# 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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### 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, ...
689 views

### 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 $O$ ...
268 views

### 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....
94 views

### 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 ...
46 views

### 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 ...
64 views

### 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. ...
117 views

### 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 ...
68 views

### 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 ...
59 views

### 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 ...
94 views

### 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 ...
775 views

### 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 ...
94 views

### 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 ...
160 views

### 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 ...
151 views

### 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 ...
218 views

### How to draw a sine wave by slicing a cylinder? 88 views

### 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 ...
1 vote
358 views

### 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 ...
76 views

### 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 ...
1k views

### 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. ...
218 views

### 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 ...
117 views

### 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 ...
1 vote
89 views

### 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 ...
549 views

### 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 ...
90 views

### 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 ...
603 views

### 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 ...
90 views

### 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 ...
70 views