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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 in the set
The set ![{\mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7B%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{g \in \mathbb{Z}[i]}](https://s0.wp.com/latex.php?latex=%7Bg+%5Cin+%5Cmathbb%7BZ%7D%5Bi%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
Ceres (bottom left), the Moon and Earth, shown to scale [Image NASA].
Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].
characterizes the intrinsic geometry of a surface.
Continue reading ‘Gaussian Curvature: the Theorema Egregium‘
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].](https://thatsmaths.files.wordpress.com/2015/01/steiner-5point.jpg)
A solution of Steiner 5-point problem with soap film [from Courant and Robbins].
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.
Continue reading ‘Waring’s Problem & Lagrange’s Four-Square Theorem’
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’
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 (front)
Continue reading ‘Gauss’s Great Triangle and the Shape of Space’
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].](https://thatsmaths.files.wordpress.com/2014/05/hypar-roof-mid-res.jpg)
Warszawa Ochota railway station, a hypar structure
[Image Wikimedia Commons].
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’
Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane 
It is remarkable that the entire structure of the space 
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’
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 
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)’
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.
![\displaystyle \mathbb{Z}[i] \equiv \{ m + i n : m, n \in \mathbb{Z} \} \,.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cmathbb%7BZ%7D%5Bi%5D+%5Cequiv+%5C%7B+m+%2B+i+n+%3A+m%2C+n+%5Cin+%5Cmathbb%7BZ%7D+%5C%7D+%5C%2C.+&bg=ffffff&fg=000000&s=0&c=20201002)
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