The secret life of the triangle

Image Credit: Michael Taylor

Think of a triangle. What immediately comes to mind? For me it was a chunk of Toblerone. You may have thought of a Doritos tortilla chip, a hang glider, an A-frame, a step-ladder or even the great and mysterious Pyramids of Giza, or the one at the entrance of the Louvre. Perhaps you thought more abstractly and imagined the mysterious Bermuda Triangle and lost ships, a tetrahedron , an isosceles, or an equilateral? The triangle has many secrets. I want to tell you a story about my new found love for the triangle. My journey started with Pythagorus, a simple right-angled triangle and Fermat’s Last Theorem. My triangle soon turned into a generator of waves and even a tool for measuring the curvature of the universe. But before I begin, let’s try an experiment. Look around your room and look for a triangle. Easy, right? Considering that it is one of the most basic of shapes in geometry, why are they almost nowehere to be seen? Where did all the triangles go? This is their (unfinished) story.

A triangle is a polygon with three corners (vertices) and three sides (edges). In 2D Euclidean geometry any 3 non-collinear (not in a straight line) points determine a unique triangle – and they come in many varieties – equilaterals, isosceles, scalenes, right-angled, acute and obtuse-angled, depending on how many equal length sides or angles they have, and on the sizes of their angles:

Image Credit: Michael Taylor

There is a lot of diversity. In fact there is an infinite number of different triangles. There is one case though where the species has only one member and that is the equilateral. It will always have three 60 degree angles no matter how big or small it is. Each visually-different (by size) equilaterial is simply a self-similar and zoomed version of another. The most interesting species though is the Pythagorean triple right-angled triangle whose sides are all whole numbers. It was this triangle that took me on a journey I could never have imagined. Pythagorus, the father of trigonometry, discovered that the square of the length of the longest side of a right-angled triangle (the hypotenuse) c was equal to the sum of the squares of the lengths of the other two sides (called the “legs”) a and b through his formula c²=a²+b²:

Image Credit: Michael Taylor

So, for example, 5²=3²+4² since 25=9+16. A nice whole number triplet (3,4,5). There are many more Pythagorean triples. For right-angled triangles whose hypotenuse is less than 100 in length, there are 16 of them: (3 , 4 , 5) ,  (5, 12, 13),  (7, 24, 25), (8, 15, 17), (9, 40, 41), (11, 60, 61), (12, 35, 37), (13, 84, 85), (16, 63, 65), (20, 21, 29), (28, 45, 53),  (33, 56, 65), (36, 77, 85), (39, 80, 89), (48, 55, 73) and (65, 72, 97)! There is a sub-species of pythagorean triple triangles called Heronion triangles after Hero of Alexandria – the father of alternative energy. It was Hero who invented the windwheel for harnessing the power of the wind on land and also the aelolipile for generating mechanical energy from steam. Heronian triangles are right-angled triangles whose area is a whole number. It turns out that all Pythagorean triples are Heronian but not all Heronians are Pythagorean triples. For example, the right-angled triangle (3,4,5) is a Pythagorean triple (5²=3²+4²) and also Heronian (area=6). But the triangle (13,14,15) is Heronian (area=84) but not a Pygthagorean triple (15²≠13²+14²). In the 3rd century BC, the Greek mathematician Diophantus wrote a collection of 130 algebraic problems giving numerical solutions of determinate equations (those with a unique solution) in his book Arithmetica. Problem II.8 of the Arithmetica asks how to split a given square number into two other squares? The answer had to wait almost two millennia until the year 1637 when Pierre de Fermat wrote (in Latin)  in the column of his copy of the Arithmetica next to Diophantus’ sum-of-squares problem:

“Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet” — Pierre de Fermat, c.1637

which roughly translates as “I have discovered a truly marvelous proof that it is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers. This margin is too narrow to contain it“. Wow. His “proof” was lost for all antiquity with his passing. However, in 1994, the English mathematician Andrew Wiles, after 7 years of secrecy, published the proof of this, Fermat’s Last Theorem. So, Pythagorus’s Theorem works for powers of two but not for higher powers. In 2007, mathematician Sam Daoud plotted Pythagorean triples for right-angled triangles with non-hypotenuse side-lengths up to 4500 – a staggering total of 11730 value pairs. What is remarkable is that there are some obvious and some less-obvious patterns which are explainable. For example, whenever the legs (a,b) of a Pythagorean triple appear in the plot, all integer multiples of (a,b) must also appear in the plot. This produces the appearance of lines radiating from the origin. There are also sets of parabolic patterns with a high density of points and all their foci at the origin:

Image Credit: Adapted from Sam Daoud

But what about triangles generating waves? My story now leaves ancient Greece and Medieval France, winding forward another 200 years and crosses the Atlantic Ocean to the USA in between 1880 and 1884. It was then that engineer Josiah Willard Gibbs developed Vector Calculus. A vector is a mathematical object having two properties: length and direction. Imagine a vector of length 1 unit lying along the x-axis. If it is rotated anti-clockwise, it will sweep out the arc of a circle of radius 1 whose angle is measured from the x-axis. After rotating 2π radians (360 degrees) it will have traced out a full circle and will be back to where it started. Whenever the tip of the vector is NOT aligned with the x-axis or the y-axis, then the vector is the hypotenuse (length=1) of a right-angled triangle centered at the origin. It turns out that the (x,y) coordinates of the tip of such vectors are equal to the Sine and Cosine (respectively) of the angle it makes with the positive x-axis. In this way, every point will have a unit vector associated with it. The circle traced out by the unit vector is called the unit circle. The numbers in brackets are the (x,y) coordinates of the tip of the unit vector.

