Why is Perigal's dissection a proof of Pythagoras' Theorem?

Perigal's diagram

The diagram on the south side of Perigal's tomb.

The diagram redrawn.

Pythagoras' Theorem asserts that for a right-angled triangle, the area of the square on the hypotenuse (the longest side) is the sum of the areas of the squares on the smaller sides. Many people are familiar with at least one proof or another. Those which people often remember best, because the basic reasoning is most direct, are dissection proofs, which cut the larger square up into regions which can be reassembled to make up the smaller squares. The diagram on Perigal's gravestone illustrates a dissection proof apparently discovered by Perigal himself. It is distinguished mostly by its pleasing symmetry.

Perigal's proof shows that the larger square can be cut up and reassembled to make up the smaller ones. It is not so easy to see exactly why this is so.

Perigal's dissection when two sides are equal. In this case it is not difficult to show that the two green squares and all the red triangles are congruent.

It is not easy to justify Perigal's dissection in general (as will be done later on). But there is one case where it is easy to do so, and that is where the two smaller sides of the triangle are equal (and the ratio of the hypotenuse to side is ). This is the case apparently pictured on Babylonian tablets dating to about 1600 B.C. (thus among the earliest examples of geometrical diagrams known, and perhaps the earliest extant examples of mathematical reasoning). It is also that discussed by Plato in the dialogue Meno.

Why is Perigal's dissection a proof of Pythagoras' Theorem?

It may not be immediately evident from Perigal's own diagram how to carry out the details of a proof. The following pictures should explain how it goes.

1. The construction of the diagram

• Draw the triangle and the squares on its sides (Fig 1).
  
• Place a cross in the larger square on a side, centred at the centre of that square and aligned horizontally and vertically (Fig 2).
  
• Cross the square on the hypotenuse starting from halfway along it, running parallel to the short side (Fig 3).

Figure 1

Figure 2

Figure 3

• Draw three other similar lines by rotating this one (Fig 4).
  
• Erase the tips of these lines (Fig 5) ...
  
• ... to obtain Perigal's figure (Fig 6).

The quadrilaterals are congruent
It must then be shown that this actually does dissect the larger square into pieces that can be reassembled to make up the smaller ones.

• The shape in Fig 7 above is a parallelogram.
  
• Because of this and rotational symmetry, all the red segments in Fig 8 have the same length.
  
• Finally, in Fig 9, all corresponding sides are parallel. This implies that the marked quadrilaterals (and so, by symmetry, all the quadrilaterals) are congruent.

Figure 7

Figure 8

Figure 9

3. The squares are congruent

• In Fig 10 the red segments on the left are both equal to the green one on the right, hence equal to each other.
  
• Subtracting equal segments demonstrates that the squares are equal, as in Fig 11. This finishes the proof.

Figure 10

Figure 11

A generalization

A few years after Perigal died, it was observed that his proof is only one of an infinite family of similar ones. The way to obtain them starts with a Pythagoras tiling.

A Pythagoras tiling covers the plane with periodic copies of the squares on the sides of the right triangle.

Then overlay this with a tiling constructed from copies of the square on the hypotenuse. Different dissections correspond to different placements of this second tiling. Perigal's dissection is just one of the possibilities. One advantage of placing Perigal's figure in such a lattice tiling is that it makes checking the validity of his dissection much more transparent. Congruences can be verified by lattice translation.

Moving the tiny red control node gives different decompositions of the square on the hypothenuse into pieces which come from the squares on the sides.

One of the dissections in the family is the one attributed to Thabit Ibn Qurra, a ninth century Arabic mathematician. It has been discovered independently many times.

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Figure 4

Figure 5