Questions tagged [dissection]

Problems that involve partitioning a geometric figure into smaller pieces with certain conditions on them (equal area, equal shape, possible to be rearranged into another given figure, etc.)

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Cutting $n$ circular cakes of different radii but equal heights into $p>n$ equal shares

You probably know the following problem: We have two circular cakes of the same height but unknown and potentially different radii, and we want to cut them into two equal shares. Each cut can only cut one piece of one cake. What is the minimal…
Hippalectryon
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Dividing an equilateral triangle into N equal (possibly non-connected) parts

It’s easy to divide an equilateral triangle into $n^2$ , $2n^2$ , $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an awesome way to do it — see the picture below (note…
Grigory M
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How many "prime" rectangle tilings are there?

Given two tilings of a rectangle by other rectangles, say that they are equivalent if there is a bijection from the edges, vertices, and faces of the tilings which preserves inclusion. For instance, the following two tilings are equivalent (some…
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Can a figure be divided into 2 and 3 but not 6 equal parts?

Is there a two dimensional shape (living in a plane) that can be divided into $2$ and $3$ but not $6$ equal parts of same size and shape? This question is a simpler take on this puzzling.SE question . If such a shape exists, the $3$ parts can't be…
Eod J.
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Do side-rational triangles of the same area admit side-rational dissections?

Call a polygon side-rational if the lengths of all its sides are rational. Call a dissection of a polygon side-rational if all of the polygons within the dissection are side-rational. Then my question is as in the title: Do any two rational…
Steven Stadnicki
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A triangle cannot be dissected into fewer than 9 convex pentagons. Is there a simple proof of this claim?

I want to dissect a triangle into $n$ strictly convex pentagonal pieces. This is possible with $n=9$ : (One construction of the above arrangement is to remove a single vertex from the dodecahedral graph.) Consider the obvious graph associated with…
RavenclawPrefect
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Can an equilateral triangle be dissected into 5 congruent convex pieces?

There is a rather surprising dissection of an equilateral triangle into 5 congruent pieces: However, these pieces aren't very "nice", consisting of 2 or 6 connected components depending on how one counts single-point overlaps. I expect that the…
RavenclawPrefect
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Splitting equilateral triangle into 5 equal parts

Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable from each other by a rigid motion) parts?
Jaska
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Union of two disjoint congruent polygons is centrally symmetric. Must the polygons differ by a 180 degree rotation?

Let $P$ be a polygon with $180^\circ$ rotational symmetry. Let $O$ be the center of $P$ and suppose $P$ is dissected into congruent polygons $A$ and $B$ . Must the $180^\circ$ rotation around $O$ send $A$ to $B$ ? This conundrum appears to have…
greenturtle3141
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A regular tetrahedron can be dissected into $1,2,3,4,6,8,12,$ or $24$ congruent pieces. Is this it?

By placing a tetrahedron on a face and making vertical cuts centered at the "top" vertex, it is easy to dissect the tetrahedron into $1, 2, 3,$ or $6$ congruent pieces. By cutting the tetrahedron into four identical pyramids meeting at the center,…
RavenclawPrefect
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Dissection of equilateral triangle into square with flipping pieces - variation of Henry Dudeney's problem

Is it possible to divide the equilateral triangle into 4 pieces and assemble a square with these four pieces, provided that one or two pieces are flipped over to the other side ? In other words, I want a mirror reflection of one or two pieces. If…
Przemyslaw Remin
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Generalizing the Borsuk problem: How much can we shrink a planar set of diameter 1 by cutting it into $k$ pieces?

Borsuk's problem asks whether a bounded set in $\mathbb{R}^n$ can be split into $n+1$ sets of strictly smaller diameter. While true when $n=1,2,3$ , it fails in dimension $64$ and higher; I believe all other $n$ are open as of this writing. However,…
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If $a^3+b^3=1$, is there a set that can be divided into two similar copies of itself, scaled by $a$ and $b$ respectively?

Let $a$ and $b$ be positive reals. If $a+b=1$ , any line segment can be divided into two smaller line segments, whose ratio the the original is $a$ and $b$ respectively. If $a^2+b^2=1$ , then there exists a triangle with side-lengths $a,b,1$ , which…
Kepler's Triangle
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Is it always possible to cut out a piece of the square with $\frac{1}{5}$ of its area?

Let there be a square that has $n+1$ notches on each edge (corners included) to divide each edge into $n$ equal parts. We can make cuts on the square from notch to notch. Is it always possible to cut out a connected piece with area $\frac{1}{5}$ the…
mathlander
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Are there "close" solutions to Hilbert's third problem?

Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra $P,Q$ of equal volume are equidecomposable by cutting $P$ into finitely many polyhedral pieces and rearranging them to obtain $Q$ . (In the two-dimensional case,…
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