# Questions tagged [discrete-geometry]

Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

1,482 questions
Filter by
Sorted by
Tagged with
40 views

### The closest ellipse and circle to a given triangle - 2

We add a little more to The closest ellipse to a given triangle. The above linked discussion used the Hausdorff distance to quantify how close two planar convex regions are. In an earlier post - ...
• 4,319
23 views

• 55
1 vote
34 views

### Bisectors and partitioning lines for convex regions defined with respect to the moment of inertia

Ref: Mathematical Omnibus by Fuchs and Tabachnikov, Lecture 11. Consider any planar convex region C. A line l may be called an inertia bisector of C if it divides C into 2 pieces each of which has ...
• 4,319
210 views

### Regular $n$-gon with diagonals: bounds on area of largest cell?

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet). I've been trying to find, in terms of $n$, bounds on the area of the largest cell, ...
• 1,143
184 views

### How much of an aperiodic tiling is needed to force aperiodicity?

Consider an aperiodic tiling. By definition, there is a $C$ such that, for any box of side $C$, the part of the tiling contained in the box can be continued to the whole plane only in a non-periodic ...
• 18.1k
215 views

### How many distances are required to calculate all distances among $n$ points in the Euclidean plane?

I want to know all the pairwise distances between points $P_1,P_2,\ldots,P_n$ in the Euclidean plane (or equivalently, I want to reconstruct the set of points up to congruence). Let's say I have an ...
• 513
523 views

### How to characterize the regularity of a polygon?

In my research, I've recently started to play with Voronoi tessellations. I currently have a Python code that creates the tessellation and I am trying to color the polygonal regions according to their ...
• 141
56 views

### Number of tiles inside a region of a hyperbolic tiling

Let $\mathbb{D}$ be the Poincare disc model of hyperbolic geometry with $\{p,q\}$ tiling on it. In this, paper authors calculated the number of tiles on any circular region centered at a vertex or ...
• 169
68 views

Let $T\neq \emptyset$ be a finite subset of $\mathbb{Z}\times\mathbb{Z}$. We say that $\mathbb{Z}^2 = \mathbb{Z}\times\mathbb{Z}$ is parquettable by $T$ if there is a partition $\frak P$ of $\mathbb{Z}... 1 vote 1 answer 118 views ### What properties are preserved by quasi-isometries Recently, I came across the notion of quasi-isometries, while thinking of "discrete spaces which are surrogates for approximate continuous ones". What (metric)/geometric properties are ... • 4,831 2 votes 0 answers 106 views ### Can a laser hit all the mirrors out of order? For this question, a "cycle" is a sequence of distinct points$X = (x_1,x_2,\cdots,x_k)\in\mathbb{R}^3$which defines a piecewise linear path starting at$x_1$and visiting the points in ... • 317 4 votes 0 answers 134 views ### On moments of inertia of planar and 3D convex bodies The following observation can be readily proved using the perpendicular axes theorem and intermediate value theorem: "Given any planar figure C, through any point on it, there is at least one ... • 4,319 4 votes 2 answers 224 views ### Discrete isoperimetric problems It is well-known that among all planar curves, the circle — invariant under$O(2)$— has the best isoperimetric ratio. Similarly, among all$n$-gons, the regular$n$-gon — invariant under the dihedral ... • 479 1 vote 1 answer 48 views ### Relation of MSTs in the Euclidean plane to Delaunay triangulations It is known that the Minimum Spanning Tree (MST) of a finite set of points in the Euclidean plane is contained in the point set's Delaunay triangulation, but is that all that can be said about their ... • 11.7k 5 votes 1 answer 172 views ### Which pyramids fill space? Let us define a pyramid as a convex polyhedron with one quadrilateral face and four triangular faces. Question: How many pyramids (or families of pyramids) are known that can fill 3D space without ... • 4,319 15 votes 0 answers 208 views ### Is the dodecahedron flexible (as a polytope with fixed edge-lengths)? Consider the (regular) dodecahedron$D\subset\Bbb R^3$. I want to continuously deform it so that throughout the deformation it stays a convex polytope, it stays a combinatorial dodecahedron (i.e. its ... • 11.2k 6 votes 0 answers 73 views ### Tiling the plane with mutually non-congruent equal area rectangles Question: Is it possible to tile the plane with mutually non-congruent rectangles all of equal area? Note 1: If the answer is "yes" then, there could be constrained versions of the question ... • 4,319 1 vote 0 answers 74 views ### Lattice packing Let$\Lambda$be a lattice in$R^n$and$R>0$a real number. Consider the number$N$of points in$\Lambda$of norm less than$R$. Let$R$goes to infinity. What can be said about the asymptotic ... • 229 3 votes 0 answers 181 views ### Approximating any$d$-dimensional convex shape that occupies a constant fraction of its bounding box with a polytope having$\mathrm{poly}(d)$facets Given any convex set$A\in\mathbb{R}^d$, we denote by$V(A)$its$d$-volume. Furthermore, given any two convex sets$A_1,A_2\in\mathbb{R}^d$, we denote by$V_{A_1,A_2}$the$d$-volume of the symmetric ... • 1,539 3 votes 1 answer 98 views ### Distance of average of points to center of minimum enclosing ball Let$v_1, ..., v_n$be distinct points in$\{0,1\}^d$with the same norm$\|v_i\|_2=k$(i.e each$v_i$has$k$ones). Let$A=\frac{1}{n}\sum_{i=1}^n v_i$be their average, and let$C$be the center of ... • 432 7 votes 0 answers 141 views ### Approximating any convex shape in$\mathbb{R}^d$with a polytope having$\mathrm{poly}(d)$facets We denote by$V(A)$the$d$-volume of any convex set$A$. Furthermore, given any two convex sets$A,B\in\mathbb{R}^d$, we denote by$V_{A,B}$the$d$-volume of the symmetric difference$V\left(A \...
• 1,539
Given the area $A$ of ​​a strictly convex polygon with $n$ vertices with integer Cartesian coordinates, there are usually several non-equivalent polygons. The relationship between the area, the number ...