Articles of platonic solids

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 having started ith any shape. I have a cube, and I need to cut/file it […]

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 distributed to approximate a sphere. Same thing with the triangles in the second picture. So how is […]

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 principally group theory. Here´s the definition of Platonic solid Wikipedia Platonic solids

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 the standard chordal metric. The simplest functions I have found posessing these symmetries are rational functions, with zeroes at the vertices of a certain inscribed […]

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” inscribed in Dodecahedron. But since my demonstration will be on “Black-Board” rather than “Slide-Show”. Can anybody suggest me a simple way […]

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 polygons themselves allow them to faces of regular convex polyhedron? Is it something in their angles? Their number of sides? Also, why […]

Bisecting line segments in a tetrahedron.

Suppose that $OABC$ is a regular tetrahedron with base $ABC$. Suppose further that $T$ is the mid-edge of $AC$, $Q$ is the mid-edge of $OB$, $P$ is the mid-edge of $OA$, and $U$ is the mid-edge of $CB$. How can one show that $QT$ bisects $PU$?

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 is I probably have it and will give it, but if you could help that would be great. I am […]

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 experience in asking well formed questions, much less answering them.) By “see” I mean something like this: you point a camera […]

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? Is this special, or are there an arbitrary number of assorted n-gons sets that may be joined together to create regular sphere-like surfaces?