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

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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 ...
6
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3answers
673 views

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance Part II

For some context see Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance As per Noah's answer and JBL's comment this was false as stated. However, I think the following ...
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1answer
418 views

Random Walks in $Z^2$/$Z^2$-intrinsic characterization of Euclidean distance

Problem: Consider a random walk on the lattice $\mathbb{Z}^2$ where on each iteration a particle either stays at its current location or moves to a neighboring vertex with probability 1/5. We start ...
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2answers
257 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, ...
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3answers
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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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5answers
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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 ...
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2answers
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fit 4 circles within a square

If I have a square and want to place four equally large circles within this square, how large can the maximum radius be (compared to the lenght of the side of the square)? Just an answer would be ok, ...
5
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1answer
408 views

Isometric embedding of a positively curved polyhedral surface

Suppose you have a 2-dimensional polyhedral surface with specified lengths for the edges so that the vertices all have positive curvature. I believe this has a unique isometric embedding into ...
5
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2answers
502 views

Maximal area coverable by $k$ disjoint isosceles triangles contained in a triangle of area 1.

Given a triangle $\Delta$ of unit area, how much area can always be covered by $k$ isosceles triangles contained in $\Delta$ and intersecting at most at their boundaries? The answer is easy for ...
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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 ...
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3answers
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What is the minimum N for which there exist N points in the plane that cannot be covered by any number of non-overlapping closed unit discs?

This problem was posed in March 2010 at G4G9 in a talk by the Japanese mathematician Hirokazu "Iwahiro" Iwasawa. He claims there is a simple proof that N > 10, though he did not share it with the ...
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0answers
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Is there any symmetric unshellable triangulation of a tetrahedron?

M. Rudin gave a triangulation of a tetrahedron with the property that after any small tetrahedron removed, the remaining part is not homeomorphic to a ball (An unshellable triangulation of a ...
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2answers
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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 $\varepsilon$ ...
59
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5answers
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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 ...
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2answers
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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) ...
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1answer
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Is it possible to dissect a disk into congruent pieces, so that a neighborhood of the origin is contained within a single piece?

Problem: is it possible to dissect the interior of a circle into a finite number of congruent pieces (mirror images are fine) such that some neighbourhood of the origin is contained in just one of ...
5
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4answers
966 views

Coloring Points in the Plane

Suppose one wants to color the points in the plane so any two points at distance one apart are different colors. How many colors are needed? I heard this problem when I was a kid. Back then the most ...
15
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1answer
718 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 ...
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6answers
887 views

Combinatorial distance ≡ Euclidean distance

Definition: A polytope has property X iff there is a function f:N+ → R+ such that for each pair of vertices vi, vj the following holds: disteuclidean(vi, vj) = f(distcombinatorial(vi, vj)) with ...
4
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4answers
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Number of spanning trees in a grid

Given a $\sqrt{n}\times\sqrt{n}$ piece of the integer $\mathbb{Z}^2$ grid, define a graph by joining any two of these points at unit distance apart. How many spanning trees does this graph have ...
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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 ...
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2answers
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The Sylvester Gallai Theorem and Sections of Varieties with “Simple Topology”.

The Sylvester-Gallai theorem asserts that for every collection of points in the plane, not all on a line, there is a line containing exactly two of the points. One high dimensional extension ...
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12answers
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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 ...
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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 ...