Square Dissection
Gardner showed how to dissect a square into eight and nine acute scalene triangles.
W. Gosper discovered a dissection of a unit square into 10 acute isosceles triangles, illustrated above (pers. comm. to Ed Pegg, Jr., Oct 25, 2002). The coordinates can be found from solving the four simultaneous equations
for the four unknowns (x_1,y_1,x_2,r_3) and picking the solutions for which 0<x_1,y_1,x_2,r_3<1. The solutions are roots of 12th order polynomials with numerical values given approximately by
Pegg has constructed a dissection of a square into 22 acute isosceles triangles.
Guy (1989) asks if it is possible to triangulate a square with integer side lengths such that the resulting triangles have integer side lengths (Trott 2004, p. 104).
See also
Mrs. Perkins's Quilt, Perfect Square Dissection, Square, Square Packing, Square Tiling, Triangle DissectionExplore with Wolfram|Alpha
More things to try:
References
Guy, R. K. In Number Theory and Applications (Ed. R. A. Mollin). Dordrecht, Netherlands: Kluwer, 1989.Pegg, E. Jr. "22 Acutes." http://www.mathpuzzle.com/22acuteisos.gif.Trott, M. The Mathematica GuideBook for Programming. New York: Springer-Verlag, p. 104, 2004. http://www.mathematicaguidebooks.org/.
Referenced on Wolfram|Alpha
Square DissectionCite this as:
Weisstein, Eric W. "Square Dissection." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SquareDissection.html