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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49 views

What is the fewest number of squares required to cover a 11×13 rectangle? [closed]

I can't understand this. I found a solution with 6 squares, but I don't know if it is right and how to explain it.
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78 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 ...
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1answer
69 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 ...
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152 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 ...
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66 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 ...
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Haberdasher problem of Henry Dudeney - is 4-piece hinged dissection of equilateral triangle into square possible?

Is it possible to divide the equilateral triangle into 4 pieces to build a square with those four pieces, provided that one or two pieces are flipped over to the other side? If possible, I wish to ...
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76 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 ...
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Dividing an equilateral triangle into equal parts [duplicate]

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,...
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158 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 ...
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1answer
49 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,...
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290 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, ...
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1answer
104 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 ...
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2answers
233 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 ...
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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", ...
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1answer
42 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 $m$ $n$-gons of equal area? For the $3$-gon, or the triangle, you can divide an edge into $m$ equal ...
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1answer
38 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 ...
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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 ...
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1answer
112 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 (...
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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 ...
2
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1answer
33 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 ...
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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 ...
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1answer
50 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 ...
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86 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 ...
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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 ...
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1answer
317 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 ...
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1answer
83 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?
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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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1answer
273 views

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" - ...
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2answers
592 views

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 \...
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1answer
78 views

Cut a disk into $N$ pieces to best pack into a square

I have a 3d printer with a square boundary. I'd like to print something with a circular base. It occured to me that I could print a circle with a diameter bigger than the width of my square if I broke ...
5
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1answer
69 views

Is it possible to cut the unit disk in $5$ "small" parts?

Let $D = \{(x,y) \in \Bbb R^2 \mid x^2+y^2 \leq 1\}$ be the unit disk. Is it possible to find five subsets $A_1, \dots, A_5 \subset D$ such that they cover $D$ and they all have diameter at most $1$? ...
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1answer
3k views

How do you divide a regular hexagon into 5 equal parts?

I am looking for a easy way for dividing a regular hexagon into 5 equal parts and preferably equal shapes or continuing shapes to make it easy to see the regions. The way that I found is dividing ...
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1answer
201 views

Dissecting a circle with an irregular rectangular grid

Can a circular disc be 'dissected' by a rectangular grid into a finite number of pieces in such a way that the individual pieces of the circle can be grouped into regions of equal area? Clearly this ...
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5answers
11k views

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 ...
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1answer
636 views

Cutting an equilateral triangle into $n$ equal pieces

We have an equilateral triangle and we want to cut it into $n$ equal triangular pieces. For which $n$ is it possible? My Attempt: I found these possible numbers: $2,3,4,6$. I also proved every $n$ of ...
5
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1answer
152 views

Jigsaw-style proofs of the Pythagorean theorem with non-square squares

The two squares on the legs of a right triangle can be chopped up (or "dissected") into several pieces that can be reassembled jigsaw-style into a square congruent to that whose side is the hypotenuse....
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31 views

Can I find the dissections of a figure based on symmetry?

Our teacher gave us a figure, and challenged us to dissect into exactly 4 shapes, that were congruent, in as many ways as possible. I won't reveal details of the specific shape. I am wondering if ...
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1answer
710 views

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 ...
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2answers
10k views

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 ...
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0answers
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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 ...
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2answers
7k views

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?