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Dissection


Any two rectilinear figures with equal area can be dissected into a finite number of pieces to form each other. This is the Wallace-Bolyai-Gerwien theorem. For minimal dissections of a triangle, pentagon, and octagon into a square, see Stewart (1987, pp. 169-170) and Ball and Coxeter (1987, pp. 89-91). The triangle to square dissection (haberdasher's problem) is particularly interesting because it can be built from hinged pieces which can be folded and unfolded to yield the two shapes (Gardner 1961; Stewart 1987, p. 169; Pappas 1989; Steinhaus 1999, pp. 3-4; Wells 1991, pp. 61-62).

DissectionTriangleSquare

Laczkovich (1988) proved that the circle can be squared in a finite number of dissections (∼10^(50)). Furthermore, any shape whose boundary is composed of smoothly curving pieces can be dissected into a square.

The situation becomes considerably more difficult moving from two dimensions to three dimensions. In general, a polyhedron cannot be dissected into other polyhedra of a specified type. A cube can be dissected into n^3 cubes, where n is any integer. In 1900, Dehn proved that not every prism can be dissected into a tetrahedron (Lenhard 1962, Ball and Coxeter 1987). The third of Hilbert's problems asks for the determination of two tetrahedra which are not equidecomposable by dissection into congruent tetrahedra directly or by adjoining congruent tetrahedra. Dehn (1900, 1902) showed this could not be done, and Kagan (1903) obtained the same result independently shortly thereafter. A quantity growing out of Dehn's work which can be used to analyze the possibility of performing a given solid dissection is the Dehn invariant.

The table below is an updated version of the one given in Gardner (1991, p. 50). Many of the improvements are due to G. Theobald (Frederickson 1997). The minimum number of pieces known to dissect a regular n-gon (where n is a number in the first column) into a k-gon (where k is a number is the bottom row) is read off by the intersection of the corresponding row and column. In the table, {n} denotes a regular n-gon, GR a golden rectangle, GC a Greek cross, LC a Latin cross, {5/2} a five-point star (solid pentagram), {6/2} a six-point star (i.e., hexagram or filled star of David), and {8/3} the solid octagram.

There is some debate as to the permissibility of flipping pieces. While it is reasonable to prefer an unflipped dissection over a flipped one if both use the same number of pieces, it is also reasonable to separately list the best known dissections flipped and unflipped when the number of pieces differ (G. Frederickson, pers. comm. to G. Theobald). The following table therefore indicates such dissections as flipped/unflipped if a dissections involving one or more flipped pieces is known that uses fewer pieces that the best known unflipped dissection.

{3}{4}{5}{6}{7}{8}{9}{10}{12}GRGCLC{5/2}{6/2}
{4}4
{5}66
{6}557
{7}8798
{8}758/9810/11
{9}891010/111312
{10}7798/9111013
{12}86106111013/1411/12
GR436576967
GC547799111065
LC5586881010757
{5/2}779911101461271010
{6/2}558698119958810
{8/3}8898/9126131212710111310

Wells (1991) gives several attractive dissections of the regular dodecagon. The best-known dissections of one regular convex n-gon into another are shown for n=3, 4, 5, 6, 7, 8, 9, 10, and 12 in the following illustrations due to Theobald.

Dissections3-7Dissections8-9Dissections10Dissections12

The best-known dissections of regular concave polygons are illustrated below for {5/2}, {6/2}, and {8/3} (Theobald).

Dissections52Dissections62Dissections83

The best-known dissections of various crosses are illustrated below (Theobald).

DissectionsCrosses

The best-known dissections of the golden rectangle are illustrated below (Theobald).

DissectionsRectangle

See also

Banach-Tarski Paradox, Blanche's Dissection, Cundy and Rollett's Egg, Decagon, Dehn Invariant, Diabolical Cube, Dissection Fallacy, Dissection Proof, Dissection Puzzles, Dodecagon, Ehrhart Polynomial, Equidecomposable, Equilateral Triangle, Golden Rectangle, Heptagon, Hexagon, Hexagram, Hilbert's Problems, Latin Cross, Maltese Cross, Nonagon, Octagon, Octagram, Pentagon, Pentagram, Polyhedron Dissection, Pythagorean Square Puzzle, Pythagorean Theorem, Rep-Tile, Soma Cube, Square, Star of Lakshmi, Swastika, T-Puzzle, Tangram, Wallace-Bolyai-Gerwien Theorem

