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.

## Posts Tagged 'Gauss'

### The Steiner Minimal Tree

Published January 29, 2015 Occasional Leave a CommentTags: Algebra, Algorithms, Gauss, Maps, Topology

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

Published October 23, 2014 Occasional Leave a CommentTags: Gauss, Number Theory, Primes, Ramanujan

**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

Published October 9, 2014 Occasional Leave a CommentTags: 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’

### Gauss’s Great Triangle and the Shape of Space

Published July 10, 2014 Occasional Leave a CommentTags: 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.

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### The High-Power Hypar

Published May 29, 2014 Occasional Leave a CommentTags: 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.

### The Prime Number Theorem

Published February 27, 2014 Occasional Leave a CommentTags: 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’

### Hyperbolic Triangles and the Gauss-Bonnet Theorem

Published October 31, 2013 Occasional Leave a CommentTags: 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.

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