Discrete geometry includes the study of covering, illumination, packing, convex bodies, convex polytopes, and other metric geometry.

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51
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4answers
5k views

Why is a circle in a plane surrounded by 6 other circles

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other number? I'm ...
6
votes
2answers
129 views

Squaring the plane, with consecutive integer squares. And a related arrangement

Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would ...
5
votes
3answers
228 views

Number of point subsets that can be covered by a disk

Given $n$ distinct points in the (real) plane, how many distinct non-empty subsets of these points can be covered by some (closed) disk? I conjecture that if no three points are collinear and no four ...
28
votes
1answer
1k views

Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly ...
23
votes
4answers
489 views

square cake with raisins

Alice bakes a square cake, with $n$ raisins (= points). Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins. Note that a single ...
8
votes
4answers
789 views

Every polygon has an interior diagonal

How does one prove that in every polygon (with at least 4 sides, not necessarily convex), that it is possible to draw a segment from one vertex to another that lies entirely inside the polygon. In ...
3
votes
1answer
313 views

Which internal angles can a lattice polygon have?

I am wondering if for a lattice polygon an internal angle can take any value? If no which ones not and why? I guess there will be some restrictions due to the discrete nature of the grid but I am ...
2
votes
1answer
57 views

Embedding finite (discrete) metric spaces to Eulidean space as isometrically as possible

Let $X = \{1, 2, 3, ..., k\}$ with the discrete metric (distance is 1 for every pair of points). How can this be embedded into $\mathbb{R}^n$ (with the usual metric) such that the embedding would be ...
2
votes
0answers
95 views

Twisted tori: discrete and continuous

Taking the advice of Mariano Suárez-Alvarez, I moved this question from MO to MSE: Motivation Let me introduce twisted (discrete) tori: Consider the Cartesian graph product $\mathcal{C}_n = C_n ...
2
votes
1answer
166 views

Brouwer's fixed point theorem implies Sperner's lemma

I'm looking for a proof that Brouwer's fixed point theorem implies Sperner's lemma. The proof should be understandable by an undergraduate. Thanks in advance!
0
votes
1answer
40 views

Number of faces of $n$ congruent disks

If I have $n$ disks, all of the same radius, how many faces (i.e. maximally connected regions) can the induced arrangement have? For example for 3 disks, it could have 7 bounded faces, but what is ...