Questions tagged [scissors-congruence]

Equidecomposability of polyhedra under cutting-and-pasting along faces, and generalizations. Hilbert's third problem, Dehn's invariants. Homological developments motivated by these issues.

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5
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1answer
300 views

Cutting the unit square into pieces with rational length sides

The following questions seem related to the still open question whether there is a point(s) whose distances from the 4 corners of a unit square are all rational. To cut a unit square into n (a finite ...
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0answers
176 views

On dissecting a triangle into another triangle

It is easy to see that an equilateral triangle can be cut into 2 identical 30-60-90 degrees right triangles which can then be patched together to form a 30-30-120 degrees triangle. So, via 2 ...
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2answers
750 views

Which right square pyramids are scissors congruent to a cube?

Consider a right square pyramid whose base has side length $2r$ and whose height is $h$. Let the dihedral angle between the base and each triangular side be $\theta$, and the dihedral angle between ...
6
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0answers
195 views

Complexity of scissors congruence?

Where is the complexity of the problem 'Given two bounded compact convex integral polyhedra in $\mathbb R^n$ presented by $O(poly(n))$ integer valued halfspaces given by linear inequalities with ...
10
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1answer
2k views

Intuition behind the Dehn Invariant

EDIT: as pointed-out below, this has been posted on math.stackexchange. I'll leave it up to the community whether or not to delete this question, but I do think there is room for a more technical ...
7
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1answer
385 views

Convex Polyhedra Scissors Congruence Problem

I am currently writing a geometry paper "Rectifications of Convex Polyhedra" and I am confused to have discovered what appears to be a remarkable discrete geometric fact: Conjecture: Let $P$ be a ...
9
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2answers
404 views

Transfers on Bloch groups and scissors congruence groups

I have a couple of questions concerning existence and description of transfers for Bloch groups and scissors congruence groups/pre-Bloch groups. To fix notation and recall definitions: From the ...
4
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1answer
280 views

A construction related to scissors congruence

I was thinking about the following some time ago. My question is whether such things have been studied before. Let $E_n$ be the abelian group with a generator for each (bounded) euclidean polytope of ...
6
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2answers
439 views

Equidecomposable graphs, unimodality and asymptotics

I will call two graphs $G$ and $H$, $r$-equidecomposable (in analogy with Hilbert's third problem) if they can be written as unions of disjoint subgraphs $$G\cong \bigsqcup_{i=1}^r G_i\quad ,\quad H\...