Tagged Questions

1
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0answers
21 views

Convex cone as sum of simplices?

In 3D a pyramid with a square base can be decomposed into the sum of two tetrahedra, i.e. two 3-simplices. I am dealing with a homogeneous N-dimensional system of inequalities and my solution is a ...
3
votes
1answer
58 views

Maximal volume for given surface area of an $n$-hedron

Is there a term for a polyhedron with $n$ faces (or, similarly, $n$ vertices) that maximises the enclosed volume for a given surface area (equivalently, minimises the surface area for a given volume)? ...
0
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0answers
31 views

Distance to a convex polyhedron: about different approaches

I know there are a lot of litterature out there about convex polyhedra and distance computation, but I don't quite catch which one has the best computational complexity in practice and in theory. I ...
2
votes
1answer
45 views

Which polyhedra have an even number of faces touching each vertex?

A two-coloring of the faces of a polyhedron is always possible when an even number of faces meet at each vertex. http://www.georgehart.com/virtual-polyhedra/colorings.html Is there a name for ...
0
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0answers
29 views

Maximum Number of Divisions in Octahedron into congruent parts?

I am trying to divide octahedron into congruent parts. I found octahedron inside tetrahedron sided by four smaller tetrahedrons. I found some division here to 12 congruent parts. I can divide ...
2
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1answer
54 views

Is there a forumla to find out if n faces can be made into a 'regular polyhedron'?

I'm not too sure about the exact terminology since Wikipedia is throwing me all over the place. I'm looking for a formula to find out if for n faces a 'regular polyhedron' can exist. In case that's ...
4
votes
1answer
56 views

Are the face–centroid pyramids of a convex congruent-faced polyhedron congruent?

Let a convex polyhedron $P$ be given, all of whose faces are congruent. Consider any pyramid formed by a face of $P$ as its base and the centroid of $P$ as its vertex. Allowing congruence to admit ...
8
votes
2answers
163 views

insphere/circumsphere ratio of a polyhedron the same as its dual polyhedron?

Is the $r/R$ ratio for any polyhedron always the same as the $r/R$ ratio of the dual of that polyhedron? Given any polyhedron, we can find the biggest sphere that fits inside it (its insphere) and ...
1
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0answers
28 views

Geometric Interpretation of $h_1(P)=f_{d-1}(P)-d$ for a polytope

In our lecture "Discrete Geometry 1", we are examining lineare realtions between the components of the f-vector and the h-vector of a polytope, in particular the Euler-Poincaré formula and the ...
12
votes
1answer
286 views

Floret Tessellation of a Sphere

I'm a programmer looking to create a 3D model of a Floret Tessellation of a sphere, like the one in this picture Class III 8,11 floret planar net (source) If anyone could point me in the right ...
2
votes
1answer
90 views

Altitude of tetrahedron?

I'm really curious to know any relationships between the altitude of a tetrahedron and how the foot of this altitude splits the base triangle. For example if you have a tetrahedron PABC with apex P, ...
2
votes
1answer
86 views

Unfolding Polyhedra

I'm interested in learning more on unfolding polyhedra. Are there any known algorithms that unfold polyhedra into nets? I'm interested in writing code on this in either MATLAB, Python, or C#. On ...
0
votes
1answer
54 views

Classification of, Information Regarding Particular Family of Polyhedra:

Let $n\ge4$. Let $n$ vertices be distributed on a spherical boundary. Let the vertices lie on this boundary as would electrons on a spherical boundary. That is, they are distributed "equally" by ...
0
votes
1answer
86 views

Relationship between angles in tetrahedron

Let's say I have a tetrahedron like this in image: Do angles $CAD$ and $CBD$ equals in general tetrahedron?
11
votes
1answer
215 views

What hexahedra have faces with areas of exactly 1, 2, 3, 4, 5, and 6 units?

I tried for a while, not very hard, to construct a polyhedron with exactly six faces, whose areas were respectively 1, 2, 3, 4, 5, and 6 units. I did not meet with any success. Still, it seems that ...
1
vote
1answer
50 views

Surface of a Ideal Tetrahedron in Hyperbolic Space H3

The hyperbolic space $\mathbb H3$, has a boundary $\mathbb CP1$. A ideal tetrahedron in $\mathbb H3$, is a tetrahedron, where the four vertices are on the boundary $\mathbb CP1$. The four vertices ...
1
vote
1answer
111 views

Graph Isomorphisms, Delaunay Triangulation on a sphere, and Kulikowski's Theorem

Suppose I have a collection of $n$ non-collinear points on a sphere, $\left\lbrace P_i\right\rbrace_{i=1}^n $. And I construct a mapping from this collection of points to the Delaunay Triangulation ...
1
vote
2answers
490 views

Given a tetrahedron, how to find the outward surface normals for each side?

Say I have a triangle in $3$D space. I can get the surface normal by calculating the vector cross product of two of the edges. But, lets say I make this a tetrahedron. How can I work out the outward ...
2
votes
1answer
66 views

Is $\mathbb R^7$ minimally sufficient for embedding 3 tetrahedra - ABCD, ABEF, and CEGH - of equal edge length?

I've got 8 points - A, B, C, D, E, F, G, H - and I need three specific sets of four (ABCD, ABEF, and CEGH) to describe tetrahedra of equal edge length in some multidimensional space. I can embed ABCD ...
1
vote
1answer
155 views

Regular polygons that touching to a sphere surface

What is the possible number of n sided polygons(every face is the same regular polygon) that touching their corners to sphere surface and also touching each other ? I would like to know the relation ...
3
votes
1answer
289 views

Conventional ordering of faces of regular polyhedron?

e.g. For an icosahedron defined as follows: Diagram: A regular icosahedron (courtesy of Microsoft Visio): We define position and orientation w.r.t. this body's frame of reference as follows: ...
8
votes
3answers
218 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 ...
1
vote
1answer
99 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
votes
1answer
333 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
votes
2answers
513 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 ...
3
votes
0answers
58 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
votes
0answers
147 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 ...
8
votes
2answers
203 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
317 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 ...
2
votes
1answer
121 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
101 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, ...
1
vote
2answers
214 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
62 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 ...
1
vote
3answers
73 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 ...
1
vote
1answer
88 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
109 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.
0
votes
2answers
256 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?
5
votes
3answers
151 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?
1
vote
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
votes
0answers
111 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
290 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
2
votes
1answer
263 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
135 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 ...
4
votes
1answer
116 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 ...
2
votes
2answers
396 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 ...
5
votes
3answers
1k 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.
3
votes
0answers
302 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
356 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 ...
4
votes
2answers
170 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 ...
7
votes
1answer
320 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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