Research and Engineering Explained: GPS and Fractals

by | Dec 4, 2020 | Coupi Research & Publications

“Think not of what you see, but what it took to produce what you see” -Benoit Mandelbrot

This is the third in a series of blogs on research and engineering in design, invention, and problem-solving for industrial products and processes, something that lays the groundwork for all that we do here at Coupi.

Fractuals

In 1975, the term “Fractal” was coined by Benoit Mandelbrot. These fractal shapes repeat patterns of self-similar shapes. The Latin origin, fractus, means an irregular surface like that of a broken stone, Mandelbrot referred to his work as “the study of roughness”. His basic research showed that these fractals are considered to be shapes of “Sacred geometry” since they hold properties that can be applied to reality. He then published a book called “The Fractal Geometry of Nature” in 1982. Little did Mandelbrot know that these fractals would go on to revolutionize how cell phones work. 

Nathan Cohen, a radio astronomer, had listened to a lecture by Mandelbrot and decided to use fractal mathematics to alter his radio antenna to get a better signal while minimizing its size. This was spurred by his landlord not allowing large antennas on his building, so he began experimenting with bending the wire into fractal patterns. By creating these fractal patterns, it reduced the size while also increasing the signal quality. Then in 1988, he then published an applied research paper detailing a variety of fractal antenna designs. These designs utilize a combination of capacitors and inductors and allow them to have different resonances depending on the fractal design. While the overall electrical length is longer, these antennas themselves are physically smaller due to the self-similar shapes.

“Menger Sponge depth 5” by fdecomite is licensed under CC BY 2.0 

This research came at a time when cell phone companies were trying to give customers more features, such as bluetooth and wi-fi. Each of these services runs on a different frequency, which meant that they would each need their own individual antenna.  Before this breakthrough, it was not possible to have smartphones as compact and powerful as they are now since they would require a multitude of antennas. Fractal antennas allow phones to process the transmission of a wider range of electromagnetic frequencies and reduce the total number of antennas required. The fractal used in cell phones today is called the sierpinski carpet, which is a plane fractal first described by Wacław Sierpiński in 1916. 

Another modern technology that required the merging of discoveries through basic and applied research is the Global Positioning System (GPS). It relies on geodesy, the doppler effect and doppler tracking, and the theory of relativity. 

GPS

People have been trying to accurately measure the Earth’s geometric shape, a field of study called Geodesy, for thousands of years. One of the earliest attempts to measure the Earth’s size was done by a Greek philosopher and scholar, Eratosthenes, who utilized two wells in different cities, Swenett and Alexandria,  and the distance between them, along with the angle of the sun hitting the wells to calculate the circumference of the Earth. Though it was not until the last hundred years that technology had advanced far enough to make accurate calculations of the size of the Earth and the locations on it. 

GPS stands for “Global Positioning System” though it is usually associated with the American system NAVSTAR. The origin of GPS comes from the Sputnik era in 1957 where scientists figured out how to track the satellite with shifts in its radio signal, otherwise known as the Doppler Effect. Part of this basic research was based on similar ground-based radio-navigation systems developed in the early 1940’s. GPS today was heavily influenced by a myriad of research done in the 1960’s and 70’s. In the 1960’s the United States Navy conducted satellite navigation experiments to track US submarines carrying nuclear missiles, they utilized six satellites orbiting the poles and the doppler effect to pinpoint submarine locations in a matter of minutes. 

This then led to a system called TRANSIT which was a constellation of 5 satellites that could provide a navigational point once per hour. A large limitation with accuracy in GPS is the relativity effect, which was initially described by Albert Einstein’s theory of general relativity between 1907 and 1915. Since time slows in a strong gravitational field that means that clocks on earth would be slower than the ones aboard the satellites in space and cause calculation issues. The solution to this was the Timation satellite in 1967 that had a clock aboard that was able to compensate for the relativity shift from the satellite relative to the Earth. 

Up until the 1970’s tracking was still not very accurate. It wasn’t until the Cold War arms race that there was a justification for the billions of dollars it would take to create improved technology through applied and developmental research. In order to overcome the limitations of previous systems, a project began in 1973. The Department of Defense took inspiration from the Navy scientists and decided to use satellites to support their planned navigation system. This then launched the first Navigation System with Timing and Ranging (NAVSTAR) satellite in 1978. This was a system built of 24 satellites that became fully operational in 1993 and was allowed for civilian and commercial use after that year. 

Since then, GPS has worked by having these satellites orbiting the planet. At any given location on earth there is a line of sight to at least 4 satellites at a given time. Each GPS satellite then broadcasts a signal with its position and timestamp. Your cell phone or other GPS device will then receive all four signals and can calculate your position with an accuracy of 10 to 15 meters. 

Without GPS technology we would not have Google Maps, driving services like Uber, or even augmented reality games such as Pokemon Go. 

Photo by Pixabay

Innovation Creates Economic Impacts

In 1967 the annual Research and Development (R&D) Expenditure in the U.S. was $24 Billion. Of this 13% went to Basic research, 21% to applied research, and 66% to developmental research. This number has changed significantly, with the amount spent on R&D totaling $495.1 Billion in 2015. From that amount 16.9% went to Basic research, 19.6% went to Applied research, and the remaining 63.5% to developmental research. The total R&D funding has increased by 2063% in 48 years. 

According to a preliminary study presented to the National Space-Based Positioning, Navigation, and Timing (PNT) board, the GPS has contributed more than $68 billion to the U.S economy alone. Cellular and smart phones generated $3.3 trillion in 2019 and has created a total of 11 million jobs. 

Without the discovery of fractals, or the many decades of basic and applied research it took to make GPS, we would not have the technology we do today. The computer, phone, or tablet you are using to read this blog likely has one or even both of these innovations inside of it. So, in the words of Benoit Mandelbrot, “Think not of what you see, but what it took to produce what you see”. 

To learn more about Fractals and GPS technology:

“Fractals Among Us.” Hackaday, 18 Aug. 2016, hackaday.com/2016/08/18/fractals-among-us/

“How GPS Receivers Work.” HowStuffWorks, 25 Sept. 2006, electronics.howstuffworks.com/gadgets/travel/gps.htm 

Oracle, The. “Sacred Geometry: How Cell Phones Work Using Fractals.” The Oracle’s Library, 22 May 2015, www.theoracleslibrary.com/2015/05/19/sacred-geometry-how-cell-phones-work-using-fractals/.

Sargent, John F. “U.S. Research and Development Funding and Performance: Fact Sheet.” Congressional Research Service, 29 June 2018. https://fas.org/sgp/crs/misc/R44307.pdf

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