Mathematics Stack Exchange
Mathematics Stack Exchange

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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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…
RavenclawPrefect
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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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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…
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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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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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 $\frac{1}{5}$…
mathlander
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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,…
RavenclawPrefect
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Is there only one way to divide an equilateral triangle into congruent fourths?

Suppose we wish to divide an equilateral triangle into fourths, such that each piece is congruent. (Let's also require connectedness.) One way to do this is to connect the medians, forming one inverted triangle in the center and three at the…
Akiva Weinberger
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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,…
RavenclawPrefect
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Transforming a 8x8, 4x4 and 1x1 square into a 9x9 square

Good day to all of you! I have a puzzle which I just cannot solve. I attached a photo of it. The task is to transform the shape on the left into a 9x9 square (on the right) using ONLY 2 "cuts" - dividing it into 3 separate objects. Mirroring,…
David Kosztyu
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Can you divide a square into 5 equal area regions

Given this shape: Is it possible to divide the cyan area into 5 equal area shapes such that: Each shape is the same Each shape has an edge touching the red square Each shape has an edge touching the outside. No diagonal lines. Its reasonably easy…
Kent Fredric
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Dissecting a polyomino to tile a copy rotated $45^\circ$ and scaled by $\sqrt{2}$

I came up with the following conjecture the other day, and was wondering if the result was well-known or even true: Define $f(P)$ for a polyomino $P$ (without holes) to be the least number of total pieces in which two copies of $P$ can be split such…
Apple
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How many squares in a rectangle?

I almost wish I'd never thought of this problem... I was tearing my hair out over it all night. Suppose we have a rectangle with side lengths $a$ and $b$, $a,b \in \mathbb Z$, $GCD(a,b)=1$, and $b \gt a$. I want to pack squares in this rectangle in…
Franklin Pezzuti Dyer
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