Questions tagged [dissection]

Question 1

Is there some "unifying explanation" of existence of this interesting dissection of a side-1 regular octagon into six side-1 quadrilaterals (two squares and four rhombi) and the following ...

Question 2

I want to dissect a triangle into $n$ strictly convex pentagonal pieces. This is possible with $n=9$: (One construction of the above arrangement is to remove a single vertex from the dodecahedral ...

Question 3

I am an elementary school teacher in South Korea with a strong interest in developing intuitive, visual methods for teaching fundamental geometric concepts—especially the Pythagorean Theorem. I have ...

Question 4

I would sincerely appreciate any critical feedback or evaluation on the following result. Here's the context. Many of you are probably familiar with dissection proofs of the Pythagorean Theorem, such ...

Question 5

I am looking for a way to divide a 2D polygon (possibly with a complex shape, like an "H". These are actually floor layouts of buildings) into several connected sub-regions, where each ...

Question 6

I just saw this question on the HNQ, and it made me wonder. Is there any way that you could divide a cylinder into a finite number of pieces such that the pieces could be reassembled into three ...

Question 7

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 $E$, ...

Question 8

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 ...

Question 9

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, ...

Question 10

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 $O$ ...

Question 11

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....

Question 12

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 ...

Question 13

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 ...

Question 14

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. ...

Question 15

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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Questions tagged [dissection]

Coincidences related to a dissection of the regular octagon

A triangle cannot be dissected into fewer than 9 convex pentagons. Is there a simple proof of this claim?

Area Conservation in Geometric Dissections for Pythagorean Theorem Visualizations

Generalized Pythagorean Dissection Tiling Result

How to partition a polygon into contiguous regions with given area ratios?

Can a cylinder be split in such a way that it can be reassembled into 3 equal cones?

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

Can a dodecagon be cut into $n$ congruent pieces for any $n$ not of the form $1,2,3,4,6,8,12k^2,24k^2$?

Can $n$ squares each be dissected into identical polygons and then re-assembled into a a single larger square

Union of two disjoint congruent polygons is centrally symmetric. Must the polygons differ by a 180 degree rotation?

Square to octagon dissection - how to cut the square?

Hexagon to Rectangle dissection: 3 pieces minimal?

Does this kind of partition have a name?

Is it always possible to cut out a piece of the triangle with $\frac{1}{3}$ the area?

Is it always possible to cut out a piece of the triangle with half the area?

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