Finite or discrete collections of geometric objects. Packings, tilings, polyhedra, polytopes, intersection, arrangements, rigidity.

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Gaussian prime spirals

Imagine a particle in the complex plane, starting at $c_0$, a Gaussian integer, moving initially $\pm$ in the horizontal or vertical directions. When it hits a Gaussian prime, it turns left ...
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Is there a dense subset of the real plane with all pairwise distances rational?

I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
55
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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 ...
46
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Can an arbitrary collection of circles of total area 1/2 fit into a circle of area 1?

Assume the circles are actually open disks, otherwise two circles each of area $\frac{1}{4}$ wouldn't fit into the circle of area 1. This seems like it should be true, thinking about packing ...
45
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If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?

Update: The answer to the title question is not necessarily, as pointed out by Tapio and Willie. I would be more interested in lower bounds. Monsky's famous and amazingly tricky proof says that if we ...
45
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What is the status of the Gauss Circle Problem?

For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) ...
42
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2answers
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How many unit cylinders can touch a unit ball?

What is the maximum number $k$ of unit-radius cylinders with mutually disjoint interiors that can touch a unit ball? By a cylinder I mean a set congruent to the Cartesian product of a line and a ...
42
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two tetrahedra in R^4

It is relatively easy to show (see below) that if we have two equilateral triangles of side 1 in $R^3$, such that their union has diameter 1, then they must share a vertex. I wonder whether we have an ...
39
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Can a discrete set of the plane of uniform density intersect all large triangles?

Let S be a discrete subset of the Euclidean plane such that the number of points in a large disc is approximatively equal to the area of the disc. Does the complement of S necessarily contain ...
37
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8answers
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A sudden smiley? :-)

This is a vague question, and I will no doubt be (properly!) chastised for posing it. I would like to generate a set $S$ of points in $\mathbb{R}^3$—$|S|$ finite or infinite—which has the ...
35
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6answers
996 views

Tetris-like falling sticky disks

Suppose unit-radius disks fall vertically from $y=+\infty$, one by one, and create a random jumble of disks above the $x$-axis. When a falling disk hits another, it stops and sticks there. Otherwise, ...
34
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3answers
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What fraction of the integer lattice can be seen from the origin?

Consider the integer lattice points in the positive quadrant $Q$ of $\mathbb{Z}^2$. Say that a point $(x,y)$ of $Q$ is visible from the origin if the segment from $(0,0)$ to $(x,y) \in Q$ passes ...
32
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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
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2answers
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What is the oriented Fano plane?

One way to remember the multiplication table of the octonions is to use the following diagram (which I got from John Baez's online paper): if $(e_i,e_j,e_k)$ is one of the lines listed according to ...
30
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1answer
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Pach's “Animals”: What if genus $> 0$ ?

Janos Pach asked a deep question 23 years ago (1988) that remains unsolved today: Can every animal—a topological ball in $\mathbb{R^3}$ composed of unit cubes glued face-to-face—be ...
28
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1answer
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Can the sphere be partitioned into small congruent cells?

On the unit $2$-sphere ${\mathbb S}^2$ furnished with the geodesic distance, a subset homeomorphic to a planar disk is called a cell. A finite family of cells is a tiling if their interiors are ...
27
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Can every $\mathbb{Z}^2$ disk be pinball-reached?

Let every point of $\mathbb{Z}^2$ be surrounded by a mirrored disk of radius $r < \frac{1}{2}$, except leave the origin $(0,0)$ unoccupied by a disk. Q. Is it the case that every disk can be ...
27
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Can we cover the unit square by these rectangles?

The following question was a research exercise (i.e. an open problem) in R. Graham, D.E. Knuth, and O. Patashnik, "Concrete Mathematics", 1988, chapter 1. It is easy to show that $$\sum_{1 \leq k } ...
27
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1answer
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Can we find lattice polyhedra with faces of area 1,2,3,…?

I asked this question two months ago on MSE, where it earned the rare Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days. Perhaps justifiably so! Here I repeat it with ...
26
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2answers
855 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 ...
25
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6answers
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When shorter means smaller?

Assume a convex figure $F\subset \mathbb R^2$ satisfies the following property: if $f:F\to \mathbb R^2$ is a distance-non-increasing map then its image $f(F)$ is congruent to a subset of $F$. Is it ...
24
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2answers
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chromatic number of the hyperbolic plane

A notorious problem in combinatorics is the following: If we color $\mathbb{R}^2$ so that no pair of points at unit distance get the same color, what is the fewest number of colors required? This ...
23
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2answers
714 views

The kissing number of a square, cube, hypercube?

How many nonoverlapping unit squares can (nonoverlappingly) touch one unit square? By "nonoverlapping" I mean: not sharing an interior point. By "touch" I mean: sharing a boundary point. ...
22
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3answers
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Can a unit square be cut into rectangles that tile a rectangle with irrational sides?

