Articles of solid geometry

Is an unit-cube polyhedron? What about other platonic solids?

Definitions According to my linear programming course and screenshot here (Finnish), a polyhedron is such that it can be constrained by a finite amount of inequalities such that $$P=\{\bar x\in \mathbb R^n | A \bar x\geq \bar b\}, A\in \mathbb R^{m\times n},\bar b\in \mathbb R^m$$ and a convex function $f(x)$ must satisfy $$f(\lambda \bar x+(1-\lambda)\bar […]

Number of unique cubes with one red cube in every $1*1*4$ segment

This is an offshoot of this question. A $4*4*4$ cube must have exactly one red cube in every $1*1*4$ segment of the cube. By “segment” I mean any row, column or depth. There will thus be $16$ red cubes in total. How many unique cubes are there which have this property? A cube with this […]

What is circumradius $R$ of the great disnub dirhombidodecahedron, or Skilling's figure?

The vertices of a uniform polyhedron all lie on a sphere. Out of curiosity, I looked at the circumradius $R$ of the $75$ polyhedra (non-prism) in the list (which assumed side $a=1$). For irrational $R$, almost all were roots of quadratics, quartics, and a few sextics that can factor over a square root: $\sqrt{2},\sqrt{3},\sqrt{5},\sqrt{7},\sqrt{11}$ (and […]

Ellipsoid but not quite

I have an ellipsoid centered at the origin. https://en.wikipedia.org/wiki/Ellipsoid Assume $a,b,c$ are expressed in $mm$. Say I want to cover it with a uniform coat/layer which is $d$ mm thick (uniformly). I just realized that in the general case, the new body/solid is not an ellipsoid. I wonder: How can I calculate the volume of […]

Volume of a pyramid, using an integral

I have a calculus exam tomorrow and this is a possible question. However, I don’t know how to handle this question. Suppose you have 3 points in space: $p_1=(a,0,0)$, $p_2=(0,b,0)$ and $p_3=(0,0,c)$, $a,b,c \gt 0$. If we connect these points we get a pyramid in the first octant, with the origin as a top. (i) […]

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 […]

Any other Caltrops?

This question has been edited. The regular tetrahedron is a caltrop. When it lands on a face, one vertex points straight up, ready to jab the foot of anyone stepping on it. Define a caltrop as a polyhedron with the same number of vertices and faces such that each vertex is at distance 1 from […]

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

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 Freudenthal and van der Waerden in 1947. Unfortunately, the paper is in a rather obscure journal , and also is written in […]

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 it should exist, because the space of possibilities is so large and so weakly constrained. Perhaps […]