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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Do cut-length-minimizing equidissections exist?
Suppose $A,B$ are polygons of equal area. By the Wallace-Bolyai-Gerwien theorem, $A$ and $B$ are equidissectable: we can make finitely many straight-line cuts in $A$ and rearrange the resulting pieces ...
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Looking for clarification of C-H Sah's definition of abstract scissors congruence
In C-H Sah's book Hilbert's third problem: scissors congruence, the author defines the data for abstract scissors congruence in order to prove Zylev's theorem by combinatorial means in great ...
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Importance of third homology of $\operatorname{SL}_{2}$ over a field
$\DeclareMathOperator\SL{SL}$I am reading some papers about the third homology of linear groups. In particular for the $\SL_{2}$ over a field. Why is it important to study these homologies?
I have ...
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Partition of polygons into 'congruent sets of polygons'
Definition: Two finite sets of polygons $A$ and $B$ are congruent if we can match polygons in $A$ in a one-one manner with polygons in $B$ with each matched pair of polygons mutually congruent.
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Scissor congruence for foliated polygons
Given two polygons of equal area with horizontal foliations, can one describe the obstruction (if there is any but I suspect the answer to be yes) to scissor-equivalence respecting the horizontal ...
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Cutting of a regular polygon into congruent pieces
Question. For which $N$ it is possible to cut a regular $N$-gon into congruent pieces such that the center of the regular polygon lies strictly inside one of the pieces? For $N=3,4$ there are trivial ...
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On a possible variant of Monsky's theorem
See Wikipedia for Monsky's theorem which states: it is not possible to dissect a square into an odd number of triangles all of equal area.
Questions: Are there quadrilaterals that allow partition into ...
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Cutting polygons into mutually similar and non-congruent pieces
It is well-known that a square can be cut into a finite number of squares all of mutually different sides (hence mutually non-congruent) - for example, see https://en.wikipedia.org/wiki/...
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On congruent partitions of planar regions
Given any integer $n$, any rectangular region or any sector of a disc (including the full disk as a boundary case) can be cut into $n$ mutually congruent pieces - by equally spaced parallel lines and ...
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Is Sydler's theorem concerning Dehn invariants constructive?
Sydler proved something of a converse to Dehn's negative resolution
of Hilbert's 3rd problem. To quote Wikipedia, Sydler showed that
"every two Euclidean polyhedra with the same volumes and Dehn ...
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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 ...
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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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Partitioning polygons into acute isosceles triangles
Question: Given an $N$-vertex polygon (not necessarily convex). It is to be cut into the least number of acute isosceles triangles.
Based on this MathSE discussion, one can think of a method to get $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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\...
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