thanks
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2answers
353 views

Which unfolding of an icosahedron has the least number of edges to be glued?

Does every unfolding of an icosahedron has the same number of edges to be glued to construct it back to the solid? If yes, what are those numbers for Platonic solids? If no, which unfoldings have the ...
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3answers
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Why are there 12 pentagons and 20 hexagons on a soccer ball?

Edge-attaching many hexagons results in a plane. Edge-attaching pentagons yields a dodecahedron. Is there some insight into why the alternation of pentagons and hexagons yields an approximated ...
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1answer
368 views

Find the Polyhedron Enclosed by Multiple Faces

I have two faces ($S_1$ and $S_2$), each bounded by a series of Vertex $V$. These two faces may or may not intersect. In addition to that there are two vertical faces $S_3$ and $S_4$, that connect ...
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2answers
2k views

Tetrahedron inside a sphere

What's the largest regular tetrahedron (side length x) you can fit inside a sphere with a radius of one?
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3answers
13k views

Height of a tetrahedron

How do I calculate: The height of a regular tetrahedron, side length 1. Just to be completely clear, by height I mean if you placed the shape on a table, how high up would the highest point be from ...
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1answer
285 views

Tetrahedron volume

How to calculate volume of tetrahedron given lengths of all it's edges?
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1answer
340 views

What property of certain regular polygons allows them to be faces of the Platonic Solids?

It appears to me that only Triangles, Squares, and Pentagons are able to "tessellate" (is that the proper word in this context?) to become regular 3D convex polytopes. What property of those regular ...