Image Credit: Michael Taylor

This is the key to understanding how the right-angle triangle generates waves. If, on a new plot, you graph the angle (θ) that the unit vector sweeps through during a full cycle along the x-axis, and the x-coordinate of the tip of the unit vector along the y-axis, then what you get is a cosine function between 0 and 2π. If you continue sweeping the unit vector round the circle then the cosine function continues its next cycle. Every additional full sweep of the unit vector corresponds to one further complete cycle of the cosine function. Likewise, if you graph the angle (θ) that the unit vector sweeps through during a full cycle along the x-axis, and the y-coordinate of the tip of the unit vector along the y-axis, then what you get is a sine function between 0 and 2π:

Image Credit: Michael Taylor

We can now see how Pythagorus’s right-angle triangles with unit vector hypotenuses generate periodic sine and cosine waves. When I went to school in England, we learned a mnemonicSOHCAHTOA. It was a way to remember how to work out the sine, cosine or tangent of one of the acute angles in a right-angled triangle:

Image Credit: Michael Taylor

As we can see, it wasn’t an accident that right-angled triangles and sine or cosine waves are related. On the contrary. One follows from the other, as night follows day. But what about those other weird “trig” functions we hear about or come into contact with? You know the ones I mean – Tan, Cot, Cosec and Sec. Guess what, they also come from the right-angled triangle and the unit circle. It is pure geometry. Here is how:

Image Credit: Michael Taylor

Even if you are not familiar with (or can’t for the life of you remember) the trigonometric identities, knowing only that:

  • cot(θ)=1/tan(θ)
  • cosec(θ)=1/sin(θ)
  • sec(θ)=1/cos(θ)

then you should be able to use the diagram above (together with Pythagorus’s Theorem and SOHCAHTOA) to work out the whole keboodle of relations in the following table:

Image Credit: Michael Taylor

Ok, so now we have a pretty good trigonometric toolbox for working with flat 2D triangles. Now things really become magical. The story, until now has spanned the Atlantic Ocean and nearly two millenia. The latest chapter was written on the verge of the new millenium on the 1st of August, 1999 by 4 cosmologists: N. Bahcall (Princeton), J.P. Ostriker (Princeton), S. Perlmutter (LBNL), and P. J. Steinhardt (Princeton) who submitted a landmark paper to the free online repository arxiv physics entitled “The Cosmic Triangle: Revealing the State of the Universe“. Their triangle, plotted as the intersection of three axes representing the amount of matter in the universe, its expansion rate and its curvature, allowed for a way of representing the past, present, and future status of our cosmos. Using recent astrophysical observations, the cosmic triangle suggests that our current universe is lightweight, accelerating, and flat:

Image Credit: After The Cosmic Triangle, Bahall et al 1999.

“Figure 1: The Cosmic Triangle represents the three key cosmological parameters Ωm, ΩΛ, and Ωk – where each point in the triangle satisfies the sum rule m+Λ+k = 1. The blue horizontal line (marked Flat) corresponds to a flat universe (m+Λ = 1), separating an open universe from a closed one. The red line, nearly along the Λ=0 line, separates a universe that will expand forever (approximately Λ>0) from one that will eventually recollapse (approximately Λ<0). And the yellow, nearly vertical line separates a universe with an expansion rate that is currently decelerating from one that is accelerating. [CMB=Cosmic Microwave Background and CDM=Cold Dark Matter]” — The Cosmic Triangle, Bahall et al 1999.

This is not a triangle in the sense that Pythagorus drew them. Rather, it is a parameter triangle encompassing the values of three axis variables. It is wonderful and complex. But, believe it or not, there is a beautiful connection with 2D triangles here. We all know that the 3 angles inside a triangle add up to 180 degrees. But draw a triangle on the surface of a ball or on a saddle, then you will get a surprise – the angles no longer add up to 180 degrees! Surfaces like a sphere have positive curvature – its triangles have angles that can add up to 270 degrees. Surfaces like the saddle have negative curvature – their triangles have angles whose sum is less than 180 degrees. On non-flat surfaces, SOHCAHTOA no longer works –  there are no Pythagorean triples or Heronians. It sounds like a dismall failure for Euclidean geometry. Well yes and no. It is true that the spacetime continuum is wonderfully warped all over the place by the influence of the gravity exerted by every massive object in the cosmos (like balls on a rubber sheet). But the converse means that, if we were able to “draw” a large enough triangle in space by using precise positions of stars or other light-emitting sources, then if Pythagorus’s Theorem holds true – then we can conclude that the universe is flat (at least on the scale of the triangle). The results of Bahall et al (1999) suggest that this is approximately the case. Like setting down an emormous set square on a giant piece of paper full of creases, our universe may just well be a Pythagorean triple. How big its hypotenuse is or its legs are, nobody knows (yet).

As my own story comes to a close, triangles are still full of surprises. While they are hardly to be seen, it seems we can’t live without them. Who hasn’t used a step-ladder or felt safe in the knowledge that we drive over bridges strengthed with A-frame steel struts? There are still many unsolved mysteries. For example, why is it that there are so many optical illusions based on impossible triangles?

Image Credit: Michael Taylor

And, why was triangular geometry almost absent from the Vitruvian man even though Da Vinci was a brilliant geometrist and engineer?

Image Credit: Michael Taylor

Grab a bag of Doritos or a bar of Toblerone and your pen and paper. Give trigonometry a try. There are still many unwritten chapters in the story of the triangle. Who knows what you may discover? Perhaps you will write one of them?

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