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Square Dissection


Gardner showed how to dissect a square into eight and nine acute scalene triangles.

SquareDissectionIsosceles10

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

x_1^2+y_1^2=1
(1)
x_1^2+(1-y_1)^2=x_2^2
(2)
2(r_3-1)^2=(1-x_2)^2
(3)
(r_3-x_2)^2+(r_3-1)^2=(x_2-x_1)^2+(1-y_1)^2
(4)

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

x_1=0.64514...
(5)
y_1=0.76406...
(6)
x_2=0.68693...
(7)
r_3=0.77862....
(8)

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 Packing, Triangle Dissection

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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 Dissection

Cite this as:

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

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