## Proofs of Pythagoras' Theorem using translations

Conceptually, one of the most attractive ways to' prove Pythagoras' Theorem is to find a partition the three squares into smaller regions, with the property that the partition of the large square is assembled from regions in the partitions of the two other squares. Such proofs are among the oldest known. Some of them turn out to be closely related to the straightforward `algebraic' proofs.

There is in fact a general and elementary result about areas in the plane (apparently proven first around 1900) which asserts that if two polygonal figures in the plane have the same area, then one can find piecewise congruent decomposition of the two figures. That is to say, on grounds of general principle one knows that if Pythagoras' Theorem is true, one can find the sort of partitions we are looking for.

### Reference

Heinz Hopf, Springer Lecture Notes in Mathematics #1000.

David Hilbert, Foundations of Geometry, Open House, 1994.

### Proof by algebra

This has apparently been found independently many times.

On each side we have a square of side a + b. On the left inscribed in it is a square of side c, and four copies of the original right triangle. The area of the square on the left is therefore

c2 + 4 (area of triangle)

On the right we have the two smaller squares of Pythagoras' Theorem and the same four triangles. The area is therefore

a2 + b2 + 4 (area of triangle)

If we subtract the area of the four triangles from each of these expressions, we get

c2 = a2 + b2

### A related dissection

The algebraic argument above leads to a curious dissection of the large square into regions which can be assembled to give the small squares - what might be called a dissection proof. It starts by laying one diagram over the other, and then shifts pieces around.

### A Chinese version

This diagram, perhaps even with its own (different) colouring scheme, comes from a classic Chinese work The Chou Pei Suan Ching, probably dating to about 300 B.C.

The algebra here asserts that

c2 = 4(ab/2) + (b-a)2 = a2 + b2.

Tufte's book (Envisioning Information, p. 84) has a representation in the colours attributed to the figure in about 200 A.D. (This is as far as I know the earliest known use of colour in mathematical diagrams.)

This proof also gives rise implicitly to a dissection proof, attributed (see Heath, volume I, p.64) to Thabit ibn Qurra from about 900 A.D.

Here is a version of this explained by Michael Buckland, an undergraduate, to a Geometry course at UBC:

Click in control node to adjust size and reset the figure. Click anywhere else to start and stop the animation.

Reference Jean Claude Martzloff, A history of Chinese mathematics, Springer 1997.

### A symmetric partition

One of the most elegant proofs by dissection paritions the large square by first translating a copy of the small square to its centre, then partitioning the residue in a rotationally invariant manner. The proof ought to be straightforward from the figure on the right.

Reference:

... in Mr. Perigal's elegant proof ... the four parts into which he has divided the greater side are equal in all respects, so that the division of the squares is symmetrical. In this consists chiefly the elegance of the construction, which is in this respect ... unique. J. W. L. Glaisher in an addendum to Perigal's note.

### The Pythagorean tilings

It was apparently Paul Mahlo who first pointed out that one could obtain a continuous family of dissection proofs of Pythagoras' Theorem by overlaying two tilings of the plane. This was discovered independently by at least a few others, notably Major MacMahon and Felix Bernstein.

Both coloured nodes move.

References:

Felix Bernstein, `Der Pythagoraische Lehrsatz', Zeitschriften fur Mathematischen und Naturwissenschaftlichen Unterricht 55 (1924), pages 204 - 207.

K. O. Friedrichs, From Pythagoras to Einstein, M. A. A. New Mathematical Library, pages 8 - 12.

Major P. MacMahon, `Pythagoras's Theorem as a repeating pattern', Nature 109 (1922), page 479.

Paul Mahlo, `Topologische Untersuchungen uber Zerlegung in ebene und spharische Polygone', Dissertation, Halle, 1908.

Greg N. Frederickson, Dissections: Plane & Fancy, Cambridge University Press, 1997, pages 28 - 31. (I wish to thank Frederickson for pointing out the work of Mahlo, MacMahon, and Bernstein.)

### Leonardo's proof

The following figure is attributed (Heath, p. 365, Loomis, p. 129) by legend to Leonardi da Vinci.

### Exercise

One way to decompose the large square into two regions, each one matching areas with one of the smaller squares. Of course the rectangles in the large square are not themselves squares. According to the principle asserted at the beginning of this page, however, one can decompose each of these rectangles and each of the smaller squares into matching congruent regions. Describe how to do this. This is not so easy, because the number of regions required will depend on the exact triangle one starts with. It will help to refer to the discussion of shears, where a similar question is answered for parallelograms with the same base but different heights.