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55
votes
5answers
2k views

Does every polyomino tile R^n for some n?

This is a question posed by Adam Chalcraft. I am posting it here because I think it deserves wider circulation, and because maybe someone already knows the answer. A polyomino is usually defined to ...
53
votes
1answer
2k views

How many cubes cover a bigger cube?

How many $n$-dimensional unit cubes are needed to cover a cube with side lengths $1+\epsilon$ for some $\epsilon>0$? For n=1, the answer is obviously two. For n=2, the drawing below ...
45
votes
2answers
2k views

vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct. This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...
32
votes
0answers
459 views

Extending a line-arrangement so that the bounded components of its complement are triangles

Given a finite collection of lines $L_1,\dots,L_m$ in ${\bf{R}}^2$, let $R_1,\dots,R_n$ be the connected components of ${\bf{R}}^2 \setminus (L_1 \cup \dots \cup L_m)$, and say that {$L_1,\dots,L_m$} ...
30
votes
4answers
1k views

Tiling A Rectangle With A Hint of Magic

Here's a a famous problem: If a rectangle $R$ is tiled by rectangles $T_i$ each of which has at least one integer sidelength, then the tiled rectangle $R$ has at least one integer side length. ...
27
votes
3answers
2k views

Is the ratio Perimeter/Area for a finite union of unit squares at most 4?

Update: As I have just learned, this is called Keleti's perimeter area conjecture. Prove that if H is the union of a finite number of unit squares in the plane, then the ratio of the perimeter and ...
26
votes
2answers
853 views

Bodies of constant width?

In two-dimensional case one can generalize figures of constant width as figures which can rotate in a covex polygon. Here is one example which can be used to drill triangular holes: I would like to ...
26
votes
2answers
606 views

Careless packing

The sequence $\frac{1}{2}, \frac{1}{2\cdot 2}, \frac{1}{3\cdot 4}, \frac{1}{4\cdot 6}, \frac{1}{5\cdot 8}, \frac{1}{6\cdot 10},\ldots$ has a curious property, as follows: a) the series with these ...
24
votes
3answers
1k views

Can one recover the smooth Gauss Bonnet theorem from the combinatorial Gauss Bonnet theorem as an appropriate limit?

First let me state two known theorems. Theorem 1 (for smooth manifolds): Let $(M,g)$ be a smooth compact two dimensional Riemannian manifold. Then $$ \int \frac{K}{2 \pi} dA = \chi (M) $$ where $K$ ...
22
votes
3answers
2k views

Covering a circle with red and blue arcs

We have a circle and two families of $n$ red arcs and $n$ blue arcs, positioned on the circle so that every two arcs of different colors intersect. Can one show that there is a point in the perimeter ...
22
votes
2answers
934 views

A geometric Ramsey problem

The following problem seems like one to which the answer could well be known: if so, I'd be interested to have a reference. How large does n have to be such that among any n points in the plane you ...
21
votes
3answers
620 views

Largest possible volume of the convex hull of a curve of unit length

What is the largest possible volume of the convex hull of an open/closed curve of unit length in $\mathbb{R}^3$?
21
votes
1answer
643 views

Sane bound on number of moves for Maker-Breaker game on $\mathbb R^2$ for $\{0,1,2,3,4\}$

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 ...
20
votes
3answers
5k views

Cutting a rectangle into an odd number of congruent pieces

We are interested in tiling a rectangle with copies of single tile (rotations and reflexions are allowed). This is very easy to do, by cutting the rectangle into smaller rectangles. What happens when ...
19
votes
3answers
749 views

Visibility of vertices in polyhedra

Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
19
votes
8answers
1k views

Higher-dimensional Catalan numbers?

One could imagine defining various notions of higher-dimensional Catalan numbers, by generalizing objects they count. For example, because the Catalan numbers count the triangulations of convex ...
19
votes
1answer
454 views

Voronoi cell of lattices with the same profile

Definition 1. Given a body $V$ in $\mathbb R^n$, the function $p_V\colon \mathbb R_+\to \mathbb R_+$ $$p_V(r)=\mathop{\rm vol} [V\cap B_r(0)]$$ will be called profile of $V$. Definition 2. Define ...
17
votes
1answer
520 views

Monomer-Dimer tatami tilings need better relationships with other math. Summary of results.

