Gauss’ Work on Geometry and GeodesyAlex Sellers and Sydney Hauver

Overview

● Who was Gauss● How to Measure the World● Surface Integrals and Divergence Theorem● Differential geometry● Class Activity/Map Projections● Gaussian Curvature● Gauss-Bonnet Theorem● Geodesy● Magnetism● Summary● Questions

Who Was Gauss (1777-1855)

● Contributions to many fields:○ number theory, algebra, statistics, analysis, differential geometry,

geodesy, geophysics, mechanics, electrostatics, magnetic fields astronomy, matrix theory, and optics.

● Nick-Names:○ Princeps mathematicorum( "the foremost of mathematicians")○ "the greatest mathematician since antiquity",

● Bottom Line:● Gauss had an exceptional influence in many fields of mathematics and

science, and is ranked among history's most influential mathematicians● Most Well Known for:● From Brunswick, summed 1-101 as 5050● It was only Carl Gauss who gave proofs of the fundamental theorem that are

still considered valid, by making use of the geometrical interpretation of complex numbers that was unknown to Euler

Measuring the World

● Gauss and his journeys with French explorer Aimé Bonpland○ Their many groundbreaking

ways of taking the world's measure

● Combination of a triangle and a pentangle give 15-gon

● Zimmerman hired land surveying

Surface Integrals and the Divergence Theorem

In 1813 Gauss used divergence theorem in considering the gravitational attraction of an elliptical spheroid

But Gauss went further than Lagrange in showing how to calculate an integral with respect to dS in the case where the surface S is given parametrically by three functions x = x(p, q), y = y(p, q),z = z(p, q). Using a geometrical argument, he demonstrated that:

How to calculate curvature

● Curvature is a local property on a surface S. ● To be defined, it is clear that the curvature may vary from

point to point

Differential Geometry

● Gauss was finally able by 1827 to put on paper the results of his thoughts of over a quarter century on the subject of curved surfaces. Gauss noted in the abstract of his work Disquisitiones generales circa superficies curvas (General Investigations of Curved Surfaces)○ He realized how curvature could be calculated in terms of an analytic

description of the surface in question○ A sphere of radius r has Gaussian curvature 1/r2 everywhere, and a flat

plane and a cylinder have Gaussian curvature 0 everywhere. The Gaussian curvature can also be negative, as in the case of a hyperboloid or the inside of a torus.

Gauss-Chern-Bonnet theorem

▪ Gauss, he also argued that the precise nature of physical space could not be determined but only by “experience”

▪ Paths with zero geodesic curvature

Gaussian Curvature and How to Measure It

● Geometry in which Euclid’s parallel postulate did not hold

● Gauss established a relationship between curvature and the sum of the angles of a triangle on the surface

Curvature and the Theorema Egregium

Bottom line: plane can be developed onto a cylinder, the curvature of the cylinder equals that of the plane, namely, 0.

How to Visualize Gaussian Curvature Class Activity

Goode homolosine projection (or interrupted Goode homolosine projection) is a pseudocylindrical, equal-area, composite map projection used for world maps

Geodesy

Accurately measuring and understanding three of Earth’s fundamental properties:

▪ Geometric shape▪ Orientation in space▪ Gravitational field

History

In 1801 Gauss became famous for determining the next appearance of the Ceres from behind the sun with only three observations. Others tried but he was by far the most successful.

Method of Least Squares

Given more sets of equations than there are unknowns, try to make the best guess. Approximation

Method of Least Squares

Approximate the unknowns by reducing the residual error from each equation. For linear least squares it becomes a simple matrix transpose:

Practice problem

Uses of Least Squares

Legendre published his method in 1805 and used it on existing data to calculate the shape of the Earth. Geodesists at the time were stunned and eager to use it.

Gauss published his more rigorous and complete method in 1809.

Triangulation and surveying

The survey of Hannover was a long endeavor that took 14 years but led to the development of the Heliotrope

"All the measurements in the world are not worth one

theorem by which the science of eternal truth is genuinely

advanced" - Gauss

Gauss and Wilhelm Weber

After surveying, Gauss became close friend with Wilhelm Weber, a physics professor at the University of Göttingen. The two were interested in electricity and magnetism

First Electromagnetic Telegraph

Gauss and Weber developed the first electromagnetic telegraph and, at the time, was the longest telegraph in history

Electricity and Magnetism

Gauss’s law for magnetism:

A magnetic field with zero divergence does not change in magnitude or direction, essentially stating there is no existence of a monopole

Summary

● Who was Gauss● How to Measure the World● Differential Geometry● Class Activity/Map Projections● Gaussian Curvature● Gauss-Bonnet Theorem● Geodesy● Magnetism

Questions

Works Cited

▪ The Gauss-Bonnet-Chern Theorem on Riemannian Manifolds Yin Li 28 Nov 2011▪ Snyder, John P. (1993). Flattening the earth: two thousand years of map projections. University of Chicago

Press. p. 1. ISBN 0-226-76746-9.▪ Thomas Banchoff; Terence Gaffney; Clint McCrory; Daniel Dreibelbis (1982). Cusps of Gauss Mappings.

Research Notes in Mathematics. 55. London: Pitman Publisher Ltd. ISBN 0-273-08536-0. Retrieved 10 April 2018.

▪ Lecture on Measuring Curvature, Dr. Robert Kusner, 3 April 2018.▪ http://bolvan.ph.utexas.edu/~vadim/classes/17f/divrot.pdf http://www2.sjs.org/raulston/mvc.10/topic.6.lab.1.htm▪ https://www.magcraft.com/johann-carl-friedrich-gauss▪ https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/

pioneers/carl-friedrich-gauss https://thatsmaths.com/2014/07/10/gausss-great-triangle-and-the-shape-of-space/▪ https://nationalmaglab.org/education/magnet-academy/history-of-electricity-magnetism/

museum/gauss-weber-telegraph▪ http://wlym.com/archive/pedagogicals/geodesy.html

https://www.encyclopediaofmath.org/index.php/Gauss,_Carl_Friedrich