**33**

votes

**0**answers

515 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 {$L_1,\dots,L_m$} ...

**25**

votes

**0**answers

537 views

### 3-colorings of the unit distance graph of $\Bbb R^3$

Let $\Gamma$ be the unit distance graph of $\Bbb R^3$: points $(x,y)$ form an edge if $|x,y|=1$.
Let $(A,B,C,D)$ be a unit side rhombus in the plane, with a transcendental diagonal, e.g. $A = ...

**17**

votes

**0**answers

275 views

### Large almost equilateral sets in finite-dimensional Banach spaces

Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set ...

**15**

votes

**0**answers

1k views

### Covers of $Z^k$

This is a question related to covers of $Z^\infty$. Is it possible to cover $Z^k$, $k>1$, with the $l_1$-metric by a constant (not depending on $k$) number of collections of subsets $U^0,...,U^c$ ...

**14**

votes

**0**answers

300 views

### Knight's tours in higher dimensions

I wonder if Knight's Tours have been explored in higher dimensions,
using the following definition of a knight move.
In dimension $d=2$, the knight moves left/right and forward/back
one step and two ...

**14**

votes

**0**answers

781 views

### Hrushovski's Construction

Zilber expressed a conjecture for $\aleph_{1}$- categorical theories (In the 80s).
Zilber's Conjecture: The geometry of any $\aleph_{1}$- categorical structure is one of the following:
(a) Trivial ...

**13**

votes

**0**answers

2k 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

**0**answers

255 views

### Grothendieck on polyhedra over finite fields

In Grothendieck's Sketch of a Programme he spends a few pages discussing polyhedra over arbitrary rings and concludes with some intriguing remarks on specializing polyhedra over their "most singular ...

**12**

votes

**0**answers

374 views

### Drawings of complete graphs with $Z(n)$ crossings

Hill conjectured that the minimum number of crossings in a drawing of the complete graph $K_n$ in the plane is exactly
$$Z(n) = \frac{1}{4} \bigg\lfloor\frac{n}{2}\bigg\rfloor ...

**12**

votes

**0**answers

437 views

### Does every connected set that is not a line segment cross some dyadic square?

A dyadic square is a subset of $R^2$ of the form $x + 2^{-n} [0,1]^2$ with $x \in 2^{-m} Z^2$, for integers $m,n \geq 0$. We say that a set $A$ crosses a square $S$ if there exists a connected subset ...

**11**

votes

**0**answers

385 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 ...

**11**

votes

**0**answers

285 views

### Update to Shephard's “Twenty Problems on Convex Polyhedra”

Forty-three years ago, Geoffrey Shephard published an influential list of open problems
on convex polyhedra.
Progress has been made on several of his problems, and perhaps some have been completely ...

**10**

votes

**0**answers

118 views

### Maximum height of intersection of triangles

I'd like some advice regarding the following question, which I have been struggling with for long time.
Let's call the shaded region in the below $S_3$. It is the union of three congruent isosceles ...

**10**

votes

**0**answers

134 views

### Self-avoiding random walks that always turn

I am wondering if the statistics of self-avoiding random lattice-walks
on $\mathbb{Z}^2$
that turn left or right at each step (i.e., they cannot continue the
direction of the preceding step) have been ...

**10**

votes

**0**answers

309 views

### A combinatorial approximation functor sSet->qCat

Let $sSet_J$ denote the category of simplicial sets equipped with the Joyal model structure. Simply by the fact that $sSet_J$ is locally presentable and its class of anodynes ($\neq \mathbf{Cof} \cap ...

**9**

votes

**0**answers

229 views

### Neighborly family of coins

Here is a puzzle:
Find 5 identical coins. Can you arrange them so that every coin is touching every other coin?
The solution is here. The hint is: use the third dimension.
My questions are ...

**9**

votes

**0**answers

313 views

### How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

Title question description: Select two lattices $\Lambda_1$ and $\Lambda_2$ (here a lattice=additive free abelian group without accumulation points) of maximal rank two in the real plane. We normalize ...

**8**

votes

**0**answers

212 views

### Orthonormal bases of R^3 with components lying in the golden field

Greg Egan proved an interesting theorem about unit vectors in $\mathbb{R}^3$ whose components actually lie in the 'golden field' $\mathbb{Q}[\sqrt{5}]$. He found it in our studies of twin ...

