Questions related to polyhedra and their properties.

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8
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3answers
341 views

Three-dimensional art galleries

The well-known art gallery problem starts with an "art gallery" (a simple polygon in the plane, not necessarily convex) and asks for the minimum number of "guards" (points on the polygon) required to ...
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1answer
141 views

Question about Euler's polyhedral formula in a proof of minimum distances

I am confused by a step made in a proof of the following result. Let $f_{2}^{\text{min}}(n)$ denote the maximum number of times the minimum distance can occur among n points in the plane. Then ...
-1
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1answer
890 views

Icosahedron coordinates

Wikipedia says (link)that cartesian coordinates of icosahedron are: (0, ±1, ± φ) (±1, ± φ, 0) (± φ, 0, ±1) Where φ = (1 + √5) / 2 is golden ratio ≈ 1.618. I ...
0
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1answer
444 views

How to calculate vertices of dodecahedron using W|A?

Can I calculate for example with wolfram alpha coordinates of vertices of dodecahedron? I know coordinates of center point of dodecahedron (center of gravity) and the height of dodecahedron. Thanks ...
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2answers
1k views

What are the vertices of a regular tetrahedron embeded in a sphere of radius R

Imagine you had a sphere of radius R centered at the origin. What are the coordinates of the vertices of the regular tetrahedron which is circumscribed by the sphere? One of the vertices of the ...
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0answers
122 views

Grothendieck's schematic point of view of regular polyhedra

I am asking this question out of curiousity. Wikipedia lists Grothendieck's main mathematical achievements the last of which is "schematic point of view, or "arithmetics" for regular polyhedra and ...
3
votes
0answers
107 views

Represent numbers on clock by polyhedrons

The "1" is replaced with a four-sided object, then the next one could be a five sided object, then six (the cube), but then after that, it is either a five-sided pyramid, or a eight-sided die. ...
2
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0answers
175 views

The structure theorem of Tropical geometry

The Structure Theorem of Tropical geometry states that "Let $X$ be an irreducible $d$-dimensional subvariety of $\mathbb T^n$ . Then $\operatorname{trop}(X)$ is the support of a balanced weighted ...
2
votes
2answers
377 views

On gluing regular tetrahedra together to form a ring

The above are part of the articles and some background of it. And there is one claim in the articles, saying that " in order to show that no trivial ring can be formed, it is sufficient to show ...
5
votes
3answers
454 views

Platonic Solids

It´s a theorem that there exist only five platonic solids ( up to similarity). I was searching some proofs of this, but I could not. I want to see some proof of this, specially one that uses ...
0
votes
1answer
235 views

The difference between polyhedral complex and support of a polyhedral complex?

A polyhedral complex is a collection of polyhedra such that intersection of any two polyhedron is a face of of both the polyhedron or empty. Support of a polyhedral complex is the set of all points ...
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votes
4answers
1k views

How to cut a cube into an icosahedron?

Edit: Originally I asked this about a using a cube, but it is not a requirement to start with a cube, just how to end up with an icosahedron as on of the answers showed how to make dodecahedron a ...
11
votes
2answers
324 views

How to prove there are exactly eight convex deltahedra?

A deltahedron is a polyhedron whose faces are equilateral triangles. It is well-known that there are exactly eight convex deltahedra, and it is easy to find out that this was first proved by ...
2
votes
1answer
594 views

Calculating the area of a cross-section of a tetrahedron

Ok, I completely revised my question. For those interested about my purpose with this question, see the older versions. So, I would like to calculate the area of a cross-section of a tetrahedron. The ...
1
vote
1answer
69 views

Determine if two polyhedrals are the same shape and if so, map their vertices

I have a polyhedron and want to determine whether it is combinatorially equivalent to another polyhedron. I know how many faces comprise each polyhedron and for each face, I know all of its vertices, ...
2
votes
1answer
190 views

convex polyhedron edge property

I have a convex polyhedron (with integral nodes). I only calculate in euclidian spaces. Let N be the set of nodes, c the center (arithmetic mean) of the polyhedron. I now want to determine if a line ...
2
votes
1answer
144 views

The set of distances between three points chosen with uniform probability on a finite interval

I pick three numbers $(x_1, x_2, x_3)$, where the value each $x_i$ is a real number selected with uniform probability on the interval $[J, K]$. I then plot a point in three-dimensions where {$x, y, ...
2
votes
0answers
142 views

3-manifold theorem reference request or proof

The following is a theorem of which I have great interest in but cannot find anything about on the internet, Every 3-manifold of finite volume comes from identifying sides of some polyhedron I'm ...
2
votes
2answers
106 views

Polyhedra from Cayley Graphs

I was playing around with the Cayley graphs for some simple groups today and stumbled across something interesting, but can't quite figure out if there's something deeper going on. Here's what I did: ...
3
votes
1answer
202 views

What's the correct name for a geometric solid that's a beanbag?

enter image description hereMy daughter is in the first grade, and I'm having a good deal of fun trying to determine the shapes of irregular geometric solids. I'm stuck on the good, old beanbag. ...
1
vote
2answers
353 views

Find points of a regular tetrahedron

I'm given one of the vertices of a regular tetrahedron and the radius of the circumsphere. I also know the center point of the circumsphere. How can I find the remaining three vertices? (It was ...
0
votes
1answer
80 views

Icosahedral group. Need the angles of all diagonals.

I want to implement the icosahedral pointgroup. For that I need all angles of the lines between two opposite vertices, between the midpoints of two opposite faces and between the midpoints of two ...
5
votes
0answers
92 views

volume of polyhedra which correspond to icosahedral fullerenes

There is, I believe, a sequence of polyhedra whose shape approaches that of the icosahedron (they all have twelve pentagonal faces and the rest hexagons), and starts: regular dodecahedron (C$_{12}$, ...
2
votes
3answers
103 views

Finding the number of edges that connect to a single vertex in a dodecahedron

Please note my geometry background is very weak (high school geometry is all I have), so I would appreciate it if someone could explain it in very layman terms how to do this. I am trying to solve ...
2
votes
1answer
106 views

Determining if two tetrahedra in $\mathbb R^3$ are identical or have reflection symmetry

I have two tetrahedra in $\mathbb R^3$, $T_1$ and $T_2$, and access to the coordinates of their vertices. $T_1$ and $T_2$ are tetrahedra in the sense that they each have four vertices, each vertex is ...
2
votes
1answer
137 views

Convex polyhedron with five, six, or seven vertices at distinct corners of a cube

What are the names of the convex polyhedron with five, six, or seven vertices, where all vertices lie at distinct corners of a cube? I'm particularly interested in the five vertex case.
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1answer
715 views

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 ...
0
votes
2answers
363 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?
6
votes
3answers
177 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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0answers
45 views

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 ...
0
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0answers
130 views

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 ...
3
votes
1answer
441 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
9
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1answer
178 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 ...
2
votes
1answer
656 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 ...
2
votes
1answer
191 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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vote
1answer
179 views

Are there formulae to determine close-packing polyhedra?

Is there a formula to determine which polyhedra will tessellate in 3D without any spaces?
8
votes
4answers
8k 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 ...
4
votes
1answer
148 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 ...
3
votes
3answers
131 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 ...
3
votes
2answers
533 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 ...
6
votes
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.
5
votes
1answer
434 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 ...
11
votes
2answers
476 views

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 ...
6
votes
2answers
238 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 ...
0
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1answer
90 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 $c$ = ...
0
votes
0answers
248 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 ...
7
votes
2answers
468 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 ...
8
votes
2answers
272 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 ...
8
votes
1answer
293 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 ...
3
votes
3answers
220 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