Equivalent and Equidecomposable Figures definition of Equivalent and Equidecomposable Figures in the Free Online Encyclopedia.

Equivalent and Equidecomposable Figures

Equivalent figures are plane or three-dimensional figures of identical area or volume. Two figures are equidecomposable if they can be subdivided into an equal number of pairwise congruent parts.

The concept of equidecomposability is usually applied only to polygons and polyhedrons. Equidecomposable figures are equivalent. The Hungarian mathematician J. Bolyai, in 1832, and the German mathematician P. Gerwin, in 1833, proved that equivalent polygons are equidecomposable, a result known as the Bolyai-Gerwin theorem. Any polygon can therefore be made into a square by dividing it into parts, which are then rearranged. The concept of equidecomposability is the basis of the decomposition method used to calculate the areas of polygons. Thus, a parallelogram can be reduced to a rectangle by subdivision and rearrangement, a triangle can be reduced to a parallelogram, and a trapezoid can be reduced to a triangle. The concept of equidecomposability is equivalent to that of equicomplementability, which is the basis of the completion method, wherein congruent parts are added to two figures in such a way that the resulting figures are congruent.

Equivalent polyhedrons are not always equidecomposable. For this reason, the method of exhaustion or some other method of disguised integration, for example, Cavalieri’s principle, is used to derive the formula for the volume of a triangular pyramid. According to Dehn’s theorem, a cube and a regular tetrahedron of equal volume are not equidecomposable. This theorem was proved by the German mathematician M. Dehn in 1901 and provides a negative answer to Hilbert’s third problem. In his proof, Dehn constructed a system of additive invariants that must be equal if the polyhedrons are to be equidecomposable. He showed that some of the invariants take on different values for the cube and for the regular tetrahedron of equal volume. These results were extended by the Swiss mathematician H. Hadwiger and his students. In particular, J. P. Siedler showed that the equality of the Dehn invariants of two polyhedrons is not only necessary but also sufficient for the polyhedrons to be equidecomposable.

REFERENCES

Problemy Gil’berta: Sb. Moscow, 1969.
Boltianskii, V. G. Ravnovelikie i ravnosostavlennye figury. Moscow, 1956.
Entsiklopediia elementarnoi matematiki, book 5. Moscow, 1966.

Mentioned in	?		Encyclopedia browser	?		Full browser	?
No references found			equitable equitangential curve equitant equitation equitempered scale Equites Equity equity crude equivalence equivalence class equivalence class partitioning equivalence classes equivalence element equivalence gate equivalence law of ordered sampling equivalence partitioning equivalence point equivalence principle Equivalence Relation equivalence transformation equivalence zone Equivalence, Principle of Equivalent equivalent absorption area equivalent airspeed Equivalent and Equidecomposable Figures equivalent angles equivalent annual rate equivalent barotropic model equivalent bending moment equivalent binary digits equivalent blackbody temperature equivalent circuit equivalent conductance equivalent continued fractions equivalent continuous sound level equivalent diameter equivalent duct diameter equivalent electrons equivalent elements equivalent embedment length Equivalent Equations equivalent evaporation equivalent focal length equivalent footcandle equivalent four-wire system equivalent head wind equivalent height equivalent inequalities equivalent magnetic length equivalent map projection			Equivalences Equivalences Equivalences Equivalencia Equivalencia de rendimiento equivalencies equivalencies equivalencies equivalencies equivalency equivalency equivalency equivalency equivalent equivalent equivalent equivalent equivalent Equivalent (chemistry) Equivalent (chemistry) Equivalent (chemistry) Equivalent a Temps Plein equivalent absorption area Equivalent Advertising Value Equivalent Agri-Environmental Planning Equivalent Air Density Depth Equivalent air depth equivalent air speed Equivalent airspeed Equivalent airspeed Equivalent and Equidecomposable Figures equivalent angles Equivalent Annual Annuities Equivalent annual annuity Equivalent Annual Average Daily Traffic Equivalent annual benefit Equivalent annual cash flow Equivalent Annual Cash Flows Equivalent annual cost Equivalent annual cost Equivalent annual cost Equivalent Annual Costs Equivalent Annual Rate Equivalent Annual Rate Equivalent Annual Value Equivalent Anode Dark Current Input Equivalent average Equivalent Axle Load Equivalent Background Input Equivalent Bandwidth equivalent barotropic model Equivalent baseband signal Equivalent baseband signal Equivalent Baseline Hours equivalent bending moment Equivalent Benefit Accrual Rate Equivalent Billing Unit equivalent binary digits equivalent blackbody temperature