Questions tagged [discrete-geometry]

Discrete geometry includes the study of covering, illumination, packing, convex bodies, convex polytopes, and other metric geometry.

Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth. The subject focuses on the combinatorial properties of these objects, such as how they intersect one another, or how they may be arranged to cover a larger object.

Discrete geometry has large overlap with convex geometry and computational geometry, and is closely related to subjects such as finite geometry, combinatorial optimization, digital geometry, discrete differential geometry, geometric graph theory, toric geometry, and combinatorial topology.

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Can a row of five equilateral triangles tile a big equilateral triangle?

Can rotations and translations of this shape perfectly tile some equilateral triangle? I've now also asked this question on mathoverflow. Notes: Obviously I'm ignoring the triangle of side $0$. Because the area of the triangle has to be a…
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Why is a circle in a plane surrounded by 6 other circles?

When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers? I'm looking for an answer deeper than "there are…
John Smith
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Why can't three unit regular triangles cover a unit square?

A square with edge length $1$ has area $1$. An equilateral triangle with edge length $1$ has area $\sqrt{3}/4 \approx 0.433$. So three such triangles have area $\approx 1.3$, but it requires four such triangles to cover the unit square, e.g.:    …
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Can all circles of radius $1/n$ be packed in a unit disk, excluding the circle of radius $1/1$?

This problem occurred to me when I came across a similar problem where the radii were taken over only the primes. That question was unanswered, but it seems to me infinitely many circles of radius $1/2, 1/3, 1/4...$ can fit into a unit disk. The…
Rob Bland
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How few disks are needed to cover a square efficiently?

A unit square can be covered by a single disk of area $\pi/2$. Let us call the ratio of the square's area to that of the covering disks (i.e. the sum of the areas of the disks) the efficiency of the covering, so that in the base case with one disk…
John Bentin
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Dividing a square into equal-area rectangles

How many ways are there to tile an $n\times n$ square with exactly $n$ rectangles, each of which has integer sides and area $n$? The sequence $C(n)$ begins 1, 2, 2, 9, 2, 46, 2, 250, 37. Clearly $C(p) = 2$ for prime $p$. The value $C(8) = 250$ was…
MJD
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Prove Existence of a Circle

There are two circles with radius $1$, $c_{A}$ and ${c}_{B}$. They intersect at two points $U$ and $V$. $A$ and $B$ are two regular $n$-gons such that $n > 3$, which are inscribed into $c_{A}$ and ${c}_{B}$ so that $U$ and $V$ are vertices of $A$…
Axas Bit
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Pick's Theorem on a triangular (or hex) grid

Pick's theorem says that given a square grid consisting of all points in the plane with integer coordinates, and a polygon without holes and non selt-intersecting whose vertices are grid points, its area is given by: $$i + \frac{b}{2} - 1$$ where…
mau
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Number of ways to stack LEGO bricks

One of the most surprising combinatorial formulas I know of counts the number of LEGO towers built from $n$ "$1 \times 2$" blocks subject to four rules: The bricks lie in a single plane. Each brick is offset by 1 stud (as in a brick wall). The…
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Circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them

I have a serious problem with this problem: Is it possible to Draw circles on the plane such that every line intersects at least one of them but no line intersects more that 100 of them !? Any help or suggestion would be helpful.
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square cake with raisins

Alice bakes a square cake, with $n$ raisins (= points). Bob cuts $p$ square pieces. They are axis-aligned, interior-disjoint, and each piece must contain at least $2$ raisins. Note that a single raisin can be shared by two pieces (if it is on their…
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Non-self-intersecting "Robot Walks"

Question 208 on Project Euler describes walks of "robots" the move in parts of a circular arc: A [$5$-]robot moves in a series of one-fifth circular arcs (72°), with a free choice of a clockwise or an anticlockwise arc for each step, but no turning…
Peter Kagey
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Convex polygons that do not tile the plane individually, but together they do

I am looking for two convex polygons $P,Q \subset \Bbb R^2$ such that $P$ does not tile the plane, $Q$ does not tile the plane, but if we allowed to use $P,Q$ together, then we can tile the plane. Here I do not require the tilings to be lattice…
Alphonse
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Minimum distance required to travel to "see" all points on a hypercube

You begin on a hypercube of dimension N at the origin i.e. $(0,0,0,0,...,0)$ When at the origin you are able to "see" one and only one step away from you. So from the origin you can see vertices $(1,0,0,0..,0), (0,1,0,...,0),... (0,0,0,...,1)$ What…
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Circle enclosing all but one of $n$ points

It looks like a simple question but it turns out rather difficult to me. Here is the question: Given $n$ points on the plane, can we always draw a circle that includes exactly $n-1$ of them?
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