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.
179
questions
59
votes
1
answer
2k
views
Why do all the Platonic Solids exist?
In three dimensions it is quite easy to prove that there exist at most five Platonic Solids. Each has to have at least three polygons meeting at each vertex, and the angles of these polygons have to ...
- 16.3k
52
votes
9
answers
12k
views
How many fair dice exist?
We know a coin is a fair die with a 50-50 probability for two alternatives. Similarly, all five Platonic solids are fair dice. That makes six solids that can be fair dice, but can there be more? One ...
- 6,813
34
votes
4
answers
41k
views
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 sphere?...
- 1,800
22
votes
11
answers
11k
views
Cleverest construction of a dodecahedron / icosahedron?
One can show, as an elementary application of Euler's formula, that there are at most five regular convex polytopes in 3-space. The tetrahedron, cube, and octahedron all admit very intuitive ...
22
votes
5
answers
4k
views
How many faces of a solid can one "see"?
What is the maximum number of faces of totally convex solid that one can "see" from a point?
...and, more importantly, how can I ask this question better? (I'm a college student with little ...
- 333
20
votes
1
answer
1k
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 ...
- 14.5k
13
votes
3
answers
502
views
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 ...
- 17.3k
12
votes
4
answers
3k
views
How many $6$-sided and $8$-sided standard dice exist? [closed]
Standard means that the sum of the opposite sides is $7$ for $6$-sided die and $9$ for $8$-sided die.
- 273
12
votes
5
answers
966
views
How does this proof of the regular dodecahedron's existence fail?
On Tim Gowers' webpage he has an example "proof" of the regular dodecahedron's existence which he claims contains a flaw.
He writes
Of course, I have not written the above proof in a ...
- 16.3k
12
votes
1
answer
734
views
Inscribing Platonic solids in each other: why can't you put a dodecahedron in an octahedron?
Given convex polyhedra $P, Q$, say that one can inscribe $P$ in $Q$ if we can find points on the surface of $Q$ whose convex hull is similar to $P$.
If we restrict $P, Q$ to be Platonic solids, we can ...
- 17.3k
12
votes
1
answer
522
views
Fitting a dodecahedron inside a cube
I'm afraid I'm not fantastic at maths and am struggling with a problem.
I am a woodworker and have been asked to cut a solid dodecahedron from a 3 inch cube of wood. I am struggling to figure out what ...
- 121
11
votes
7
answers
1k
views
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 ...
- 25.1k
11
votes
4
answers
2k
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 ...
- 341
10
votes
2
answers
4k
views
Five Cubes in Dodecahedron
I will demonstrate why the group of rotational symmetries of a Dodecahedron is $A_5$. For that, we have to find five nice objects, on which the group of symmetries acts.
One object is "Cubes" ...
- 10.3k
10
votes
4
answers
8k
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 ...
- 9,675
10
votes
2
answers
209
views
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: ...
- 2,616
10
votes
3
answers
3k
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 ...
- 1,817
9
votes
1
answer
3k
views
Is there a value for $\pi$ that relates to triangles?
So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which ...
9
votes
3
answers
883
views
Making a regular tetrahedron out of concrete
I'm trying to make the following tetrahedron made of concrete just for fun:
Each edge is a beam with a triangular cross section.
I imagine the easiest way is to make 6 identical truncated triangular ...
- 225
9
votes
1
answer
434
views
Polyhedral symmetry in the Riemann sphere
I found an interesting set of constants while exploring the properties of nonconstant functions that are invariant under the symmetry groups of regular polyhedra in the Riemann sphere, endowed with ...
- 5,811
9
votes
4
answers
652
views
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 ...
- 2,441
8
votes
1
answer
1k
views
Tetrahedron packing in Cube
I'm thinking about following solid geometry problem.
Q: Suppose you have a box of "cube" shape with edge length 1.
Then, How many regular tetrahedrons(with edge length 1) can be in the box?
So, this ...
- 531
8
votes
1
answer
631
views
Platonic solids and charged particles
It is known that there are five Platonic solids:
If, lets say, there are 4 particles with the same electricity charge and whose movement is constrained to be on a sphere, resulting forces will ...
- 16k
7
votes
3
answers
758
views
Conceptual reason for why the volume of an ocahedron is four times the volume of a tetrahedron
The image below shows that a regular octahedron can be scaled by a factor of $2$ (resulting in a $2^3$ factor in volume) and decomposed as six octahedra and eight tetrahedra.
If $V_o$ and $V_t$ ...
- 5,052
7
votes
1
answer
381
views
Equations for interior of Platonic solids
It is well-known that for Platonic solids:
The interior of cube a.k.a. hexahedron can be described with inequality $\max\{|x|,|y|,|z|\}<a$.
The interior of octahedron is $|x|+|y|+|z|<a$.
But ...
7
votes
1
answer
208
views
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?
...
- 81
7
votes
2
answers
696
views
Why are polyhedra related to the prime numbers 2, 3 and 5, but not to the prime number 7?
Just take a quick glance at all the numbers in these Wikipedia pages on polyhedra:
http://en.wikipedia.org/wiki/Platonic_solid
http://en.wikipedia.org/wiki/Archimedean_solid
http://en.wikipedia.org/...
