Posts Tagged 'Geometry'

Archimedes and the Volume of a Sphere

One of the most remarkable and important mathematical results obtained by Archimedes was the determination of the volume of a sphere. Archimedes used a technique of sub-dividing the volume into slices of known cross-sectional area and adding up, or integrating, the volumes of the slices. This was essentially an application of a technique that was — close to two thousand years later — formulated as integral calculus.

SphConCyl-5

Cone, sphere and cylinder on the same base. The volumes are in the ratios  1 : 2 : 3 [image from mathigon.org].

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Some Fundamental Theorems of Maths

Every branch of mathematics has key results that are so
important that they are dubbed fundamental theorems.

The customary view of mathematical research is that of establishing the truth of propositions or theorems by rigorous deduction from axioms and definitions. Mathematics is founded upon axioms, basic assumptions that are taken as true. Logical reasoning is then used to deduce the consequences of those axioms with each major result designated as a theorem.

As each new theorem is proved, it provides a basis for the establishment of further results. The most important and fruitful theorem in each area of maths is often named as the fundamental theorem of that area. Thus, we have the fundamental theorems of arithmetic, algebra and so on. For example, the fundamental theorem of calculus gives the relationship between differential calculus and integral calculus.

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Symplectic Geometry

Albert-EinsteinFor many decades, a search has been under way to find a theory of everything, that accounts for all the fundamental physical forces, including gravity. The dictum “physics is geometry” is a guiding principle of modern theoretical physics. Einstein’s General Theory of Relativity, which emerged just one hundred years ago, is a crowning example of this synergy. He showed how matter distorts the geometry of space and this geometry determines the motion of matter. The central idea is encapsulated in an epigram of John A Wheeler:

\displaystyle \mbox{Matter tells space how to curve. Space tells matter how to move.}

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Chase and Escape: Pursuit Problems

Jolly-RogerFrom cheetahs chasing gazelles, through coastguards saving shipwrecked sailors, to missiles launched at enemy aircraft, strategies of pursuit and evasion play a role in many areas of life (and death). From pre-historic times we have been solving such pursuit problems. The survival of our early ancestors depended on their ability to acquire food. This involved chasing and killing animals, and success depended on an understanding of relative speeds and optimal pursuit paths.

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Gaussian Curvature: the Theorema Egregium

ShapeOfUniverse

Surfaces of positive curvature (top), negative curvature (middle) and vanishing curvature (bottom) [image credit: NASA].

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 his work on geodesy, surveying and map-making, which involved taking measurements on the surface of the Earth. The total curvature — or Gaussian curvature — depends only on measurements within the surface and Gauss showed that its value is independent of the coordinate system used. This is his Theorema Egregium. The Gaussian curvature {K} characterizes the intrinsic geometry of a surface.

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A Trapezoidal Prism on the Serpentine

Walking in Hyde Park recently, I spied what appeared to be a huge red pyramid in the middle of the Serpentine. On closer approach, and with a changing angle of view, it became clear that it was prismatic in shape, composed of numerous barrels in red, blue and purple.

Christo-Mastaba-00

Changing perspective on approach to the Mastaba

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The Miraculous Spiral on Booterstown Strand

We all know what a spiral looks like. Or do we? Ask your friends to describe one and they will probably trace out the form of a winding staircase. But that is actually a helix, a curve in three-dimensional space. A spiral is confined to a plane – it is a flat curve. In general terms, a spiral is formed by a point moving around a fixed centre while its distance increases or decreases as it revolves [see TM145, or search for “thatsmaths” at irishtimes.com].

Terra-Nova-Bootertown-C

The spiral sandbank on Booterstown strand (satellite image digitally enhanced by Andrew Lynch).

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