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**53**

votes

**1**answer

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### 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

**1**answer

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

**1**answer

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

**2**answers

658 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

**2**answers

529 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

**0**answers

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

**3**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**3**answers

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

**1**answer

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 ...

**4**

votes

**4**answers

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

**3**answers

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

**3**answers

916 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

**2**answers

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

**2**answers

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

**4**answers

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

**1**answer

703 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

**1**answer

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

**0**answers

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

**2**answers

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

**3**answers

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

**4**answers

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

**3**answers

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### 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

**3**answers

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 ...