# 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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### Proof verification: A cube and a tetrahedron are not scissors congruent?

I am familiar with a proof that the cube and tetrahedron are not scissors congruent along the following lines: Given a polyhedron or collection of polyhedra $\mathcal{P}$ whose edges form a set $E$, ...
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### Can a dodecagon be cut into $n$ congruent pieces for any $n$ not of the form $1,2,3,4,6,8,12k^2,24k^2$?

Suppose I want to cut a regular dodecagon into $n$ congruent simply-connected pieces. For which $n$ is this possible? I can cut it into 24 right triangles, by cutting from the center to each vertex ...
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### Can $n$ squares each be dissected into identical polygons and then re-assembled into a a single larger square

Suppose you have $n$ unit squares. Can you dissect each square into polygons such that all the polygons are identical, and then re-arrange the polygons into a single big square of area $n$? Rotations, ...
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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 $O$ ...
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### Square to octagon dissection - how to cut the square?

How to cut the square which tessellates to octagon using straightedge and compass? What are the exact measures of colored sides? What is the angle marked with red color? Edit (I added vertices): Edit....
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### Hexagon to Rectangle dissection: 3 pieces minimal?

A hexagon can be divided into 3 pieces to make a rectangle. Can we prove 3 pieces is minimal? For a equilateral triangle to square dissection, it's thought that 4 pieces is minimal. We can prove that ...
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### Does this kind of partition have a name?

Note: Reposting from OR Stackexchange as advised there. Consider a convex polyhedron $A$. Assume we have subsets $A_1,\ldots,A_n$ of $A$ that are themselves covex polyhedra and are mutually disjoint ...
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### Is it always possible to cut out a piece of the triangle with $\frac{1}{3}$ the area?

This is a part $3$ of a sequence of questions starting with my highly upvoted question (at the time of writing, my third-best post). Feel free to extend this series using other polygons and fractions. ...
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### Is it always possible to cut out a piece of the triangle with half the area?

This is a sequel to my highly upvoted question (at the time of writing, my third-best post). Let there be an equilateral triangle that has $n+1$ notches on each edge (corners included) to divide each ...
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### Dissect a square into two contiguous congruent shapes. Must the dissection be rotationally symmetric?

Every dissection I can think of that cuts a square into two contiguous congruent shapes seems to be rotationally symmetric. (Allowing disjoint shapes allows for dissections that aren't.) Is there a ...
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### Can dissections always be 'adjusted' to polygonal dissections?

Given two polygons, a dissection is a decomposition of the two polygons into a finite set of "nice" pieces with disjoint interiors, along with a bijection between the sets of pieces where a ...
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### Confusion regarding Tarski's circle-squaring problem

Wikipedia describes Tarski's circle-squaring problem like this: Tarski's circle-squaring problem is the challenge, posed by Alfred Tarski in 1925, to take a disc in the plane, cut it into finitely ...
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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 ...
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### Other solutions to cubing the cube variation

It's known that a cube can't be divided into smaller cubes of distinct sizes. So for fun, I defined a "wannabe cube" as a cuboid whose dimensions are (not equal, but) consecutive integers ...
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### What is the "most proportionate" perfect squared square?

A perfect squared square is a squared square with all elements of different sizes with at least two elements. The "proportionateness" of a squared square is the ratio of the size of the size ...
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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 ...
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### Can a scaled $L$-tromino be cut into two congruent polyominoes?

The $L$-tromino can trivially be cut into two congruent trapezoidal pieces: It can also be trivially cut into three squares, and into four other $L$-trominoes of half the side length. I am curious ...
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### Can the following figures be divided into 6 equal parts?

This question asks whether a figure can be divided into $2$ and $3$ equal parts, but not $6$. It is in turn based off of an earlier puzzling.SE question. One natural approach is to consider the case ...
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### On partitioning triangles and pentagons [closed]

Is there any triangle that can be cut into 5 mutually congruent pieces? If the answer is "yes" how does one characterize such triangles? What if we restrict the pieces to be convex? Is ...
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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. ...
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### Cut two squares into rectangles to reassemble a single square.

This problem would belong to puzzling SE, except that I suspect it to be impossible. So I post it here, to see if someone can provide an argument proving the impossibility. The problem: Is it ...
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### Is there a known method to dissect Jessen's icosahedron and rearrange to form a cube?

It can be shown that Jessen's icosahedron is scissors-congruent to a cube, meaning that it can be sliced into smaller polyhedral pieces that can be rearranged to form a solid cube. What is the ...
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### Prismatoid plane section with straightedge and compass

I came up with a task I'm out of ideas how to do a solution. Perhaps I'm not paying attention to an obvious thing, but still. All I came up with is to build orthogonal projections but that's not a ...
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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 ...
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### Two dissection problems for rectangles

Let us consider two integer rectangles (that is, with sides of integer length) $S$ and $T$ of the same area. Then, obviously, $S$ can be dissected into several integer rectangles $A_1$, ..., $A_n$ (we ...
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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 ...
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### Dividing a disk of diameter 1 into pieces of smaller diameter

