Pythagoras' Theorem by Tessellation

This is a subtle and beautiful proof. It's quite easy to get an insight into why it works. The applet below is supposed to serve this purpose. The Pythagorean Theorem claims that a² + b² = c², where a and b are sides whereas c is the hypotenuse of a right-angled triangle. For the sake of the proof, we tasselate the plane into two grids. One is composed of the repeated pattern of two touching squares with sides a and b. The second grid consists of squares of side c parallel (and perpendicular) to the hypotenuse of the given triangle. These two grids have a common feature: they both are invariant under translation to the vector c parallel to the hypotenuse and of the same length. Note that this translation is equivalent to first translating the plane horizontally to the length of one side (say, a) and then vertically to the length of another (b). The insight might be gained when the two grids are superimposed as in the applet. Because of the above mentioned property the second grid cuts the first one into parts that may be combined in at least two different ways: to form two squares a² + b² or the bigger square c². (This is by no means a rigorous proof. See if you can fill in the details.)

The applet may be in two states. In the first you modify a right-angled triangle. In the second you drag the second grid to observe various "proofs" of the Pythagorean Theorem. All the operations are performed by dragging the mouse.


Please try switching to IE 11 (Windows) or Safari (Mac), for no other browser nowadays runs Java applets. If asked whether to allow the applet to load, click Yes - the applet is signed with a security certificate from a trusted company.


What if applet does not run?

Hubert Shutrick came up with a proof by dissection as depicted below:

Proof by dissection, Hubert Shutrick

The above applet leads to a continuum multitude of proofs by dissection. Here's the one related to Hubert Shutrick's construction:

Proof by dissection, Hubert Shutrick

Reference

  1. K. O. Friedrichs, From Pythagoras to Einstein, MAA, 1965

Related material
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  • Dancing Rectangles Model Auxetic Behavior
  • A Hinged Realization of a Plane Tessellation
  • A Semi-regular Tessellation on Hinges A
  • A Semi-regular Tessellation on Hinges B
  • A Semi-regular Tessellation on Hinges C
  • Escher's Theorem
  • Napoleon Theorem by Plane Tessellation
  • Parallelogram Law: A Tessellation
  • Simple Quadrilaterals Tessellate the Plane
  • Pythagorean Theorem By Plane Tessellation
  • Pythagorean Theorem By Hexagonal Tessellation
  • Hinged Greek Cross Tessellation
  • Pythagorean Theorem: A Variant of Proof by Tessellation
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