Portions of this entry contributed by Gavin Theobald

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References

Ball, W. W. R. and Coxeter, H. S. M. Mathematical Recreations and Essays, 13th ed. New York: Dover, pp. 87-94, 1987.Coffin, S. T. The Puzzling World of Polyhedral Dissections. New York: Oxford University Press, 1990.Coffin, S. T. and Rausch, J. R. The Puzzling World of Polyhedral Dissections CD-ROM. Puzzle World Productions, 1998.Cundy, H. and Rollett, A. Ch. 2 in Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., 1989.Dehn, M. "Über raumgleiche Polyeder." Nachr. Königl. Ges. der Wiss. zu Göttingen f. d. Jahr 1900, 345-354, 1900.Dehn, M. "Über den Rauminhalt." Math. Ann. 55, 465-478, 1902.Eppstein, D. "Dissection." http://www.ics.uci.edu/~eppstein/junkyard/dissect.html.Eppstein, D. "Dissection Tiling." http://www.ics.uci.edu/~eppstein/junkyard/distile/.Eriksson, K. "Splitting a Polygon into Two Congruent Pieces." Amer. Math. Monthly 103, 393-400, 1996.Frederickson, G. Dissections: Plane and Fancy. New York: Cambridge University Press, 1997.Frederickson, G. N. Hinged Dissections: Swinging & Twisting. New York: Cambridge University Press, 2002.Gardner, M. "Mathematical Games: About Henry Ernest Dudeney, A Brilliant Creator of Puzzles." Sci. Amer. 198, 108-112, Jun. 1958.Gardner, M. The Second Scientific American Book of Mathematical Puzzles & Diversions: A New Selection. New York: Simon and Schuster, 1961.Gardner, M. "Paper Cutting." Ch. 5 in Martin Gardner's New Mathematical Diversions from Scientific American. New York: Simon and Schuster, pp. 58-69, 1966.Gardner, M. The Unexpected Hanging and Other Mathematical Diversions. Chicago, IL: Chicago University Press, 1991.Hunter, J. A. H. and Madachy, J. S. Mathematical Diversions. New York: Dover, pp. 65-67, 1975.Kagan, B. "Über die Transformation der Polyeder." Math. Ann. 57, 421-424, 1903.Keil, J. M. "Polygon Decomposition." Ch. 11 in Handbook of Computational Geometry (Ed. J.-R. Sack and J. Urrutia). Amsterdam, Netherlands: North-Holland, pp. 491-518, 2000.Kraitchik, M. "Dissection of Plane Figures." §8.1 in Mathematical Recreations. New York: W. W. Norton, pp. 193-198, 1942.Laczkovich, M. "Von Neumann's Paradox with Translation." Fund. Math. 131, 1-12, 1988.Lenhard, H.-C. "Über fünf neue Tetraeder, die einem Würfel äquivalent sind." Elemente Math. 17, 108-109, 1962.Lindgren, H. "Geometric Dissections." Austral. Math. Teacher 7, 7-10, 1951.Lindgren, H. "Geometric Dissections." Austral. Math. Teacher 9, 17-21, 1953.Lindgren, H. "Going One Better in Geometric Dissections." Math. Gaz. 45, 94-97, 1961.Lindgren, H. Recreational Problems in Geometric Dissection and How to Solve Them. New York: Dover, 1972.Madachy, J. S. "Geometric Dissection." Ch. 1 in Madachy's Mathematical Recreations. New York: Dover, pp. 15-33, 1979.Pappas, T. "A Triangle to a Square." The Joy of Mathematics. San Carlos, CA: Wide World Publ./Tetra, pp. 9 and 230, 1989.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Stewart, I. The Problems of Mathematics, 2nd ed. Oxford, England: Oxford University Press, 1987.Theobald, G. "Geometric Dissections." http://home.btconnect.com/GavinTheobald/Index.html.Weisstein, E. W. "Books about Dissections." http://www.ericweisstein.com/encyclopedias/books/Dissections.html.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 56-57 and 243-244, 1991.

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Theobald, Gavin and Weisstein, Eric W. "Dissection." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Dissection.html

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