We are all familiar with splitting natural numbers into prime components. This decomposition is unique, except for the order of the factors. We can apply the idea of prime components to many more general sets of numbers. The Gaussian integers are all the complex numbers with integer real and imaginary parts, that is, all numbers … Continue reading Gaussian Primes
Tag: Gauss
Gauss Predicts the Orbit of Ceres
On the first day of a new century, January 1, 1801, astronomer Giuseppe Piazzi discovered a new celestial object, the minor planet Ceres. He made about 20 observations from his observatory in Palermo before the object was lost in the glare of the Sun in early February. Later in the year, several astronomers tried without … Continue reading Gauss Predicts the Orbit of Ceres
Gaussian Curvature: the Theorema Egregium
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 … Continue reading Gaussian Curvature: the Theorema Egregium
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 … Continue reading The Steiner Minimal Tree
Waring’s Problem & Lagrange’s Four-Square Theorem
$latex \displaystyle \mathrm{num}\ = \square+\square+\square+\square &fg=000000$ 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 … Continue reading Waring’s Problem & Lagrange’s Four-Square Theorem
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 … 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 … Continue reading Gauss’s Great Triangle and the Shape of Space
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 … Continue reading The High-Power Hypar
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 … 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 $latex {\mathbf{H} = \{(x,y): y>0\}}&fg=000000$ together with a metric $latex \displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,. &fg=000000$ It is remarkable that the entire structure of the space $latex {(\mathbf{H},ds)}&fg=000000$ follows from the metric. The … Continue reading Hyperbolic Triangles and the Gauss-Bonnet Theorem
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 $latex {H = \{(x,y): y>0\}}&fg=000000$ together with a metric $latex \displaystyle d s^2 = \frac { d x^2 + d y^2 } { y^2 } \,. &fg=000000$ It is remarkable that the … 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 … Continue reading Geometry in and out of this World
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 … Continue reading Gauss Misses a Trick
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