# 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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### 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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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 ...
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\...