- 852
7
votes
1
answer
347
views
How does the existence of Platonic graphs imply the existence of Platonic solids?
I will use the following definitions
Platonic graph: A 3-connected planar graph with faces bounded by the same number of edges and vertices having the same number of incident edges.
(remark: the ...
- 2,033
7
votes
0
answers
353
views
How many platonic solids do exist in non-euclidean space?
The proof that there exists only five platonic solids assumes that the angle between the adjacent sides must be less than 360°, because otherwise the surfaces would be flat or even overlap. However ...
- 2,660
6
votes
3
answers
547
views
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 ...
- 8,178
6
votes
1
answer
1k
views
Which of the $43,380$ possible nets for a dodecahedron is the narrowest?
I want to fit multiple regular dodecahedron nets on to an infinitely long roll of paper. I want this to result in the largest possible dodecahedrons, for a roll of a given width.
My hunch is that the ...
- 163
6
votes
2
answers
2k
views
How to design/shape a polyhedron to be nearly spherically symmetrical, but not a platonic solid?
There are only 5 platonic solids, but take a look at these images:
How are these things designed? How are they shaped? It looks to me like those hexagons are all the same size and shape, and evenly ...
- 1,411
6
votes
1
answer
2k
views
Icosahedron and inscribed cube
We can inscribe a cube in dodecahedron (see this), where $12$ faces of dodecahedron give the $12$ edges of the cube.
Can we inscribe cube in icosahedron?
- 10.3k
6
votes
1
answer
81
views
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 ...
- 116
6
votes
1
answer
201
views
Is the adjective 'regular' necessary in the definition of Platonic solids?
The definition I mean can be found in the tag Wiki of Platonic solid tag and also in Wikipedia:
Definition 1:
A Platonic solid is a regular, convex polyhedron with congruent faces
of regular ...
- 2,033
6
votes
1
answer
596
views
Suggested name for "inflated" tetrahedron
What's the name or class of the following tetrahedron-like shape?
Sketchup Model
WebGL 3D-viewer
It's apparently some sort of (not strictly convex) shell of a tetrahedron and it's scaled spherical ...
- 341
6
votes
0
answers
230
views
About the Platonic Solids in all dimensions
I am asking about the Platonic solids in all dimensions, some reference about the proofs of many of the statements made here.
I would like to hear about how to think about higher dimensions, mainly ...
- 4,651
5
votes
2
answers
611
views
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 ...
- 141
5
votes
2
answers
861
views
A tetrahedron inside another tetrahedron. Could the contained tetrahedron have a greater perimeter then the outside one? [duplicate]
Suppose you have a tetrahedron. It doesn't have to be regular. Now suppose you have another tetrahedron contained inside the first tetrahedron. Again do not assume it is regular and do not assume that ...
- 987
5
votes
2
answers
522
views
Volume of regular octahedron
According to several sources on the internet, the volume of a regular octahedron with unit edge lengths is approximately 0.47. I haven't seen an explanation for why this is so yet. When I tried to ...
- 179
5
votes
3
answers
1k
views
Do non-convex platonic solids exist?
Consider a solid with the following properties -
It is composed of congruent, regular polygons.
At each vertex, the same number of edges and faces meet.
This is the same as the requirement for the ...
- 6,813
5
votes
2
answers
2k
views
How to find the maximum diagonal length inside a dodecahedron?
I am trying to find the maximum length of a diagonal inside a dodecahedron with a side length of $2.319914107\times10^{89}$ meters.
I am not sure if any other information than that is needed, if it ...
- 51
5
votes
3
answers
1k
views
Exactly 5 Platonic solids: Where in the proof do we need convexity and regularity?
The famous statement that
Only five convex regular polyhedra exist.
is usually proven as follows:
Let $P$ be a convex regular polyhedron with
V vertices,
E edges and
F faces.
Moreoever, ...
- 2,029
5
votes
1
answer
849
views
Can platonic solids be constructed using compass and straightedge?
By moving the concept of geometric construction into three dimensions, could one trace the 3D wireframe of any of the five platonic solids using only a compass and straightedge?
If not, what ...
- 617
4
votes
3
answers
806
views
Solids that are platonic apart from faces being irregular polyhedra.
Scouring through Wikipedia, I've found the following analogs to platonic solids that are composed of irregular faces.
Cube = Trigonal Trapezohedron
Dodecahedron = Tetartoid
Tetrahedron = Disphenoid
...
- 6,813
4
votes
4
answers
310
views
Irregular analogue of cube and octahedron.
If we take a Dodecahedron and remove the constraint that the pentagonal faces have to be regular pentagons, we get a solid called a Tetartoid. If we take the dual of that, we will end up with the ...
- 6,813
4
votes
2
answers
94
views
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 ...
4
votes
5
answers
2k
views
Angles subtended by an edge in a regular dodecahedron?
If I have a regular dodecahedron and construct lines between the center of the dodecahedron and its vertices. How do I calculate the angle between such lines, subtended by an edge?
This picture can ...
user601567
4
votes
1
answer
425
views
what abstract regular polyhedra exist?
It is well-known since Plato that there are only 5 regular polyhedra which live in 3D Euclidean space. However abstract polytopes are defined solely by their incidences, and are not confined by the ...
- 16.8k
4
votes
1
answer
89
views
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 ...
- 197