Category:Equidecomposability

This category contains results about Equidecomposability.
Definitions specific to this category can be found in Definitions/Equidecomposability.

Two objects are equidecomposable if and only if they can be broken down into identical sets of component pieces.

Equidecomposable Sets

Two sets $S, T \subset \R^n$ are said to be equidecomposable if and only if there exists a set:

$X = \set {A_1, \ldots, A_m} \subset \powerset {\R^n}$

where $\powerset {\R^n}$ is the power set of $\R^n$, such that both $S$ and $T$ are decomposable into the elements of $X$.

Equidecomposable Polyhedra

Let $n \in \N$ be a natural number.

Let $K_1$ and $K_2$ be polyhedra embedded in a Euclidean space of $n$ dimensions such that both are the union of a finite number of $n$-simplexes

Let $K_1$ and $K_2$ be the union of a finite number of polyhedra:

where:

each pair of the polyhedra $A_i$ and $A_j$, and $B_i$ and $B_j$, intersect only in $m$-simplexes where $m < n$

each $A_i$ is congruent to its corresponding $B_i$.

Then:

$K_1$ and $K_2$ are equidecomposable.