For arbitrary positive integers $m$ and $n$, if we dissect a unit square into an $m\times n$ rectangular grid of $1/m\times 1/n$ rectangles, we can reassemble these $mn$ rectangles into an $n/m\times ...
22
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Tetrahedron insphere iteration

I know that iterating the following incircle construction approaches an equilateral triangle in the limit:       Starting with any triangle $T$, one forms $T'$ by connecting ...
22
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8answers
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Points and lines in the plane

Does a positive real number $k\geq1$ exist such that for every finite set $P$ of points in the plane (with the property that no three points of $P$ lie on a common line and $|P|\geq3$), one can choose ...
21
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1answer
960 views

Building a genus-$n$ torus from cubes

I wonder if this has been studied: What is the fewest number of unit cubes from which one can build an $n$-toroid? The cubes must be glued face-to-face, and the boundary of the resulting ...
20
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6answers
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Covering a unit ball with balls half the radius

This is a direct (and obvious) generalization of the recent MO question, "Covering disks with smaller disks": How many balls of radius $\frac{1}{2}$ are needed to cover completely a ball of radius ...
20
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5answers
802 views

Iterated Circumcircle

Take three noncollinear points (a,b,c), compute the center of their circumcircle x, and replace a random one of a,b,c with x. Repeat. It seems this process may converge to a point, assuming no ...
20
votes
2answers
889 views

Forbidden mirror sequences

Let $\cal{M}$ be a finite collection of two-sided mirrors, each an open unit-length segment in $\mathbb{R^2}$, and such that the segments when closed are disjoint. A ray of light that reflects off the ...
20
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Rolling-ball game

The analyses in two recent MO questions, "Rolling a random walk on a sphere" and "Maneuvering with limited moves on $S^2$," suggest a Rolling-Ball Game, as follows. A unit-radius ball sits on a grid ...
20
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1answer
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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 ...
19
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2answers
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Erdős, Harary, Tutte's “dimension of graph”: Progress in last 48 yrs?

I just ran across this delightful paper by an amazing triumvirate: Paul Erdős, Frank Harary, and William Tutte. "On the dimension of a graph." Mathematika 12.118-122 (1965): 20. (Cambridge link) ...
19
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1answer
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Hidden points in polygons

Let $h(n)$ be the largest number of mutually invisible points that can be located in a polygon $P$ of $n$ vertices. Two points $x$ and $y$ are mutually invisible if the segment $xy$ contains a point ...
18
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What upper bounds are known for the diameter of the minimum spanning tree of $n$ uniformly random points in $[0,1]^2$?

Let $P$ be a pointset consisting of $n$ uniformly random elements of $[0,1]^2$. It is known that the diameter (greatest number of edges in any shortest path between two points) of the Delaunay ...
18
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3answers
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Research trends in geometry of numbers?

Geometry of numbers was initiated by Hermann Minkowski roughly a hundred years ago. At its heart is the relation between lattices (the group, not the poset) and convex bodies. One of its fundamental ...
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Growing random trees on a lattice $\rightarrow$ Voronoi diagrams

Imagine growing trees from $k$ seeds on a square $n \times n$ region of $\mathbb{Z}^2$. At each step, a unit-length edge $e$ between two points of $\mathbb{Z}^2$ is added. The edge $e$ is chosen ...
18
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2answers
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Trapping a convex body by a finite set of points

In $\mathbb{R}^n$, let $K$ be a convex body and $T$ a finite set of points disjoint from the interior of $K$. Say that $T$ traps $K$ if there is no continuous motion of $K$ carrying $K$ arbitrarily ...
17
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2answers
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Erdős-Szekeres for first differences

The classical Erdős-Szekeres theorem says that any sequence of $n^2+1$ real numbers contains a monotonic $(n+1)$-term subsequence. Suppose, however, that we want to find a subsequence which is not ...
17
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3answers
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Sperner Lemma Applications

I was always fascinated with this result. Sperner's lemma is a combinatorial result which can prove some pretty strong facts, as Brouwer fixed point theorem. I know at least another application of ...
17
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2answers
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Placing points on a sphere so that no 3 lie close to the same plane

Motivation I am working with arbitrary parallelopiped tilings given by projection from a higher dimensional space. The collection of tiles, and some properties of the higher dimensional space are ...
16
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4answers
705 views

Non-chaotic bouncing-ball curves

I was surprised to learn from two Mathematica Demos by Enrique Zeleny that an elastic ball bouncing in a V or in a sinusoidal channel exhibits choatic behavior:     (The Poincaré map ...
16
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6answers
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Shortest grid-graph paths with random diagonal shortcuts

Suppose you have a network of edges connecting each integer lattice point in the 2D square grid $[0,n]^2$ to each of its (at most) four neighbors, {N,S,E,W}. Within each of the $n^2$ unit cells of ...
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Cutting convex sets

Any bounded convex set of the Euclidean plane can be cut into two convex pieces of equal area and circumference. Can one cut every bounded convex set of the Euclidean plane into an arbitrary number ...
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+50

What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below). Question A. How does one arrange $n$ unit cubes ...
16
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How many unit squares can you pack into a rectangle with nearly integer side lengths?

Earlier today, somebody asked what looks like a homework problem, but admits the following reading which I think is interesting: Suppose $a_1,\dots, a_n$ are positive integers, and $\varepsilon$ ...
15
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3answers
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Gauss-Bonnet Theorem for Graphs?

One can define the Euler characteristic χ for a graph as the number of vertices minus the number of edges. Thus an n-cycle has χ = 0 and K4 has χ = –2. Is there an analog for the ...
15
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5answers
886 views

Is a rhombus rigid on a sphere or torus? And generalizations.

If a rectangle is formed from rigid bars for edges and joints at vertices, then it is flexible in the plane: it can flex to a parallelogram. On any smooth surface with a metric, one can define a ...
15
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1answer
748 views

Are there analogues of Desargues and Pappus for block designs?

Finite projective planes are fascinating objects from many perspectives. In addition to the geometric view, they can be viewed as combinatorial block designs. From the geometric perspective, there ...
15
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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 $U^0,...,U^c$ ...