A monomer-dimer tiling of a rectangular grid with $r$ rows and $c$ columns satisfies the \emph{tatami} condition if no four tiles meet at any point. (or you can think of it as the removal of a ...
15
votes
3answers
978 views

Is Euler characteristic of a simplicial complex upper bounded by a polynomial in the number of its facets ?

What is the best upper bound known on the (absolute value of) the Euler characteristic of a simplicial complex in terms of the number of its facets ? In particular, I am interested in proving or ...
15
votes
2answers
300 views

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 ...
15
votes
4answers
2k views

Why is complex projective space triangulable?

In an exercise in his algebraic topology book, Munkres asserts that $\mathbf{C}P^n$ is triangulable (i.e., there is a simplicial complex $K$ and a homeomorphism $|K| \rightarrow \mathbf{C}P^n$). Can ...
15
votes
1answer
328 views

Does every convex polyhedron have a combinatorially isomorphic counterpart whose all faces have rational areas?

Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas? Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
14
votes
3answers
1k views

Representation of vectors in $\mathbb{R}^2$ via differences of small vectors.

Is the following fact true? Let $v_1,\ldots, v_k \in \mathbb{R}^2$, $\|v_i\|\leq 1$, be vectors that add up to zero. Does there exist a permutation $\sigma\in S_k$ and vectors $w_1,\ldots, w_k ...
13
votes
2answers
1k views

How many vertices/edges/faces at most for a convex polyhedron that tiles space?

I wonder if this problem has already been examined before: Consider a convex polyhedron that tiles $\mathbb R^3$. What is the maximum of vertices/edges/faces that such a polyhedron can have? ...
13
votes
1answer
475 views

Partitioning the vertices of an n-cube with random hyperplane cuts

An evolutionary biologist asked me a question which boils down, at least in part, to what seems to me an interesting question of combinatorial/probabilistic geometry. It is an old chestnut of a ...
13
votes
0answers
1k views

Minimum tiling of a rectangle by squares

Given the $n\times m$ rectangle, I want to compute the minimum number of integer-sided squares needed to tile it (possibly of different sizes). Is there an efficient way to calculate this?
12
votes
2answers
219 views

Smallest containing simplex

Let $V_n$ be the least real number such that for every convex subset of $\mathbb{R}^n$ with hypervolume $1$ there is a containing simplex with hypervolume $V_n$. What is known about $V_n$? Is there a ...
12
votes
3answers
491 views

Is {6,3,7} an 'ultrahyperbolic' Coxeter group?

These pictures, drawn by Roice Nelson, are attempts to visualize a geometry having as symmetries the {6,3,7} Coxeter group, by which I mean the one coming from the Coxeter diagram ...
11
votes
2answers
704 views

Access to a preprint by D. N. Verma

Some work I am doing is connected with a sequence 1, 3, 40, 1225, 67956, $\dots$ which agrees with http://oeis.org/A012250 for all eight terms. The only useful information in OEIS on this sequence is ...
11
votes
3answers
377 views

Is there a nice way to “unravel” a great dodecahedron?

The great dodecahedron is a non-convex regular polyhedron bounded by 12 pentagonal faces, crossing each other, arranged in a star-shaped manner around each of its 12 vertices (see the Wikipedia page ...
11
votes
2answers
283 views

When does a set of collinearity conditions imply collinearity of all of the points?

Suppose we have a set of $n$ points $\{X_1,X_2,\dots,X_n\}$ in the real plane and $\mathcal{A}$ a family of subsets of $\{1,\dots,n\}$. By a "set of collinearity conditions for $\mathcal{A}$" we mean ...
11
votes
3answers
1k views

When are Ehrhart functions of compact convex sets polynomials?

