corresponding curves with and interchanged. If is varied for a given (i.e. on a given parabola), integer values of occur relatively frequently if is a square or a small multiple of a square. If several such values happen to lie close together, the corresponding parabolas approximately coincide, and the triples cluster in a narrow parabolic strip. For instance, , , , and ; the corresponding parabolic strip around is clearly visible in the scatter plot. The angular properties described above follow immediately from the functional form of the parabolas. The parabolas are reflected at the -axis at , and the derivative of with respect to at this point is –1; hence the incidence angle is 45°. Since the clusters, like all triples, are repeated at integer multiples, the value also corresponds to a cluster. The corresponding parabola intersects the -axis at right angles at , and hence its reflection upon interchange of and intersects the -axis at right angles at , precisely where the parabola for is reflected at the -axis. (The same is of course true for and interchanged.) Albert Fässler and others provide insights into the significance of these parabolas in the context of conformal mappings. Special cases and related equations The Platonic sequence The case of the more general construction of Pythagorean triples has been known for a long time. Proclus, in his commentary to the 47th Proposition of the first book of Euclid's Elements, describes it as follows: Certain methods for the discovery of triangles of this kind are handed down, one which they refer to Plato, and another to Pythagoras. (The latter) starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side, the hypotenuse. ...For the method of Plato argues from even numbers. It takes the given even number and makes it one of the sides about the right angle; then, bisecting this number and squaring the half, it adds unity to the square to form the hypotenuse, and subtracts unity from the square to form the other side about the right angle. ... Thus it has formed the same triangle that which was obtained by the other method. In equation