The new Pythagoras

Martin Hansen

One of mathematics' best known theorems is that which is attributed to Pythagoras.
Pythagoras, however, had a rival, named Pyfudgerous.
He too had a theorem for working out the hypotenuse of
a right-angled triangle from the length of the other two sides.

I am confident that readers will have no difficulty, whatsoever, in working out the hypotenuse of a right-angled triangle, if given measurements of the other two sides. To prove the point, let us say that one of the shorter sides in such a triangle is of length 9 cm and that the other is of length 40 cm. What is the length of the hypotenuse ?
 

Figure 1

 
Figure 1 contains the theorem that you probably used, namely:
 

 

 
Figure 1 also features a second theorem that I suspect is new to you. It too is a theorem for working out the length of the hypotenuse of a right-angled triangle from the lengths of the other two sides.
This pythagorean alternative I have named Pyfudgerous because I do not want you to trust it !
 

Test it

Try substituting in A = 9 cm and B = 40 cm to check that it gives H = 41 cm as Pythagoras' theorem just did. In fact, it does give the anticipated answer, but perhaps I have rigged it that way. ( I just might ! ) So try putting in some other Pythagorean integer triples such as 3:4:5 or 5:12:13 or 8:15:17 or 7:24:25. Integer triples are particularly useful here because we want to know it it is working exactly.
 

The truth

A quick spree of button pressing should reveal that the theorem of Pyfudgerous seems to be delivering the goods. In fact, it will always work, as can easily be proved by expanding the brackets and cancelling:
 

 

 
The truth is that the theorem of Pyfudgerous is simply the theorem of Pythagoras in an altered form. This is a sad moment. Something new and intriguing has turned out to be something old in a repackaged form.
 

The repackaging idea

This repackaging idea is not new. How one chooses to write a quadratic equation depends upon what aspect is to be emphasised. For example, I might rewrite the equation of the curve
 

 

 
in factorised form as
 

 

 
if it is the x-axis crossing points that are of interest or in the completed square form as
 

 

 
when it is the minimum point that is under consideration. Similarly, although the theorem of Pyfudgerous is not a piece of fundamentally new mathematics, it may still be of some significance.
 

Question time

Here is a problem for which Pyfudgerous, rather than Pythagoras, is tailor-made.
 
What is the length of the hypotenuse of a right-angled triangle in which the difference between the two shorter sides is 97 and the product of the two shorter sides is 1680 ?
 
( I hope that you are busily working out a solution at this point, before reading on ! )
 
The Pyfudgerean solution is effortless, for the question, in effect, states that (B - A) = 97 and AB = 1680. So, double the 1680 and add on a squared 97. Now square root and up pops the answer: H = 113. Try out this problem on a friend. I have had many a happy moment watching two simultaneous equations awkwardly being solved from which it transpires that A = 15 and B = 112 and then, via Pythagoras, being informed that H = 113.
In fact, if it is required to find A and B, there is a further clever hop in the spirit of Pyfudgerous using the fact that
 

 

 
to quickly find (B + A) = 127, which can be very easily solved simultaneously with (B - A) = 97 to give the correct solutions.
 

Picturing theorems

Figure 2 is an illustration of how the theorem of Pythagoras might have been spotted by looking hard at a tiled floor.
 

 

Figure 2

 
The grey square formed upon the hypotenuse of the black triangle contains the same number of tiles (i.e has the same area as) the two grey squares formed upon the other sides of the black triangle.
 

 

Figure 3

 
Figure 3 is a picture from which it is clear that the total area, (B + A)², is made up from four rectangles of dimension A by B plus a square of side (B - A). This is the previously used Pyfudgerean hop 
 

 

 
Likewise, Figure 4 is a visual representation of the theorem of Pyfudgerous.
 

 

Figure 4

 
The large square of side H is of the same area, H², as the four triangles, ½AB each, plus the small square, (B - A)².
Hence:
 

 

 
One of the interesting features of Figure 4 is the fact that it is a square expressed in terms of a ring of four triangles plus a smaller square. There must be some reduction factor, r, that will shrink Figure 4 into its own central square.
 

 

Figure 5

 
Figure 5 shows one such shrinkage and Figure 6 shows an infinity of such stepped shrinkages.
 

 

Figure 6

 
This is a fascinating picture. It spirals inwards and indefinite zooming in upon the centre reveals an unending, repeating structure. I am going to call Figure 6 the fractile of Pyfudgerous. ( A fractile is a fractal that tiles the plane )
The mathematical principle behind this process is outlined in Box 1.
 

 

Box 1

 
Notice that it could be applied to any ringed square such as Figure 3, for example. In fact, it can be applied to any ringed shape, as illustrated by Figures 7 and 8. Box 1 reveals that all such diagrams represent the sum to infinity of a geometric progression, a topic tackled early on in all A-level mathematics courses.
 

 

Figure 7

 

 

Figure 8

 


Mathematics Review, November 1993, Volume 4, Number 2.
© Philip Allan Publishers Limited. ISSN 0957-1280

 

 
The following Figure was submitted with the article but not published. 

 

Bonus Figure not used by the publisher

 

 

 

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