Questions tagged [platonic-solids]
A Platonic solid is a regular, convex polyhedron with congruent faces of regular polygons and the same number of faces meeting at each vertex.
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Probability of rolling a specified face for Archimedean solids
Here it is shown that (for a "suitable" mathematical definition of fairness) there are no fair $n$-sides die with odd $n$. The question originated with fairness being defined as, among other,...
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A convention for the names of the nets of the regular octahedron?
Below is a representation of the 11 nets of the octahedron.
One of them (middle row, second column) is often given the name 'Butterfly' thanks to Cahill's map projection, maybe.
It seems trivial, but ...
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Relationship between vertices and edges in platonic solids
The Question:
Given a list of the vertices in a platonic solid, is there a way to calculate which vertices are connected by an edge? I know that one could find edges using edge length or rotations but ...
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Vertices and faces of polyhedra via operators
Context
I was making a table in which, for each polyhedron, the list of its points in Cartesian coordinates could be expressed in a compact manner.
I take the data from this site.
Example with the ...
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Do these four properties imply a polyhedron is a regular icosahedron?
Suppose we have a 3D polyhedron with the following properties:
Each vertex has exactly 5 edges coming out.
Each face has exactly 3 edges.
No self-intersections.
Each vertex is locally congruent in ...
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Why is the tetrahedron special?
I've wondered why the tetrahedron is special : It is the only Platonic Solid in 3 dimensions where faces are opposite to vertices; the other Platonic Solids have face-face and vertex-vertex opposite ...
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How Much To "Twist" A Polygon To Match A Face On A Regular Polyhedron
Suppose I want to create an icosahedron by building a set of twenty triangular pyramids (aka tetrahedrons, but see below) of an appropriate size and then rotating each into position. I need to do ...
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What is analogous to the set of four main diagonals in Rotational Symmetry of Cube, if we want to find the total symmetry of cube?
I am working on a project on applications of group theory, starting from point groups and molecular symmetry.
Coming up on representations of each point groups, I started studying Representations of ...
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Rational vertices after affine transformations of polyhedra
Given a polyhedron $P$ with vertices $V_i[x_i,y_i,z_i]$, how can we determine if there exist an affine transformation which transforms the polyhedron into $P'$ with vertices $V'_i[x'_i,y'_i,z'_i]$, ...
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Is my derivation of the tetrahedral bond angle correct?
Let $T$ be a regular tetrahedron with edge length $x$.
Let $A$ be one of the faces of $T$.
Let $P$ be the plane containing $A$.
Let $L$ be the line segment from the center of $A$ to one of the ...
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Spin the octahedron while moving it through a hole
Suppose we have an octahedron with side length a, and we need to make it through a square hole. We can spin the octahedron while moving it, then what is the minimum side length of the square hole?
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Proportion of regular tetrahedron occupied by mutually tangent balls centred at its vertices
The centres of four balls of radius $1$ are the vertices of a regular tetrahedron of side length $2$. What is the proportion of the tetrahedron occupied by the balls?
At first I thought it should just ...
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Coloring the faces of the regular icosahedron, again...
The calculation of the number of ways to color the faces of the regular icosehedron by 2 different colors is given in this link: Coloring the faces of a regular icosahedron with $2$ colors
My question ...
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The barycenter of a regular tetrahedron coincides with the center of its circumsphere
This is a self-answered question, after some playing around. I would be happy to see alternative solutions.
Let $x_1,x_2,x_3,x_4 \in \mathbb{R}^3$ be
the vertices of a regular tetrahedron, i.e. $|x_i-...
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The vertices of a tetrahedron lie on a sphere
I am struggling a bit with the following (elementary) question:
How to prove that every regular tetrahedron admits a circumsphere, i.e. there exist a sphere on which all four vertices lie.
I would ...
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Common region between an icosahedron and a dodecahedron
This is admittedly one of the hard problems I've come across. It involves the common region (intersection) between two dual platonic solids: icosahedron, and dodecahedron.
The question is as follows:
...
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Volume of objects like hypercube / hypersphere : $V_{n}^{(m)}(r) = \dots$
I am looking for some general form of equation for calculating volume for specific geometry objects.
The main idea is to find :
$$
V_{n}^{(m)}(r) = \dots
$$
Where:
$V$ - volume of object
$n$ - ...
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Given the vertices of a solid face, compute the distance of a point from the face
I am writing a program where I need to compute the ordinary distance of a point from a face of a solid (imagine, for instance, a point inside a cube).
I have all the vertices of the face (x,y,z) [or (...
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Are there any other surfaces of constant width constructed from Platonic solids?
Watching videos about Reuleaux polygons, it is neat to see that the triangle is not the only polygon that creates a shape of constant width. Actually, a shape of constant width can be created from any ...
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Artist needing to determine geometric angle for sculpture based on platonic solid
Dear Mathematicians I need your help for a new sculpture!
I will attach images but first imagine 2 hexagons - where one is rotated 30 deg. They are separated by 12 equilateral triangles. I need to ...
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Number of distinct configurations of equiangular lines passing through origin
We have a $d$ dimensional space and want to place $n$ lines passing through the origin in such a way that the angles between any two pairs of lines is the same. Such a placement results in a ...
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Is there a neat way to construct the Coxeter Group $H_4$ from that of $H_3$ using the fact that $|H_4| = 14400 = 120^2 =|H_3|^2$?
Let $H_4$ and $H_3$ be the usual finite irreducible Coxeter Groups of their names: as presentations $H_4 = \langle s_1,s_2,s_3,s_4\rangle$ subject to the relations $$s_i^2 = (s_is_j)^2 = (s_1s_2)^5 = (...
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Space diagonals of a dodecahedron
I have been studying on platonic solids for a while and figuring out properties of dodecahedron. A dodecahedron with sidelength $a$ has $60$ surface diagonals and $100$ space diagonals, $10$ being ...
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Why doesn't an inscribed cube perfectly sample the surface of a sphere?
I was curious about ways to sample perfectly dispersed points on the surface of the sphere. This question had some interesting info: Is the Fibonacci lattice the very best way to evenly distribute N ...
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Isometry of cube in $\mathbb R^4$?
Find the Symmetry Group of : Tetrahedron and Cube.
I know that there is a duplicate question Symmetry group of Tetrahedron
but my professor suggested us to write coordinates of both cube and ...
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