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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Find a pair of diagonals on the hypercube $C(4)$ such that no symmetry exists between them
Question 8.6.6 from Groups, Matrices and Vector Spaces by James Carrell:
The 4-cube has eight diagonals. They are represented by the semidiagonals $\pm \mathbf{e}_1 + \pm \mathbf{e}_2 + \pm \mathbf{e}...
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How to visualize the 6 roto-reflections in the group of symmetries of a tetrahedron $S_4$?
I'm working on an applet that will calculate the product of two symmetries. (It's unfinished but here's a link to the project if you're curious.) I want the applet to show visuals to help the user ...
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How many identical building blocks can make an icosahedrally symmetrical structure?
I'm working on an applied problem that requires designing symmetrical structures. I'd like to form an arrangement with full icosahedral symmetry (point group Ih in 3D) out of building blocks with Cs ...
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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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Proof that there are 4 Kepler-Poinsot solids [duplicate]
Under the extended definition of regularirty there are 9 regular polyhedra, 5 Platonic solids and 4 Kepler-Poinsot solids (where we assume them to have a finite volume to exclude honeycombes and other ...
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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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Do solids retain their symmetry after uniform truncation?
I was looking at platonic solids and Archimedean solids. I observed that the point group of the solids still the same after uniform truncation. Examples of this is that Cube, truncated cube, ...
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A parallelogram with sides $8$ and $6$ and angle of $30^\circ$ between them is rotated around its larger side
A parallelogram with sides $8$ and $6$ and angle of $30^\circ$ between them is rotated around its larger side. Find the volume of the formed solid.
I really can't imagine what we'll get. This is what ...
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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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Minimizing the sum of reciprocal distances of points on a sphere
Consider a sphere of radius $R$, with $N$ points placed on its surface
The straight-line distance between any two points $i$ and $j$ is denoted by $d_{ij}$, where $1 \le i, j \le N$
I'm interested in ...
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Isometry subgroups of a solid that can inscribe another
Consider the (regular) dodecahedron and the cube. The latter can be inscribed in the former.
I am trying to deduce from this that the symmetry group of the cube is a subgroup of the symmetry group of ...
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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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The Homeomorphy of The Tetrahedron, The Cube, and The Octahedron
My question involves all possible dimensions, but this is the lowest dimension where these are qualitatively distinct shapes, so I phrase it in three dimensions for convenience. The analogs for higher ...
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Regular icosahedron with integer vertices
The eight points
$$
(\pm 1, \pm 1, \pm 1)
$$
are the vertices of a cube. The six points
$$
(\pm1, 0,0)\; , \; (0, \pm1, 0)\; , \; (0,0,\pm1)\; , \;
$$
are the vertices of an octahedron. The four ...
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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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Does the 120-cell have 5 inscribed 600-cells?
Another question got me thinking. The $600$-cell has $600$ tetrahedral cells, and $120$ vertices which may be viewed as elements of the binary icosahedral group $2I$, a subset of the unit quaternions $...
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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 segment from a vertex of a regular tetrahedron to the center of its circumsphere is orthogonal to the opposing face
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, which lie ...
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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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Will the dihedral angles of the Platonic solids become rational if one switches to radians?
It seems that it could be possible for the dihedral angles of the platonic solids to be rational if one were to stop using our biased degree units and use the units natural to the platonic realm: ...
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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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Nested Platonic Solids, shortest and longest distances to vertices
Given the five platonic solids nested in the order from innermost to outermost of:
Icosahedron, octahedron, tetrahedron, cube, and dodecahedron --
the problem is to determine the shortest distance and ...
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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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Inscribing a regular tetrahedron in another regular tetrahedron
Given a regular tetrahedron, make a uniformly scaled copy of it, such that both the original and the scaled copy (the scale factor is less than $1$) share the same centroid, and have the same ...
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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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Does this argument show that we do not need to define the Platonic solids as convex?
I'd be very interested in any thoughts on the following argument regarding the necessity of defining Platonic/regular polyhedra as convex. To be specific: are there any obvious flaws in the argument, ...
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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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Number of regular $n$-topes in $\mathbb{R}^n$ for $n\in\mathbb{R}$?
Ever since I've come across it, I have been puzzled by the sequence $(a_n)_{n\in\mathbb{N}_0}=(1,1,\infty,5,6,3,3,3,\cdots)$, describing the number of regular $n$-topes in $n$-dimensions, where $a(k)=...
