How the humble triangle shaped human history
You probably never thought much about triangles. Thanks to the weaknesses of the story, the triangles have spent most of the story in the shadow of the circle. Circles are considered to have almost mystical properties, mainly because the ratio of their circumference to their diameter – you call it pi – has become a mathematical celebrity. He even gets his own day in the nerd calendar. But, as I found out while searching for my book The art of more, triangles have been seriously underestimated.
You probably know some of the basics. There is the Pythagorean theorem, for example, which was actually used 1,000 years before Pythagoras was born. Ancient civilizations such as the Babylonians and Egyptians used it for surveying and to create perfectly square corners for their buildings: tie a knot in a long rope at intervals of 3, 4, and 5 units, and you can use these 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 also had other triangular tricks: “similar” triangles, for example, used to estimate the distance to a ship moored offshore.
But these are just the most basic uses of the triangle. Triangles were also the shapes that brought us the science of optics, through the work of the Arab mathematician Ibn Al-Haytham. Translations of son Optical book led to the triangular designs with a realistic perspective that began in the Renaissance, and eventually to the creation of lenses and telescopes, and all that they have brought to our understanding of the cosmos and the microscopic world.
The properties of triangles are also at the origin of the compression algorithms of the trillions of JPEG and MPEG files which circulate today in our connected world. In fact, the design and function of all electronic devices, including the generation and transmission of the electrical energy that powers them, depend on our understanding of the properties of triangles. But perhaps the most significant global impact of triangles has come with shipping.
“Navigation is nothing more than a right triangle,” French navigator Guillaume Denys said in 1683. It refers to what we call a right triangle: the properties of that shape, he said , are all a sailor needs to figure out to get around.
When ships strayed from course during a voyage, whether due to adverse winds, an island in the way, or because they were attacked by pirates, the crew used the math of right triangles to reset them. on the right track. This math is carried by words that may sound familiar from your own school: sine and cosine. Simply put, these are numbers related to the ratios of the lengths of the sides of a particular right triangle.
Medieval sailors carried tables of sines and cosines, or a sinecal quadrant, a simple tool that allowed them to be discovered along the way. 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 martelioio.
The compass rose has each quarter of the compass divided into eight “rhumbs”, which describe the direction. The first quarter, for example, has north by east, north-northeast, north by northeast, plain northeast, and so on.
The toleta de martelioio was a table of numbers related to sines and cosines, designed specifically for marine use. The numbers told sailors how to correct their course if the wind, or something else, had diverted the trip. If you know how many miles you have traveled off-road and how many rhumbs off the desired course you have traveled, the toleta then gives the distance to cover on a new heading before getting back on track. And not a sinus in sight.
In the 15th century, Prince Henry of Portugal gathered as much knowledge as he could in order to establish a school for Christian sailors that would allow his faith to dominate the exploration of the world. One of the beneficiaries of this was Christopher Columbus, who used what he had learned about triangles to try and navigate west to India – accidentally stumbling across the Americas in the process.
Modern navigation is also based on triangles. In 1972, NASA launched Landsat-1, the first satellite built to study the geography of the Earth. To insiders, it was clear that the satellite could also provide a whole new kind of map of the world, and two years later, the United States Geological Survey (USGS) mapping coordinator published an article describing a suitable mathematical projection.
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Alden Colvocoresses – Colvo for his friends – devised a map that would capture the movement of the satellite’s scanner, the satellite’s orbit, the Earth’s rotation, and how the axis of that rotation moves in a 26,000 year cycle thanks to the ‘precession’ Earth. In order to avoid distortions, the card would have the shape of a cylinder, and the surface of this cylinder would oscillate back and forth along the major axis of the cylinder.
This way there would be no disastrous distortions as the satellite data was compiled into a map. It was a bold idea. But no one at NASA or USGS knew how to do the geometric analysis required to actually build the projection.
The man who ultimately solved the complexities was called John Parr Snyder. Snyder first heard about the problem in 1976, after his wife bought him a rather cheesy birthday present for his 50th birthday: a ticket to “The Changing World of Geodetic Science”, a mapping convention. in Columbus, Ohio.
Colvo gave the speech and exposed his problem. Snyder was addicted. He spent five months of his evenings and weekends figuring it out, using his spare bedroom as an office and nothing more technical than a Texas Instruments TI-56 programmable pocket calculator. Almost immediately, the USGS gave Snyder a job.
Snyder’s “Mercator Oblique Spatial Projection” was an essential step in the construction of satellite maps of our planet. These are vital for everything in 21st century civilization, from military operations and navigation to weather forecasting, environmental conservation and climate monitoring.
Snyder’s projection gave us Google Maps, Apple Maps, your car’s GPS, and every other digital mapping technology you can think of. Its calculation involves applying 82 equations to each of the data points on the satellite image. It’s terribly complex, but suffice it to say that it involves a complex set of sines and cosines.
Thousands of years after discovering its properties for the first time, we are still harnessing the power of the triangle.
The Art of More: How Mathematics Created Civilization by Michael Brooks is out now (£ 18.99, Scribe).