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.)

19 questions
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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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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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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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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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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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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 ...
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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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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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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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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 ...
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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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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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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 ...