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Pythagoras: a tiling with the pythagorean theorem

Very little is known for certain about Pythagoras. Probably he was born on Samos around 580 BC. He founded a school in Croton, a Greek colony in what is now southern Italy.

You can play with a demonstration of the Pythagoras' theorem inspired in Euclid in the next link:

Pythagoras Theorem: Euclid's demonstration
Demonstration of Pythagoras Theorem inspired in Euclid.

Here we show Pythagoras' theorem in a tiling. This tiling can be seen in floors, for example.

Pythagorean mosaic on a kitchen in Sabayes (Huesca, Spain)
Theorem of Pythagoras, Pythagorean Theorem in a tiling | matematicasvisuales

This tiling is made using two kind of squares (green and blue). These are the squares of the two legs of a right triangle.

We can draw a chessboard tesselation which is made of squares that are the squares of the hypotenuse.

Then we can see the right triangle:

Theorem of Pythagoras, Pythagorean Theorem in a tiling | matematicasvisuales

"It is intuitively clear that the two tiles (big square and union of two smaller squares) must have the same area. Indeed, this statement is simply the Pythagorean theorem, and it has been conjectured that it was discovered by a contemplation of tiling designs. (...). From an axiomatic point of view, this proof is a particularly economical one since it shows that the two smaller squares can be dissected into a bounded number of pieces which then can be put together in such a manner that they just fill the big square." (Magnus, p. 53)

The simplest case is about an isosceles right triangle. Then the squares on the two legs are equal. These two squares are made of four congruent triangles (half a square) that fit together to form the square on the hypotenuse.

Theorem of Pythagoras, Pythagorean Theorem in a tiling | matematicasvisuales

Using a different grid we can see another proof by dissection due to Henry Perigall (1801-1898).

Theorem of Pythagoras, Pythagorean Theorem in a tiling | matematicasvisuales

In this variation, the four pieces are not congruent.

Theorem of Pythagoras, Pythagorean Theorem in a tiling | matematicasvisuales


John Stillwell, "Mathematics and its History", Springer-Verlag, New York, 2002.
Euclides, The Elements
W. Magnus, "Noneuclidean Tesselations and Their Groups", Academic Press, New York-London, 1973.
Alexander Bogomolny, Cut the Knot. Pythagorean theorem.
H.S.M. Coxeter, 'Introduction to Geometry', John Wiley and Sons, Second edition, pp. 8-9.
Martin Gardner, 'Sixth Book of Mathematical Diversions from "Scientific American"'. Scribner, 1975.
Eli Maor, "The Pythagorean theorem: a 4000-year history", Princeton University Press, United States of America, 2007.
Bill Casselman, On the dissecting table. Henry Perigall, in +Plus Magazine (
Greg N. Frederikson, "Dissections, Plane and Fancy", Cambridge University Press, United States of America, 1997.
F.J. Swetz and T.I. Kao, "Was Pythagoras chinese?", The Pennsylvania State University Press, United States of America, 1977.


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