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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...
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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...
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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 together with a metric It is remarkable that...
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Hyperbolic Triangles and the Gauss-Bonnet Theorem
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. The “straight...
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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...
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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...
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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...
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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...
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Waring’s Problem & Lagrange’s Four-Square Theorem
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...
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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...
View ArticleGaussian 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...
View ArticleGauss 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...
View ArticleGaussian Primes
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...
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