the right angle

It is often said that there are no right angles in nature, but this is not strictly true. Of course, there are no animals or plants with square corners, but the human visual field is determined by two lines that intersect at 90 degrees: the horizontal, as defined by the horizon (if you live on the coast or are aboard a ship at sea); and the vertical, as defined by gravity. The need to define these lines started when humans began to build structures in brick and stone.

Defining the vertical is easy—a plumb line does that job—but in order to define what is horizontal, it is necessary, somehow, to produce a right angle. The first civilization to do this was probably Sumer in the third millennium BC, but both the Babylonians and the Egyptians were able to construct right-angle triangles around 4,000 years ago. While the Egyptians used only the 3:4:5 triangle, they may have been aware of other number combinations, but the Babylonians certainly knew other ratios, which they calculated using a sexagesimal (base 60) system of arithmetic. This strikes me as being a seriously unwieldy way to perform calculations, but it’s why there are sixty seconds in a minute and sixty minutes in an hour, so it must have had some benefits.

Most people will be familiar with the theorem of Pythagoras—possibly the most famous theorem in the whole of mathematics—which states that in a right-angle triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides, as in this diagram, which illustrates the simplest of the ratios (3:4:5) that produce right-angle triangles:

However, I wondered whether I could work out any other ‘Pythagorean triples’. I already knew that the ratio 5:12:13 produced a right-angle triangle, but were there any more? That question turned out to be surprisingly easy to answer. First, it is not possible for the smallest value to be 2 or 4, and if the smallest value is 6, then the only combination that works is 6:8:10, in which the three numbers are not coprime (they have a common factor, 2). In other words, 6:8:10 is simply a multiple of 3:4:5 and therefore doesn’t count as a separate Pythagorean triple.

On the other hand, if I set the value of the smallest term at 7, then it’s straightforward to find two consecutive numbers, the squares of which have a difference of 49. At this point, I noticed that while the increases in the value of the smallest term were linear (3, 5, 7), the other terms increased more rapidly. Thus, 4, 12, 24. Could the middle term in the next triple be 40? It is. The left-hand column in the following table illustrates how far I went in identifying further triples, first calculating the appropriate values using the method I’ve just outlined, then confirming that the calculated values fitted the formula using the time-honoured method of long multiplication on the back of an envelope.

It seems to me that this is a series that will continue to infinity, and I could say the same about the right-hand column in the table, in which the difference between the two larger numbers is 2. This one works only when the smallest number is divisible by 4, because where the smallest number is divisible only by 2, the result is a multiple of one of the ratios in the left-hand column.

This is where things get more difficult. When I tried to find a ratio where the difference between the two largest numbers is 3, all I could come up with was multiples of other ratios that I’d previously discovered. At this point, I remembered that Jacob Bronowski had quoted, incredulously, the ratio 3367:3456:4825 in his landmark BBC TV series on the history of science, The Ascent of Man, as an indicator of the arithmetical prowess of the Babylonians 4,000 years ago. My first reaction was that this must be a multiple of a primitive Pythagorean triple, but I was amazed to find that the three terms are coprime (one of the prime factors of 4825 is 193).

This must have taken some calculating! It is clearly a primitive Pythagorean triple, but it immediately occurs to me that it may be part of a series like the simpler ratios described above. It was time to see what the internet had to say on the subject. The first page of my search included the statement that there are 16 Pythagorean triples in which the value representing the hypotenuse is less than 100. I’d already identified nine of these in the table above; here are the other seven:

I spotted immediately that the difference between the two larger numbers in the second and fourth ratios is 8, so I wondered whether they were part of the same sequence. They are, and this is another sequence that I’m assuming continues to infinity.

Notice that the difference between the two larger numbers in the first of these two tables is also 8, but the difference between the two smaller numbers is just 1, meaning that it is not part of the same sequence. However, it is difficult to identify a sequence from just one triple, especially when that triple is as large as the Babylonian example cited above.

I’ve concentrated so far on triangles where the values representing all three sides are integers, but there is what I assume to be an even larger category of Pythagorean triples involving irrational numbers. For example, if the lengths of the two sides enclosing the right angle are 2 and 3, respectively, then the length of the hypotenuse is the square root of 13, which is irrational (meaning that it cannot be represented by a fraction).

The general conclusion that I draw from my investigations is that there are an infinite number of ratios that meet the Pythagorean criteria, but I’m unable to explain why this particular juxtaposition of squares produces a right angle, or even what is special about 90 degrees that makes it the right angle.