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.Continue reading ‘The Steiner Minimal Tree’
Posts Tagged 'Gauss'
Tags: Algebra, Algorithms, Gauss, Maps, Topology
Tags: Gauss, Number Theory, Primes, Ramanujan
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.
Tags: Gauss, Number Theory, Recreational Maths
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’
Tags: Gauss, Geophysics, Maps
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.
Tags: Algebra, Gauss, Geometry, Recreational Maths
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.Continue reading ‘The High-Power Hypar’
Tags: Analysis, Arithmetic, Gauss, Number Theory, Primes
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’
Tags: Gauss, Geometry, Topology
Poincaré’s half-plane model for hyperbolic geometry comprises the upper half plane together with a metric
It is remarkable that the entire structure of the space follows from the metric.
Continue reading ‘Hyperbolic Triangles and the Gauss-Bonnet Theorem’