You’ve probably never thought much about triangles. Thanks to the foibles of history, triangles have spent most of history in the circle’s shadow. Circles are seen as having near-mystical properties, mostly because the ratio of their circumference to their diameter – you know it as pi – has become a mathematical celebrity. It even gets its own day in the nerd calendar. But, as I discovered when researching my book The Art of More, triangles have been seriously under-rated.
You probably know some of the basics. There’s Pythagoras’s Theorem, for instance, which was actually in use 1,000 years before Pythagoras was born. Ancient civilisations such as the Babylonians and the Egyptians used it for surveying, and to create perfectly square corners for their buildings: put a knot in a long rope at intervals of 3, 4 and 5 units, and you can use those knots as corner-points that create a triangle with a perfect right-angle between the sides that are 3 and 4 units long. The ancients had other triangular tricks too: ‘similar’ triangles, for instance, provide a way to estimate the distance to a ship moored offshore.
But these are only the most basic uses of the triangle. Triangles were also the shapes that brought us the science of optics, through the work of Arab mathematician Ibn Al-Haytham. Translations of his Book of Optics led to triangle-based drawings with realistic perspective that began in the Renaissance, and eventually to the creation of lenses and telescopes, and all that they brought to our understanding of the cosmos and the microscopic world.
The properties of triangles are also behind the compression algorithms of the trillions of JPEG and MPEG files that shuttle through our connected world today. In fact, the design and function of all electronics, including the generation and transmission of the electrical energy that powers them, is dependent on our understanding of the properties of triangles. But perhaps the most significant global impact of triangles came with navigation.
“Navigation is nothing more than a right triangle,” said the French mariner Guillaume Denys in 1683. He is referring to what we call a right-angled triangle: the properties of this shape, he said, are all a sailor needed to understand to get around.
When ships went off course during a journey, whether it was because of unfavourable winds, an island in the way, or because they were waylaid by pirates, the crew would use the mathematics of right-angled triangles to set them back on course. That mathematics is carried by some words that might sound familiar from your own schooldays: sine and cosine. Put simply, these are numbers related to the ratios of the lengths of the sides of any particular right-angled triangle.
Medieval sailors would carry tables of sines and cosines, or a sinecal quadrant, a simple tool that made it possible to find them out en route. But if they didn’t want to engage with sines and cosines at all, they could just use two simple tools: the compass rose and the toleta de marteloio.
The compass rose has each quarter of the compass divided into eight ‘rhumbs’, which describe direction. The first quarter, for instance, has north-by-east, north-northeast, north by northeast, plain northeast, and so on.
The toleta de marteloio was a table of numbers related to the sines and cosines, designed specifically for maritime use. The numbers told sailors how to correct their course if the wind, or something else, had diverted the journey. If you know how many miles you’ve sailed off course, and how many rhumbs off your desired heading you’ve been sailing, the toleta then gives the distance to travel on a new heading before you’re back on track. And not a sine in sight.
In the 15th Century, Prince Henry of Portugal gathered together as much of this knowledge as he could in order to establish a school for Christian sailors that would allow his faith to dominate world exploration. One of the beneficiaries of this was Christopher Columbus, who used what he had learned about triangles in order to try sailing west to the Indies – accidentally stumbling upon the Americas in the process.
Modern-day navigation also relies on triangles. In 1972, NASA launched Landsat-1, the first satellite built to study the Earth’s geography. To insiders, it was clear that the satellite could also provide an entirely new kind of world map, and two years later the cartographic coordinator for the United States Geological Survey (USGS) published a paper describing a suitable mathematical projection.
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Alden Colvocoresses – Colvo to his friends – imagined a map that would account for the movement of the satellite’s scanner, the satellite’s orbit, the rotation of the Earth, and the way the axis of that rotation evolves in a 26,000-year cycle thanks to Earth’s ‘precession’. In order to avoid distortions, the map would have the form of a cylinder, and the surface of this cylinder would oscillate back and forth along the cylinder’s long axis.
In this way, there would be no disastrous distortions as the data from the satellite was compiled into a map. It was an audacious idea. But no one at NASA or the USGS knew how to do the geometric analysis required to actually construct the projection.
The man who eventually worked out the complexities was called John Parr Snyder. Snyder first heard about the problem in 1976, after his wife bought him a rather nerdy 50th birthday present: a ticket to attend ‘The Changing World of Geodetic Science’, a mapping convention in Columbus, Ohio.
Colvo gave the keynote, and outlined his problem. Snyder was hooked. He spent five months of his evenings and weekends solving it, using his spare bedroom as a study and nothing more technical than a Texas Instruments TI-56 programmable pocket calculator. Almost immediately, the USGS gave Snyder a job.
Snyder’s ‘space oblique Mercator projection’ was an essential step towards constructing satellite maps of our planet. These are vital for everything in 21st Century civilisation, from military operations and navigation to weather forecasting, environmental conservation, and climate monitoring.
Snyder’s projection gave us Google Maps, Apple Maps, the Satnav in your car, and every other digital mapping technology you can think of. Its maths involves applying 82 equations to each of the data points on the satellite image. It’s terrifyingly complex, but suffice it to say that it involves a complex array of sines and cosines.
Thousands of years after we first discovered its properties, we are still harnessing the power of the triangle.
The Art of More: How Mathematics Created Civilisation by Michael Brooks is out now (£18.99, Scribe).