**8**

votes

**0**answers

140 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**8**

votes

**0**answers

121 views

### Diameter of simplicial complex mirrored in property of Stanley-Reisner ring?

Consider a pure finite abstract simplicial complex $\Delta$. Define its diameter as the maximal distance between any two facets, i.e., between any two faces of maximal dimension $d-1$. The distance ...

**7**

votes

**0**answers

83 views

### Points on $k$ Circles

Let $k$ be a fixed positive integer. We want to find the minimum number $f(k)$, such that for a set of finite points in the plane, if any $f(k)$ of them are on $k$ circles, then all of them are on $k$ ...

**7**

votes

**0**answers

181 views

### Minimal “basis” in $n$ dimensional unit cube

Let's
$$
B^n=\{\bar\alpha=(\alpha_1,\alpha_2,\ldots,\alpha_n)|\alpha_i\in \{0,1\}\};~~~~n=1,2,\ldots
$$
and let's
$$
C\subseteq B^n,
$$
$$
S(C)=\{\bar\alpha\oplus\bar\beta\ | \bar\alpha,\bar\beta\in ...

**7**

votes

**0**answers

85 views

### Is the equidissection spectrum closed under addition?

If a polygon can be cut into $m$ as well as into $n$ triangular pieces of equal area, can it also be cut into $m+n$ triangles of equal area?
(I'm editing after realizing that my conjecture that a ...

**7**

votes

**0**answers

342 views

### Rectangology and squareology

I thought that rectangles were simple, and squares even simpler. Until my research has led me to several questions about rectangles and squares, which I can't solve.
I started by posting this ...

**7**

votes

**0**answers

141 views

### Fractional Helly for more than one piercing

Fractional Helly Theorem says the following:
For every $0<\alpha\leq 1$ there exists $\beta = \beta(d, \alpha)$ with the following property. Let $C_1 , C_2 , ..., C_n$ be convex sets in $R^d$, $n ...

**7**

votes

**0**answers

272 views

### A question about a blue fan and a red fan and their common refinement

Is the following conjecture true?
Conjecture: Let $M_1$ be a red map and let $M_2$ be a blue map drawn in general position on $S^n$, and let $M$ be their common refinement. There is a vertex $w$ of ...

**7**

votes

**0**answers

237 views

### Coloring toroidal polyhedra with convex faces?

Consider a toroidal polyhedron, which is a topological torus, in which all faces are planar, two faces meet in at most an edge, and adjacent faces are not coplanar. The Szilassi polyhedron has 7 ...

**7**

votes

**0**answers

593 views

### An Ex functor for the contravariant homotopy structure

I'm going to slack on the background and get to the point:
Is there a good notion of an $Sd/Ex$ adjunction for $sSet/S$ equipped with
the contravariant model structure (cofibrations are monomorphisms ...

**7**

votes

**0**answers

274 views

### 3-dimensional Cayley graph

I would like to see Cayley graphs drawn in 3-dimensional Euclidean space such that the vertices are represented by points and various shadows display the actions of the generators.
For example, ...

**6**

votes

**0**answers

108 views

### How big can a family of pairwise intesecting affine spaces be?

I apologize if this question might seem to be a bit too elementary.
Let $\mathbb{P}^n$ be the projective space over $k$ - an algebraically closed field of characteristic 0. Let $1\leq l\leq n-1$, and ...

**6**

votes

**0**answers

175 views

### Linked circles in R3

Two circles in 3-D are linked iff each one passes through the interior of the other.
There are $N$ points in 3-D in general position (no four lie on a plane). Each triple of points defines a unique ...

**6**

votes

**0**answers

112 views

### Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between edges are rational multiples of $\pi$?

After reading these very interesting questions, I came up with another one:
Does every convex polyhedron have a combinatorially isomorphic counterpart whose angles between all pairs of edges meeting ...

**6**

votes

**0**answers

78 views

### Constructing a polyhedron of maximal possible volume from given bounds on areas of its faces

Consider $n$ variables $a_1,...,a_n$ ranging over $\mathbb{R}^+$. Suppose we are given $n$ pairs of positive rational numbers $(p_1,q_1),...,(p_n,q_n)$ where each pair imposes bounds on the ...

**6**

votes

**0**answers

857 views

### Interpolating points with minimum curvature constraint

I have $n$ points $p_i$ strictly interior to a rectangle $R$,
and I would like to connect them with a curve $C$ whose curvature is as low as possible.
Let $\kappa_\max(C)$ be the sharpest (largest ...