Let $F$ be an arbitrary bounded set on the plane, $n \in \mathbb{N}$. Let's define $d_n(F)$ as the minimum diameter one can ensure when cutting a set $F$ into $n$ pieces. So, here is a discussion of ...
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For which $n$ is it possible to divide an equilateral triangle into $n$ equal (i.e., obtainable from each other by a rigid motion) parts? It is easy to come up with a partition for $n \in \{1, 2, 3, 4,... • 139 11 votes 0 answers 193 views ### Are there "close" solutions to Hilbert's third problem? Hilbert's third problem (or a modern formulation thereof) asks whether two polyhedra$P,Q$of equal volume are equidecomposable by cutting$P$into finitely many polyhedral pieces and rearranging them ... • 17.7k 3 votes 1 answer 66 views ### How to minimize exposed surface for half a pie? I bought a large pie (of radius$R$). I cut off a half and gave it to my friend. This exposed an area or$2Rh$-- where$h$is the pie's thickness -- to air. I watched one Numberphile video too many,... • 14.3k 31 votes 0 answers 662 views ### 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, ... • 17.7k 5 votes 1 answer 214 views ### Is there a "nice" rep-tile of order$6$? A planar set is said to be a rep-tile if it can be tiled by congruent shapes, each similar to the original. If there are$k$such shapes, each scaled down by a factor of$\sqrt{k}$, it is said to be ... • 17.7k 15 votes 2 answers 340 views ### 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 ... • 17.7k 19 votes 2 answers 549 views ### 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", ... • 17.7k 1 vote 1 answer 69 views ### Dissecting a triangle into$n$-gons For an integer$n \ge 3$and positive integer$m$, is it possible to divide a triangle into$mn$-gons of equal area? For the$3$-gon, or the triangle, you can divide an edge into$m$equal ... • 320 2 votes 1 answer 67 views ### Is there a dissection tool available online? I want to write some pages on a website exploring geometric dissections (initially 2d, e.g. Dudeney's, Haberdasher's Problem, Archimedes' Loculus, tangrams and pentominoes, etc., but eventually moving ... • 5,057 4 votes 1 answer 104 views ### When can we prove statements of the form "This shape cannot be decomposed into$k$congruent pieces"? Dissection problems tend to be pretty hard; for instance, to my knowledge we don't know whether it's possible to dissect an equilateral triangle into$k$disjoint congruent regions for any$k$whose ... • 17.7k 4 votes 1 answer 249 views ### AMC 2003 Cut the cube into pieces. For each vertex of a cube a plane is constructed through the three vertices which are neighbors of that vertex. Into how many parts do these eight planes dissect the cube? (A) 9 (B) 13 (C) 21 (D) 27 (... • 349 2 votes 2 answers 3k views ### Algorithm for filling a rectangle with tiles of known relative size I checked existing questions but couldn't find the case I wanted. I have a set of tiles of known relative size. The tiles could be squares, circles, or rectangles. They all have the same aspect ... • 121 2 votes 1 answer 111 views ### Rectangle Dissection Into Smaller Rectangles yields shared side Let$R$be a rectangle disected into$N$smaller rectangles with sides parallel to those of$R$. Is there any known condition on N such that if that holds we can always find$2$rectangles sharing a ... • 21 1 vote 0 answers 180 views ### Doubly-true dissections: Dissecting an$n$-gon into$s_i$-gons, such that$n = \sum s_i$Dissecting a polygon into other ones is a famous subject. Many people have studied varying topics about dissection. It is well known that a regular hexagon can be dissected into two equilateral ... • 634 1 vote 1 answer 65 views ### Types of possible moves of tiling puzzle pieces This is a question about tiling puzzle jargon. What are the types of moving a tiling piece? I mean the single word or a phrase for rearrangement move of a piece. I would be grateful for proper ... 2 votes 0 answers 120 views ### Geometric dissection theory Today, i realized that one way to prove the Pythagorean Theorem is to dissect the given right-angled triangle into 2 triangles similar to it, and apply well-known properties of ratios of areas. So ... • 587 1 vote 0 answers 112 views ### Equidecomposability of a Cube into Trirectangular Tetrahedra and a given tetrahedron My original problem is: 1) Let$XYZT.X'Y'Z'T'$be a cube. Given$A\in XYY'X',B\in XYZT,C\in Y'Z'$and$D\in TT'$. Is there a way to dissect the cube into Trirectangular Tetrahedra and$ABCD$? I ... • 2,597 6 votes 1 answer 617 views ### Dissecting a square into similar 1:sqrt(2) rectangles Can you dissect a square into similar rectangles with aspect ratio 1:sqrt(2)? I have a suspicion you can't and that a proof could be constructed whereby you make one side of the square an integer ... 1 vote 1 answer 126 views ### Cutting a square into non-similar triangles [closed] Is it possible to cut a square into an infinite number of triangles, so that all of them are non-similar? • 949 0 votes 0 answers 17 views ### Find a dissection that satisfies integral inequalities Given$a>0$,$n \in \mathbb{N}$and a non-negative function$f \in L^{1}(\Omega)$satisfies $$\int_{\Omega} f(x) dx \le (n+1)a$$ Find a dissection$\left\{\Omega_j \right\}_{j=1}^{n+1}$of$\...
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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 \...