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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 ...
2
votes
1answer
311 views

When are nontopological bistellar flips manifold-preserving?

A topological bistellar flip is the term used by Dougherty, Faber, and Murphy to describe a bistellar flip that does not cause any face of a complex to be duplicated. Suppose we consider a ...
2
votes
1answer
534 views

$n$ lines in general position; there are $n-2$ small triangles

Suppose we have $n$ lines in general position in the plane. Prove that there are at least $n-2$ ''small'' triangles. Here a "small" triangle is a triangle that is not contained in any larger triangle. ...
8
votes
2answers
660 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 ...
2
votes
2answers
531 views

Counting integral points of a polytope in R^3 (the c_1 coefficient of Ehrhart polynomial)

(Sorry I'm outsider in this field.) I need to count the number of integral points in a convex polytope in $\mathbf{R}^3$. The cones in the dual fan are not necessarily regular (does it create any ...
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?
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 ...
6
votes
2answers
520 views

Presentation of the cohomology of generalized flag varieties as graded ranks of rings of symmetric polynomials

Hello! Let $n,m\geq 0$ be integers. If I understand it correctly, there is the following description of the cohomology of the complex Grassmannian $\text{Gr}(m+n;m)$: denote by $\text{Sym}(n,m)$ the ...
2
votes
1answer
183 views

Constructing affine hypersurfaces with one singularity

This is a followup to my previous poorly-worded question. Consider a finite collection of points $S \subset \mathbb N^n$ lying in a hyperplane $H$. These points define exponents of a collection of ...
4
votes
1answer
388 views

Secondary fans and Stanley Reisner ideals

Consider a collection of points $S \subset \mathbb R^d$. I would like to understand all possible fans $\Sigma$ whose support is the cone over $S$: $|\Sigma| = cone(S)$. I have heard that the ...
1
vote
0answers
232 views

Any 2-coloring of the scope of equilateral triangle must create a monochromatic rectangular triangle ?

Hello All, I'm trying to prove something that we know that is correct but can't find how to do it or any example to show how to solve it. Any 2-coloring of the scope of equilateral triangle must ...
5
votes
2answers
333 views

Up to projectivities, which configurations of four lines in $\mathbb{P}^3$ can one distinguish?

Background I am interested in the projective classification of reduced curves of degree four in $\mathbb{P}^3(\mathbb{R})$ (and more generally of degree $n+1$ in $\mathbb{P}^n(\mathbb{R})$). More ...
3
votes
1answer
202 views

Can any vertex remain when removing halfspaces from a projectively transformed polytope?

Let P be a simple polytope defined as an intersection of n halfspaces. A facet F of P, supported by halfspace H, is removable if the intersection of the remaining (n-1) halfspaces is bounded. F is ...
2
votes
1answer
372 views

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 ...
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 ...
10
votes
1answer
387 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 ...
4
votes
4answers
525 views

Upper bound for the number of subsets of N points which exhaust their convex hull

Hello. Looking at a set S of N points in the plane, I call a subset B of S "legal" if the set of points contained in the convex hull of B is exactly B itself. In other words, a subset B of S in legal ...
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 ...
8
votes
3answers
918 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 ...
4
votes
2answers
258 views

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, ...
4
votes
2answers
285 views

Characterization of combinatorial manifolds in terms of links

I need to reference the following result. Do you know a good source? The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent: a) $S$ is an $n$ manifold; b) The link of ...
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 ...
4
votes
1answer
705 views

The Join of Simplicial Sets ~Finale~

Background Let $X$ and $S$ be simplicial sets, i.e. presheaves on $\Delta$, the so-called topologist's simplex category, which is the category of finite nonempty ordinals with morphisms given by ...
2
votes
1answer
190 views

Are combinatorial configurations whose Levi graphs may be represented as covering graphs over voltage graphs realizable with pseudolines?

This question is related to this previous question. Many combinatorial configurations have Levi graphs which may be represented as derived graphs obtained from voltage graphs over a cyclic group; in a ...
2
votes
0answers
258 views

Drawing a combinatorial 3-configuration of points and lines with pseudolines

This question is related to the question of drawing a combinatorial 3-configuration of points and lines with straight lines. We only relax the condition and admit drawings with pseudolines. Let us ...
11
votes
2answers
611 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 ...
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 ...
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. ...
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 ...
3
votes
3answers
267 views

Are there infinite sets of stellations of polyhedra?

Lists of stellations of polyhedrons are given particular rules like in the book The Fifty Nine Icosahedra which follows "Miller's Rules". There seems to be no "correct" ruleset to use, so more ...