Given a lattice $L$ and a subset $P\subset \mathbb R^d$, we define for each positive integer $t$ $$f_P(L,t)=|(tP\cap L)|$$ the number of lattice points in $tP$. Let's say $P$ is nice if $f_P(L,t)$ is ...
11
votes
2answers
610 views

Drawing 3-configurations of points and lines with straight lines

It is well-known that the black-and-white coloring of the Heawood graph on 14 vertices determines a combinatorial 3-configuration with 7 "points" and 7 "lines", known as Fano plane. Similarly, any ...
11
votes
0answers
375 views

Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories.(In 80s) Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following: a) Trivial (No ...
11
votes
0answers
357 views

Reciprocity (Ehrhart-style) for real polytopes?

Is there some sense in which the well-known Ehrhart reciprocity law for rational, convex, polytopes can be extended to any convex polytope with arbitrary real vertices? In other words, given any ...
10
votes
1answer
311 views

Fano plane drawings: embedding PG(2,2) into the real plane

By a drawing of the Fano plane I mean a system of seven simple curves and seven points in the real plane such that every point lies on exactly three curves, and every curve contains exactly three ...
10
votes
2answers
445 views

Cut and Fold Polyhedron!

I have two convex polyhedra such that their sums of side areas are equal. It is true that I can cut one of them and flatten it on the plane, then fold the flattened polygon to reach the other ...
10
votes
2answers
1k views

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 ...
10
votes
3answers
595 views

Combinatorial analogues of curvature

There appear to be many "combinatorial" definitions of curvature as applied to finite simplicial (or regular CW) complexes. For instance, we have the ideas of Cheeger, Muller and Schrader, of Forman ...
10
votes
1answer
386 views

Random geometric graphs and spanners

I would grateful to learn of work mixing random geometric graphs with random graphs under the Erdős-Renyi model, and in particular concerning spanners. Select $n$ points uniformly at random from the ...
9
votes
3answers
712 views

Polytopes with few vertices.

Suppose I have a convex polytope in $\mathbb{R}^d$ which I know has few vertices (in the case which prompted this question, I seem to have a polytope in $\mathbb{R}^9$ which has sixteen vertices). Is ...
9
votes
3answers
693 views

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 ...
9
votes
1answer
237 views

Which values can attain the minimum solid angle in a simplex

Given a simplex $S$ with a vertex $v$ by the solid angle at this vertex I mean the value $\hbox{vol}(B \cap S)/\hbox{vol}(B)$ where $B$ is a small enough ball centered at $v$ (for example, in the ...
9
votes
2answers
329 views

On well separated point sets in the plane

Let us say that a finite set $A$ in the plane is $1$-separated if: 1) it has an even number of points; 2) no open ball of diameter $1$ contains more than $|A|/2$ points. For a $1$-separated set $A$ ...
9
votes
2answers
285 views

The area of the intersection of convex sets with prescribed pairwise intersections

Consider two numbers $a>b>0$. Let $A_1,A_2,A_3$ be three convex sets in ${\mathbb R}^2$ such that $\mu(A_i)=a$, $\mu(A_i\cap A_j)=b$ ($\mu$ is the usual measure on ${\mathbb R}^2$). What is the ...
9
votes
1answer
265 views

Maximum number of Vertices of Hypercube covered by Ball of radius R

Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ...
8
votes
7answers
761 views

Omniscient bots gathering on $\mathbb{Z}^2$

There are $N=n^2$ "bots" on distinct integer lattice points in the plane. Each knows the positions $p_i$ of all bots, and each has unlimited (private) memory. Each executes the same algorithm ...
8
votes
3answers
915 views

Which finite groups are generated by n involutions?

One of the interesting problems in abstract polytope theory is to determine, for a given finite group, when that group is the automorphism group of a regular abstract polytope. This is equivalent to ...
8
votes
2answers
551 views

A Problem about partitioning $S^2$

Question: Can the 2-dimensional sphere $S^2$ be partitioned into four nonempty sets such that every circle in $S^2$ passes through just three of these four sets? Here, "just three" means "exactly ...
8
votes
2answers
657 views

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 ...