Posts Tagged 'Gauss'

Gaussian Curvature: the Theorema Egregium

ShapeOfUniverse

Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

One of greatest achievements of Carl Friedrich Gauss was a theorem so startling that he gave it the name Theorema Egregium or outstanding theorem. In 1828 he published his “Disquisitiones generales circa superficies curvas”, or General investigation of curved surfaces. Gauss defined a quantity that measures the curvature of a two-dimensional surface. He was inspired by his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his Theorema Egregium. The Gaussian curvature {K} characterizes the intrinsic geometry of a surface.

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The Steiner Minimal Tree

Steiner’s minimal tree problem is this: Find the shortest possible network interconnecting a set of points in the Euclidean plane. If the points are linked directly to each other by straight line segments, we obtain the minimal spanning tree. But Steiner’s problem allows for additional points – now called Steiner points – to be added to the network, yielding Steiner’s minimal tree. This generally results in a reduction of the overall length of the network.

Solution of Steiner 5-point problem with soap film [from Courant and Robbins].

A solution of Steiner 5-point problem with soap film [from Courant and Robbins].

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Waring’s Problem & Lagrange’s Four-Square Theorem

\displaystyle \mathrm{num}\ = \square+\square+\square+\square

Introduction

We are all familiar with the problem of splitting numbers into products of primes. This process is called factorisation. The problem of expressing numbers as sums of smaller numbers has also been studied in great depth. We call such a decomposition a partition. The Indian mathematician Ramanujan proved numerous ingenious and beautiful results in partition theory.

More generally, additive number theory is concerned with the properties and behaviour of integers under addition. In particular, it considers the expression of numbers as sums of components of a particular form, such as powers. Waring’s Problem comes under this heading.

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Triangular Numbers: EYPHKA

The maths teacher was at his wits’ end. To get some respite, he set the class a task:

Add up the first one hundred numbers.

That should keep them busy for a while”, he thought. Almost at once, a boy raised his hand and called out the answer. The boy was Carl Friedrich Gauss, later dubbed the Prince of Mathematicians. Continue reading ‘Triangular Numbers: EYPHKA’

Gauss’s Great Triangle and the Shape of Space

In the 1820s Carl Friedrich Gauss carried out a surveying experiment to measure the sum of the three angles of a large triangle. Euclidean geometry tells us that this sum is always 180º or two right angles. But Gauss himself had discovered other geometries, which he called non-Euclidean. In these, the three angles of a triangle may add up to more than two right angles, or to less.

10 Deutschmark currency note

10 Deutschmark currency note (front)

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The High-Power Hypar

Maths frequently shows us surprising and illuminating connections between physical systems that are not obviously related: the analysis of one system often turns out to be ideally suited for describing another. To illustrate this, we will show how a surface in three dimensional space — the hyperbolic paraboloid, or hypar — pops up in unexpected ways and in many different contexts.

Warszawa Ochota railway station, a hypar structure [Image Wikimedia Commons].

Warszawa Ochota railway station, a hypar structure
[Image Wikimedia Commons].

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The Prime Number Theorem

God may not play dice with the Universe, but something strange is going on with the prime numbers  [Paul Erdös, paraphrasing Albert Einstein]

The prime numbers are the atoms of the natural number system. We recall that a prime number is a natural number greater than one that cannot be broken into smaller factors. Every natural number greater than one can be expressed in a unique way as a product of primes. Continue reading ‘The Prime Number Theorem’

Hyperbolic Triangles and the Gauss-Bonnet Theorem

Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane {\mathbf{H} = \{(x,y): y>0\}} together with a metric

\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.

It is remarkable that the entire structure of the space {(\mathbf{H},ds)} follows from the metric.
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Poincare’s Half-plane Model (bis)

In a previous post, we considered Poincaré’s half-plane model for hyperbolic geometry in two dimensions. The half-plane model comprises the upper half plane {H = \{(x,y): y>0\}} together with a metric

\displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,.

It is remarkable that the entire structure of the space follows from the metric.

In the earlier post, we derived the total curvature by evaluating the Riemann tensor. Here, we compute the curvature directly, using Gauss’s “Remarkable Theorem”.
Continue reading ‘Poincare’s Half-plane Model (bis)’

Geometry in and out of this World

Hyperbolic geometry is the topic of the That’s Maths column in the Irish Times this week (TM031 or  click Irish Times and search for “thatsmaths”).

Living on a Sphere

The shortest distance between two points is a straight line. This is one of the basic principles of Euclidean geometry. But we live on a spherical Earth, and we cannot travel the straight line path from Dublin to New York: we have to stick to the surface of the globe, and the geometry we need is more complicated than the plane geometry of Euclid. Spherical geometry is central for the study of geophysics and astronomy, and vital for navigation.

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Gauss Misses a Trick

Carl Friedrich Gauss is generally regarded as the greatest mathematician of all time. The profundity and scope of his work is remarkable. So, it is amazing that, while he studied non-Euclidian geometry and defined the curvature of surfaces in space, he overlooked a key connection between curvature and geometry.  As a consequence, decades passed before a model demonstrating the consistency of hyperbolic geometry emerged.

Continue reading ‘Gauss Misses a Trick’


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