Geometry At Light Speed

Geometry is a powerful tool that has been around a long time. Its formal textbook history goes back more than four thousand years, and it was an integral part of the construction of Egypt’s Pyramids, which date to 2,590 BCE or so (~4,600 years).

Many of the fundamentals of the geometry that is taught in today’s schools around the world were originally published for the first time in 13 books known as “Euclid’s Elements” around 300 BCE. “Euclid of Alexandria” was a great Geometer, but perhaps an even greater analyst, teacher, and historian. His books on the fundamentals of geometry included clarifications and refinements of knowledge that had already been in general use for thousands of years. By the time “Elements” had been published, a lot of it had been in use in Egypt, Greece, the Indus River Valley, and elsewhere for a long time.

Flickr; University of Glasgow Library, Image From February 16, 2018; Euclid’s Elements of Geometry; English Translation, 1661

It goes without saying that some sort of formal measurement and analysis approach has been in use as long as people have engaged in commerce, built durable shelters, palaces, houses of worship, or developed defensive and offensive capabilities. Probably more than 6,000 years. Everything that was more complicated than sitting in trees or caves probably involved at least some formal planning, measurement, and construction. All of that was almost certainly ongoing long before Euclid and other great mathematicians came along.

People of Asian and European descent have long known of a measure called the “Cubit”. Cubits were common units of measure based on the length of a forearm from the elbow to the tip of the middle finger, based on the anatomy of a man of medium build (about 46 centimeters, or 18 inches). In Egypt and elsewhere, “Cubit Rods” were eventually used to partially standardize measurement. But not all Cubits and Cubit Rods were equal across all cultures and civilizations; that was something that became increasingly inconvenient between cultures, and is one reason France set out to update measurement units in the middle of the 19th Century using what is now known as the Metric System (decimals). The cubit and a host of other measures were slowly replaced in the decades that followed, but the Metric System is still not universal; and the Cubit is still used in England and Ireland as the preferred measure for Hedgelaying. Good fences make good neighbors?

Hedgelaying in southern England, circa 2011; Wikipedia

For all of the improvements in our approach to measurement during the past 2,000 years, here we are close to surviving 2020, but with a lot of archaic measures that are not very useful for solving conundrums like, “where is MH370?”. Most of the scientific communities around the world adapted long ago (e.g., nano, light year, parsec, etc.), as they generally do; but the rest of us are still muddling along with tools that are old and cumbersome. What’s worse, one of the problems with those old tools is that they sometimes lock us into efforts that have no path forward; no solution. That is part of what happened during the initial effort to locate MH370’s debris field. Lead investigators were “old school” professionals and believed they needed things like “ground speed” and “compass heading” to find the plane. Neither of those things were available, and neither were needed. But instead of using the one measure that was available (light speed), the effort bogged down and eventually failed as the focus turned to guessing values for ground speed and heading. It was an impossible task and seriously delayed good intentions. The answer to the plane’s track was there in front of the entire world the whole time, but it had been calculated, measured, and recorded in light speed metrics, so it was ignored by those charged with finding it. Astronomers work with such metrics all the time; even today’s building-trades professionals increasingly use light to measure distances (e.g., Bosch Bluetooth Enabled 165 foot Laser Distance Measure), but the search for MH370 could not break free of centuries-old analog training, education, and mind-set.

The illustration below may help. Five different units of measure are used for the distance between the Kuala Lumpur Airport and the plane’s crash site. They are equivalent values and lead to exactly the same spot near a large Seamount known as Zenith Plateau. But there is an enormous difference in the approaches used to obtain the three “rate” values, when compared to the approach used to obtain the two light speed values. Once we have the latter, we can easily obtain the former; but not before that. You may recall that MH370 was unable to send “ground speed”, “heading”, etc., to those trying to track it because the pilot disabled outbound communication shortly after takeoff. As a result, RATE information, such as “per mile”, or “per kilometer” is not available. And the only way to obtain rate information for MH370 is to first solve “total distance” issues with the speed of light metric. Then, expressions like “per kilometer” follow easily.

Geometrically derived ping sequence for MH370; 2020; Mike Chillit

Think about it: if we don’t know the speed of an object in traditional analog terms like “per minute, hour, day, etc.”, we might not be able to determine where the object is or was at any given moment. That is a fact of life in an analog world: how long will it take to get to the store? How long will it take to get to the beach? RATE calculations involve distance, time, and velocity. If we don’t know an object’s “velocity per unit of time” in the analog world, we can’t say anything about its location after it moves. To complicate matters, velocity can be highly variable in the analog world.

Using the speed-of-light-constant solves the problem because time and distance at light speed are equivalent through a medium like earth’s atmosphere. If we know one, we know the other. Unlike analog velocity (e.g., 50 kilometers per hour), light travels at a constant speed.

For anyone unfamiliar with the light speed metric, light travels through  atmosphere at the rate of one kilometer every 3.33564 microseconds (µs) . That is always true when the medium is earth’s atmosphere, and means the speed of light is constant. To illustrate, MH370’s final ping had a radius of 16,276 microseconds; that is all we need to know it crashed 16,276 / 3.34 = 4,879 kilometers away from the ground point (GPS) of its tracking satellite. With just a little more basic geometry, we also know that means the plane was 2,775 kilometers south of the airport, on a heading of 177.3 degrees south southeast when it crashed.