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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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Why do we get equiangular lines from Platonic solids that have triangles as faces.
Consider placing $n$ lines in $d$ dimensional space in a way that the angles between any two pairs of lines is always the same (and they all pass through the origin). When $d=3$, we get configurations ...
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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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Interpreting the isometries of a tetrahedron
A tetrahedron has $12$ rotational symmetries and $24$ isometries in total. This means that the group of isometries is isomorphic to $S_4.$
If we denote the vertices of a tetrahedron as $1$, $2$, $3$, ...
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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 $$s_i^2 = (s_is_j)^2 = (s_1s_2)^5 = (...
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Only 2 ways to 4-color regular tetrahedron, looking for simple answer
A regular tetrahedron is a triangular pyramid whose faces are all equilateral triangles. How many distinguishable ways can we paint the four faces of a regular tetrahedron with red, blue, green, and ...
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Space diagonals of a dodecahedron
I have been studying on platonic solids for a while and figuring out properties of dodecahedron. A dodecahedron with sidelength $a$ has $60$ surface diagonals and $100$ space diagonals, $10$ being ...
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Platonic solids - topological and geometrical conditions
With V, E, F as the numbers of vertices, edges and faces of a given polyhedron and based on Euler‘s polyhedron formula
$$ V - E + F = 2 $$
it is quite simple to derive a necessary topological ...
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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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Dodecahedron faces are buttons, vertices have counters that track the use of the buttons
As a follow up to this question, I'm trying to teach invariants by creating a game. The idea is to start with a dodecahedron where each of the 20 vertices has a counter on it and each of the 12 faces ...
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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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Algebraic argument for why any $A_5$ in $S_6$ can be extended to an $S_5$ in $S_6$
It is well known that $S_5$ is a subgroup of $S_6$ in a way that acts transitively on $6$ points, a surprising fact related to the outer automorphism of $S_6$. One can see this using the icosahedron: ...
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Does Euclid's demonstration that there are only five Platonic solids need to assume convexity?
At the end of the Elements Book XIII Euclid gives a demonstration that there are only five regular solids, i.e. 'no other figure, besides the said five figures, can be constructed by equilateral and ...
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Symmetric distribution of points on a sphere
Background:
The VSEPR theory provides a useful guide to predicting molecular geometries. One implication of the theory is that the orientation of ligands in three-dimensional space is such that the ...
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How to find the order of the group of rigid motions of platonic solids in $\mathbb{R}^3$?
The following appear as exercises in Dummit and Foote's Algebra (Section $1.2$ - Dihedral Groups):
Let $G$ be the group of rigid motions in $\mathbb{R}^3$ of a tetrahedron. Show that $|G| = 12$
Let $...
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Why does a convex polyhedron being vertex-, edge-, and face-transitive imply that it is a Platonic solid?
Suppose that we have a convex polyhedron $P$, such that the symmetry group of $P$ acts transitively on its vertices, edges, and faces (that is, it is isogonal, isotoxal, and isohedral). It then ...
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Is it possible to tile dodecahedrons in a 3D grid without any spaces between them? [closed]
Is it possible to tile dodecahedrons in a 3D grid without any spaces between them? Feel free to send me web links about that.
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Uniform distribution of points on a sphere: only Platonic solids?
I'm quite sure the only way to uniformly distribute $n$ points on the sphere $S^2$ is by inscribing one of the 5 Platonic solids, thus there only exists a solution for $n=4,6,8,12,20$.
But am I right? ...
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Can an irregular tetrahedron be inscribed in a cuboid or prism?
A regular tetrahedron can be inscribed in a cube in a way that the tetrahedron edges are diagonals of the cube faces.
Is it possible to similarly inscribe any irregular tetrahedron in a cuboid or ...
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Koch-like fractal of the dodecahedron: Does this object already exist?
Platonic solids are made of squares, triangles and one of them is made of pentagons.
If the faces of the cube are taken while preserving its vertices, each face can be divided into smaller squares ...
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How to show that if two Platonic solids have the same number of edges, vertices, and faces, then they are similar in $\mathbb{R}^{3}$?
Note: It appears that some of the terms here do not have standardized definitions, so some sources may give conflicting info.
I was looking into the proof that there are only five Platonic solids in ...
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Formal counting of cube faces colorings wrt full symmetry
How many 3-colorings of cube's faces exist if we consider two colorings the same iff it's possible to rotate and/or mirror the cube such that one coloring goes to another?
I do know a similar question ...
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