Equidecomposability is also known as scissors congruence. Although it is believed that Euclid knew about equidecomposability, a theorem was not developed and proven until the early 1800's. Wolfgang Bolyai initiated the question as to whether or not polygons are equidecomposable, and in 1833, it was proven by P. Gerwien. This theorem is known as the Bolyai-Gerwien Theorem. It was later noted that William Wallace had actually proven equidecomposibility before Gerwien, in 1807. Thus, another name for this theorem is the Wallce-Bolyai-Gerwien Theorem.History The Wallace-Bolyai-Gerwien Theorem states that two polygon which have equal area are equidecomposable. Along with the theorem, we have the following five lemmas.
An interesting point about this theorem, is that the term polygon refers to not only simple closed figures, but also more complex polygons, such as figures with "holes".
- If a figure A is equidecomposable with a figure B, and figure B is equidecomposable with a figure C, then figures A and C are also equidecomposable.
- Every triangle is equidecomposable with some rectangle.
- Two parallelograms which have a common base and the same area are equidecomposable.
- Two rectangles having equal areas are equidecomposable.
- Every polygon is equidecomposable with some rectangle.
Although this theorem is correct for 2 dimensions, it does not hold for three dimensions. The question of the three dimensional equidecomposability is known as Hilbert's Third Problem. Hilbert posed this problem in 1900, and shortly after, Dehn proved that equidecomposability does not always hold for three dimensional figures.
Since the early 1800's, equidecomposability has become part of what is known as recreational mathematics. Once it was discovered that different types of polygons could be dissected and their pieces rearranged to form other polygons, the question of minimal dissection was posed.
Here is a list of the number of pieces needed for certain regualr polygons to form other regular polygons. It is important to note that the minimum number of pieces is an open question for each type of regular polygon. Therefore, some of the minimums may have recently changed from what is printed here.
Original Polygon New Polygon Number of Pieces Square Triangle 4 Pentagon Triangle 6 Square 6 Hexagon Triangle 5 Square 5 Pentagon 7 Heptagon Triangle 8 Square 7 Pentagon 9 Hexagon 8 Octagon Triangle 7 Square 5 Pentagon 9 Hexagon 8 Heptagon 11 Enneagon Triangle 8 Square 9 Pentagon 12 Hexagon 11 Heptagon 14 Octagon 13 Decagon Triangle 7 Square 7 Pentagon 10 Hexagon 9 Heptagon 11 Octagon 10 Enneagon 13 Dodecagon Triangle 8 Square 6 Pentagon 10 Hexagon 6 Heptagon 11 Octagon 10 Enneagon 14 Decagon 12 Golden Rectangle Triangle 4 Square 3 Pentagon 6 Hexagon 5 Heptagon 7 Octagon 6 Enneagon 9 Decagon 6 Dodecagon 7 Greek Cross Triangle 5 Square 4 Pentagon 7 Hexagon 7 Heptagon 9 Octagon 9 Enneagon 12 Decagon 10 Dodecagon 6 Golden Rectangle 5 Latin Cross Triangle 5 Square 5 Pentagon 8 Hexagon 6 Heptagon 8 Octagon 8 Enneagon 11 Decagon 10 Dodecagon 7 Golden Rectangle 5 Greek Cross 7 Maltese Cross Square 7 Hexagon 14 Greek Cross 8 Swastika Square 6 Hexagon 12 Greek Cross 8 Maltese Cross 9 Pentagram Triangle 7 Square 7 Pentagon 9 Hexagon 9 Heptagon 11 Octagon 10 Enneagon 14 Decagon 6 Dodecagon 12 Golden Rectangle 7 Greek Cross 10 Latin Cross 10 Hexagram Triangle 5 Square 5 Pentagon 8 Hexagon 6 Heptagon 9 Octagon 8 Enneagon 11 Decagon 9 Dodecagon 9 Golden Rectangle 5 Greek Cross 8 Latin Cross 8 Pentagram 11 Octagram Triangle 8 Square 8 Pentagon 9 Hexagon 9 Heptagon 12 Octagon 6 Enneagon 13 Decagon 12 Dodecagon 12 Golden Rectangle 7 Greek Cross 10 Latin Cross 11 Pentagram 13 Hexagram 10 In addition to minimal number of pieces, another current aspect of equidecomposability is puzzles containing several polygons. These puzzles will have several polygons already dissected and the person playing the puzzle must figure out how to put the pieces back together to form a certain polygon.
Overall, this topic in mathematics is fairly new and has become popular only recently. As time goes on, mathematicians are discovering new ways to dissect polygons, and with fewer pieces.