# Tag Info

### What is this 4D shape with Radial equilateral symmetry property?

From your very definition of the vertex set, i.e. \mathcal{Q}_d=\{\mathbf{q}=\langle q_1,q_2,...,q_d\rangle : q_i = +1, q_j = -1, q_{\not = i \not = j} = 0, \: \forall \: i,j \in [1,d] \; | \; i \...
• 3,503
1 vote
Accepted

### Question on convex polyhedra - how would I approach this?

The $n$-gonal prism has $n+2$ faces, so that settles the problem for $n\ge5$. The tetrahedron has $4$ faces and it is trivial to show that no polyhedron can have fewer faces. So the polyhedra you want ...
• 92.4k
Accepted

### Are there any other solids like the pseudo-rhombicuboctahedron (Miller's solid)?

Given the definitions of local and global uniformity, the problem has long been solved – at least for the convex case. The Johnson solids are the strictly convex polyhedra with regular polygon faces ...
• 92.4k
1 vote
Accepted

### What is this 4D shape with Radial equilateral symmetry property?

Well, the four vertices (1,-1,0,0,0), (1,0,-1,0,0), (1,0,0,-1,0), (1,0,0,0,-1) are all the same distance apart and hence form a tetrahedron. Your figure has 10 of these; besides, it has some other ...
• 11.9k

### 3D picture of the 38-sided Engel space-filling polyhedron

Now there is a computed 3D model of it with animation, interactive rotation, and space-filling arrangement of a few polyhedrons: https://community.wolfram.com/groups/-/m/t/2617634
1 vote

### How to Prove that hyperplane is facet defining?

The meaning of a "facet-defining" hyperplane of $\mathbb{R}^n$ for a polyhedron of dimension $d$ is not so obvious when $d<n$. Here I assume it is facet-defining iff the intersection of ...
• 2,422
Accepted

### 38-sided space-filling polyhedron

The images of $\left(\frac{427}{6984}, \frac{761}{6984}, \frac{1421}{6984}\right)$ under $I4_132$ indeed generate this polyhedron as their Voronoi cells. I used this description of $I4_132$ and wrote ...
• 7,662
Yes, it is possible, since the two sets of polyhedra both have Dehn invariant $0$; a result of Sydler (1965) shows that any two collections of polyhedra with the same Dehn invariant can be cut into ...