Time for pi
Happy Pi Day! We hope you have the chance to enjoy the tasty kind of pie today – in the meantime, here’s a pi “appetizer”.
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The 10 Deutschmark bill from 1993 depicting Gauss with a graph of Gaussian curve of errors.
To a surveyor, pi is a key concept that undergirds everything, because surveying occurs on a sphere. Specifically, all boundaries must take into consideration the curve of the earth (and any distortions of that curve, as well as elevation).
Surveying presents interesting mathematical challenges that have been examined for centuries. The ancient Greeks called it “geodaisia”, which literally means “division of the Earth”. Some of the greatest minds in history have studied these challenges, and one of them, Carl Gauss (1777-1855), invented a new field of mathematics and several devices in the course of his studies.
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Image of the 10 Deutschmark bill featuring Gauss’ Heliotrope and the triangulations he made with it during the Survey of Hannover.
Gauss
Back in 1817, Johann Carl Friedrich Gauss (one of the top three mathematicians in history*), was so interested in geodesy that he accepted a commission from King George IV of England (who was also King of the German Kingdom of Hannover) to triangulate the entire Kingdom.
The survey, which Gauss completed thirty years later in 1847, did not produce a very accurate map of Hannover, but it did result in advances in the mathematics of curved surfaces, including the development of curvilinear coordinates.
Order of King George IV from 9 May 1820 to the triangulation project (with the additional signature of Count Ernst zu Münster below) By Hajotthu, CC BY-SA 3.0,
By Pauli-Pirat - Own work, CC BY-SA 4.0,
Additionally, to increase the accuracy of his observations, Gauss developed a new lens, appropriately named the Gauss eyepiece. He designed this lens to deal with the blur found in when trying to focus on a distant point. Today, the term “Gaussian blur” describes an algorithm (derived from Gauss’ original calculations) that reduces image noise and detail in an image. You can find Gauss’ original calculations here.
*the other two are Newton and Archimedes
There’s more . . .
To improve pointing accuracy on distant targets Gauss also invented the heliotrope.
The following is paraphrased from LIDAR Magazine: The Surveyors’ Heliotrope: Its Rise and Demise by Silvio A. Bedini10.31.2004
In 1820, while he was making trigonometrical observations in Hannover, Gauss ran into an issue that slowed down his progress. Every time he directed his telescope toward the steeple of St. Michael’s Church at Hamburg, a distance of seven German miles (thirty-two English miles), the little round window in the upper part of the church reflected the image of the sun toward him. This interfered with his observations and wasted time.
This problem led Gauss to consider how to create a beacon that was sufficiently bright so that it could be observed even in the daytime. It occurred to him that it might be possible to use the sun’s rays for signals, and he developed the concept of capturing the rays with a mirror and reflecting them to the place to which the signal was to be given. He estimated the strength of the sun’s light as well as the amount of diminution it suffered in the atmosphere. From these calculations he concluded that a small mirror of not more than two or three inches in diameter would be adequate to reflect the sun’s image to the distance of ten or more German miles.
In 1821 he commissioned the Breithaupt Company to construct the first model of the heliotrope (named after a sun-loving flower). The device exceeded his expectations, and was extremely useful in practice, having the brightness of a first magnitude star at a distance of fifteen miles.
The heliotrope with various modifications became standard equipment for large scale triangulation for several decades.
. . . and more . . .
The most important result of Gauss’ triangulation project was a book on geodesy, which he published in two volumes between 1844 and 1847.
Gauss was a pivotal figure in mathematics, and his work with triangulation helped to launch his work on the intrinsic geometry of curved surfaces. In turn, this “paved the way for the development of differential geometry, a field that investigates the properties of curved spaces using calculus and differential equations. Differential geometry is essential in various areas of modern mathematics and physics, including general relativity, where the curvature of spacetime is described using differential geometric concepts.
Gauss’ algorithm for math lovers.
‘By challenging the notion of flat geometry and exploring the properties of curved surfaces, Gauss’s work extended the foundations of geometry beyond Euclidean principles. His contributions to differential geometry revolutionized our understanding of space, paving the way for new mathematical frameworks and challenging long-held assumptions about the nature of geometry itself.” [1]
Finally, time for pi
Gauss was the first mathematician to develop a method for deriving the digits of π. This algorithm is notable for being rapidly convergent, with only 25 iterations producing 45 million correct digits of π. Although highly accurate, this method proved very time-consuming. Fortunately, today we have computers, and just last year, pi was calculated to 105 trillion digits.[2]
As much as we’d like to, we don’t serve pie at Berntsen, but we’re proud to serve surveyors who use pi every day to make sure their calculations are accurate.