**6**

votes

**0**answers

217 views

### balls in arrangements of hyperplanes

The following theorem is from Aronov, Naiman, Pach and Sharir's
An invariant property of balls in arrangements of hyperplanes. I would like to state them and then ask if any related problem/theorem ...

**6**

votes

**0**answers

459 views

### Historical and terminological questions about Dan Kan's Ex functor and its relation to the classical case of simplicial complexes

Recall that we may define a functor $\xi:\Delta\to \operatorname{Poset}$ sending a simplex $[n]$ to the set of monotone injections $[k]\hookrightarrow [n]$ for $k\geq 0$ (effectively, $k\leq n$ as ...

**5**

votes

**0**answers

444 views

### N-balls covering n-balls

This question is a follow-on question from:
Covering a unit ball with balls half the radius
The questions are these:
Given an arbitrary dimension d, and a unit n-ball in d-dimensional Euclidean ...

**5**

votes

**0**answers

477 views

### Optimal Gear Trains

Suppose you need to slow down a turning motor so that a gear turns at
an angular velocity $\frac{a}{b}$ of that of the motor shaft, where $a$ and
$b$ are natural numbers. For example, this set of ...

**5**

votes

**0**answers

318 views

### $n$ lines in a general position and the number of empty triangles

Question. Consider $n \geq 5$ lines in a general position (i.e. no two lines are parallel and no triple intersections are allowed) in $\mathbb{R}^2$. Let $T(n)$ denote the maximal number of empty ...

**4**

votes

**0**answers

81 views

### Integer decomposition of dilated integral polytopes

For $n > 0$, let $P$ be an integral polytope, that is, the convex hull in $\mathbb{R}^n$ of points in $\mathbb{Z}^n$. Suppose that $\dim(P) = n$.
Question: Given $d > n + 2$ is it true that
$$ ...

**4**

votes

**0**answers

70 views

### How many subspaces are generated by three or more subspaces in a Hilbert space?

In the book of G. D. Birkhoff "lattice theory", it is mentioned that there are 28 subspaces that can be obtained from three subspaces in general position in a Hilbert space (using intersections and ...

**4**

votes

**0**answers

81 views

### Can we replace 2-fold cover by n rectangles with 1-fold cover by n rectangles?

Suppose that $n$ rectangles cover every point of their union exactly twice (except for points on their boundaries).
Can we partition this union into at most $n$ rectangles?
I think it's pretty ...

**4**

votes

**0**answers

252 views

### Unique Domino Tiling

Question: how does one enumerate all star-convex $2n$-vertex sublattices of the plane that have the unique domino-tiling property?
Definitions:
A subset S of the xy-plane is star-convex if there is ...

**4**

votes

**0**answers

122 views

### Graph drawing maximizing the volume of the convex hull

Given a graph $G=(V,E)$ and a length function $\ell:E\to\mathbb{R}_+$.
An embedding of the graph into the $d$-dimensional Euclidean space is a map $f:V\to\mathbb{R}^d$ such that ...

**4**

votes

**0**answers

183 views

### Mean number of $n$-simplices per $(n-2)$-simplex in a triangulated $n$-manifold

Work by Tamura (extending results by Luo and Stong) shows the following.
Theorem: For any closed 3-manifold $M$ and any rational number $4.5 < r < 6$ there is a triangulation $T$ of $M$ for ...

**4**

votes

**0**answers

139 views

### Slices of Simplices that are Simplices, Reference?

I am trying to find a reference for the following fact. It is elementary and not hard to prove, but I haven't been able to find the question treated anywhere.
Let $A$ be an $l\times n$ matrix with ...

**4**

votes

**0**answers

417 views

### Intersection of pencils in $\mathcal{R}^2$

Consider $9n$ pencils through non-collinear points $p_1, \ldots , p_{9n}$ in $R^2$ each consisting of at most $n$ concurrent lines. Define the intersection $S$ of these pencils to be the set of points ...

**3**

votes

**0**answers

85 views

### Modeling bubble rafts

If you go to images.google.com and search on "bubble rafts", you'll see various pictures of disk packings that in large patches look like the six-around-one dense packing of the plane by equal-sized ...

**3**

votes

**0**answers

74 views

### A taut string of equilateral triangles

Let $T$ be a unit edge-length equilateral triangle composed of three cylinders each
of (small) radius $r>0$. (By "small" I mean approximately $< 0.1$.)
Think of $T$ as a physical, rigid ...

**3**

votes

**0**answers

92 views

### Lattices achieving best density

Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ ...