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|Remark 22.1.4 This example suggests the following question: If we assume |
Archimedes' axiom (A) in addition, are the notions of equidecomposable and
equal content equivalent? We will see that the answer is yes in any Hilbert plane
with (P) ...
|We shall now begin to examine the problems of equidecomposability and |
equicomplementabilityforsolid bodies (polyhedra). Two polyhedra are said to be
equidecomposable if one of them can be decomposed into a finite number of
parts in ...
|... set of functions 4A2A, 465Xj equidecomposable (in 'G-equidecomposable') |
396Ya; (in 'G-r-equidecomposable') §395 (395 A); (in 'G-cr-equide- composable')
§448 (448A) equidistributed sequence (in a topological probability space) 281N,
|We show that any two rectangles of the same area are equidecomposable using |
translations alone. It is clear that a rectangle of size a x b is equidecomposable to
a rectangle of size (a/2) x (26) using two pieces and translations. Repeating this ...
|any two countably infinite sets of X are countably G-equidecomposable. In |
particular, any countably infinite subset of X is countably G-paradoxical. Exercise
2.2.1. Show that finite G-equidecomposability and countable G-
|Gardner's Theorem 8.1. Indeed. suppose that D and Q are equidecomposable |
using the translations t|,t3, t3. Replacing D by n (D) we may assume that t] is the
identity. Then t1, t2, t3 generate a discrete group. which contradicts Theorem 8.1.
|7s a regular tetrahedron in R3 equidecomposable to a cube using Lebesgue |
measurable pieces ? We may add the following question posed by the author in
Section 10.2 of . Problem 6.3. Is every polygon equidecomposable to a
|Paradoxical Sets Suppose that the group G acts on the set X. A subset A of X is G|
-paradoxical, or simply paradoxical, if it is equal to A1 ∪A2 for some pair A1, A2
of disjoint subsets of A, each of which is G-equidecomposable with A. The ...