Chartres and its cathedral

Chartres is a town of some 50 000 people lying on and above the river Eure about 100 km south west of Paris. The town is surrounded by flat agricultural land, and is dominated by its magnificent Gothic cathedral, one of the most beautiful and best preserved in France. The present cathedral was consecrated in the year 1260, replacing the old church on the site which had suffered several devastating fires. The facade is flanked by two great towers between which is featured a large and beautiful rose window. Below this window is the Royal Doorway (Portail Royal): three entrances which are decorated with statues. The arch above the right door contains statues representing figures from the past who have made great contributions to learning; these include Aristotle, Cicero, Euclid, Ptolemy, and Pythagoras. The figure of Pythagoras sits at the foot of the arch on the right hand side of the door.   
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The statue

The figure of Pythagoras appears to be writing on a small lap-top desk, but it is more likely that the object in his lap is a stringed instrument. It is curious that the figure is left-handed: there is some evidence to suggest that the proportion of left-handed mathematicians is rather more than the proportion of left-handed people in a random population.

It is fitting that Pythagoras should be given a place in this beautiful cathedral. His contributions to geometry and music find parallels in the structure of this building and the use to which it is put.

Further investigation ...

Levis-Godechott, N. (1987), Chartres, révélée par sa sculpture et ses vitraux, Zodiaque (in French).

Bunt, L. N. H., Jones, P. S., Bedient, J. D. (1976), The historical roots of elementary mathematics, Prentice-Hall.

http://www.er.uqam.ca/nobel/r14310/Ptolemy/Chartres.html































Pythagoras and music

Pythagoras lived around 540 BC. He found that the major musical tones can be produced by shortening a string by simple whole number ratios. Thus if a string of length 12 gives C in the scale, then a string of length 9 will give F, a string of length 8 will give G, and a string of length 6 will give the octave C above.

The position of the Pythagoras figure

 Investigate In the modern scale, the lengths of the strings representing descending semi-tones are in the ratio 122. Starting with the length 0.5 (octave C) use your calculator to determine the lengths of the strings for the 12 notes below (B, A#, A, G#, G, F#, F, E, D#, D, C#, C). Compare the values of Pythagoras; they are surprisingly close.

  Investigate We say that b is the harmonic mean of a and c if . Pythagoras applied the harmonic mean to the string lengths of C and G to produce E, to F and C to get A, and to A and C to get B. Use length 1 for bottom C, find these values, and compare them with the modern scale lengths.

  Project  Use your library and the Internet to discover more about the mathematics of music.

























The theorem of Pythagoras

The theorem of Pythagoras about the side lengths of a right-angled triangle must be the best-known result in geometry. With the notation of the figure, the theorem claims that the sum of the areas of the squares on the two shorter sides of the triangle is equal to the area of the square on the long side (hypotenuse). Equivalently, a2 + b2 = c2.

 Investigate Assuming the truth of Pythagoras’ theorem, find some values of a, b and c which occur as lengths of right-angled triangles.

 Investigate Four right-angled triangles are placed to form a square of side a + b, as in the diagram. Establish the formula a2 + b2 = c2 by calculating the area of this square in two different ways.

Pythagoras’ theorem enabled early builders to construct right angles with accuracy. For example, a rope loop of length 12 units could be easily placed as a 3–4–5 triangle

 






















Pythagorean triples

A set of numbers {a, b, c} which satisfies the equation a2 + b2 = c2 is called a Pythagorean triple.

We might ask whether there are formulae for a, b and c which determine all such triples. The Pythagoreans made some progress in finding such a formula, but it was Euclid (about 300 BC) who found the following simple result.

If u and v are integers, and if x = u2 – v2, y = 2uv, and z = u2 + v2, then x, y and z are integers such that x2 + y2 = z2. Furthermore, all Pythagorean triples can be generated in this way.

Detail from the fresco
‘The school of Athens’ by Raphael

 Investigate  Show that if x, y and z are defined as above, then x2 + y2 = z2.  For the Pythagorean triples you found earlier, can you find values of u and v which determine them?