Questions tagged [dissection]

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

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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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50 views

Algorithms to generate random fault-free rectangulation? [closed]

I want to generate random rectangular partition of a given $m*n$ rectangle under the constraint that it must be fault-free partition. Basically, no two adjacent rectangles share a common side and at ...
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0answers
71 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 ...
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1answer
116 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
57 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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13 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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0answers
84 views

Covering a polygon with triangles

Here is a great covering problem. For which $n$ can we partition a regular $n$-gon into finitely many triangles such that no two triangles share a side? I tried joining segments, and then deleting ...
8
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1answer
194 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
344 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 \...
2
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1answer
56 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
63 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
2k 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 ...
2
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1answer
167 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 ...
3
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0answers
453 views

cutting an equilateral triangle to $n$ equal pieces

We have an equilateral triangle and we want to cut it into $n$ equal pieces. For which $n$ is it possible? My Attempt: I found these possible numbers $2,3,4,6$ and also I proved every $n$ of the ...
5
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1answer
118 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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0answers
25 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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0answers
249 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
9k 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 ...
14
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2answers
5k 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?