# Tagged Questions

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### Maximum possible number of similar three colored triangles

I want to maximize the number of similar triangles with vertices from three fixed sets, each vertex from one set. For example if you fix two points $X,Y$, (i.e. two sets with only one member) then ...
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### Integer lattice points on a hypersphere

Is the following statement true? For every integer $n\ge2$ and every integer $k\ge0$ there exists a hypersphere in $\mathbb{R}^n$ (circle, sphere etc) containing exactly $k$ integer lattice ...
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### What properties does generalized Delaunay triangulation have?

Suppose that instead of the usual circle, we pick some other convex set D and make the Delaunay triangulation of a finite planar point set with respect to this set, i.e. connect two points if there is ...
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### What is the smallest number of subsets in such a subdivision?

Given any $30$ points in the plane, what is the smallest number of subsets in a subdivision of the set of $30$ points into subsets such that all the points in each subset are on the boundary of the ...
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### Is this bounded?

May be better to ask for help here. Let $v_{1}$, $v_{2}$, $\ldots$, $v_{m}$ be the vertices of a convex polygon in the plane and $v_{m+1}$ be a vertex in the interior of the convex polygon. Connect ...
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### Realization spaces for regular convex polytopes

Q1. Are there convex polytopes combinatorially equivalent to each of the regular polytopes that are realized with integer vertex coordinates? ...
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### Get Largest Inscribed Rectangle of a Concave Polygon

I'm looking for an algorithm to find a set of largest inscribed rectangles of a concave polygon where each rectangle must be collinear with one of the edges of the polygon. In other words, I want to ...
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### Covering a Polygon with Rectangles

I am tyring to cover a simple concave polygon with a minimum rectangles. My rectangles can be any length, but they have maximum widths, and the polygon will never have an acute angle. I thought about ...
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### Graph implemened into the plane with segments as edges and we search for matching with no edges intersecting

There are some points in the plane and some of them are connected with segments between them. We look at this structure as a graph implemented into the plane where the points are the vertices and the ...
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### When is the conical hull of a finite set of vectors a subset of the space? (and tilings)

Consider a hypercube in n-dimensions, and take some projection down to an m-dimensional subspace. Now take all vertices and m-1 dimensional facets visible from some direction outside the projection. ...
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### Point sets in Euclidean space with a small number of distinct distances

It is well known and not hard to prove that the regular simplex in n-dimensions is the only way to place n+1 points so that the distance between distinct pairs of points is always the same. My general ...
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### Isostatic graphs and the Henneberg conjecture

I have been reading "Combinatorial Rigidity" by Graver, Servatius and Servatius and I am interested in their chapter on rigidity in dimension $\geq$ 3. I have two questions. What is the current ...
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### Not quite regular polyhedra

Take a naive interpretation of regular polyhedra: All vertices (including epsilon ball) congruent All edges congruent All faces congruent We can now find interesting families by removing one ...
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### Subdivision of triangles into congruent triangles

Recently some old notes of mine have gotten me to thinking about the problem of subdividing a triangle into $N$ smaller triangles, all congruent to one another. A little thought shows the following ...
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### Approximate search space on a 5x5x5 cube with 3 different possible classes?

Hey all, I read the meta, and I realize this question might be pretty elementary for this site, but I'm having trouble computing this, and I know it won't take too much insight for someone to give me ...
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### Are Penrose tilings universal? Do aperiodic universal tilings exist?

Consider a tiling of the plane using tiles of at least two types (e.g, a Penrose tiling such as that shown at the bottom of this question, which tiles the plane with two types of tiles). List the tile ...
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### When can a folded polygon be isometrically (locally) embedded into R^3 ?

I am interested in 3-D representations of various things that naturally live in a non-simply-connected compact surface. There is the usual way of producing a compact surface of any orientable or ...
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### Centralizing four red vectors in six green sectors

Four red vectors are given, one per quadrant, $[0,90^\circ)$, $[90^\circ,180^\circ)$, etc. A rigid star of six green vectors separated by $60^\circ$ can be positioned at \$(\theta, \theta+60^\circ, ...