## The Divisonal Kalaidoscope | |
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π is a mathematical constant and a transcendental (and therefore irrational) real number. It is the ratio of a circle's circumference to its diameter in Euclidean geometry, and has many uses in mathematics, physics, and engineering. While the value of π has been computed to billions of digits, practical science and engineering will rarely require more than 10 decimal places. As an example, computing the circumference of the Earth's equator from its radius using only 10 decimal places of π yields an error of less than 0.2 millimetres. The exact value of π has an infinite decimal expansion: its decimal expansion never ends and does not repeat, since π is an irrational number. This infinite sequence of digits has fascinated mathematicians and laymen alike, and much effort over the last few centuries has been put into computing more digits in the expansion and investigating the number's properties. Despite much analytical work, and supercomputer calculations that have determined over 1 trillion digits of π, no simple pattern in the digits has ever been found (taken from Wikipidia). For more information see e.g. the Pi pages. Carl Friedrich Gauss (Gauß) (30 April 1777 - 23 February 1855) was a German mathematician and scientist of profound genius who contributed significantly to many fields, including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. Sometimes known as "the prince of mathematicians" and "greatest mathematician since antiquity", Gauss had a remarkable influence in many fields of mathematics and science, and is ranked as one of history's most influential mathematicians. Gauss was a child prodigy, and there are many anecdotes which tell of his astounding precocity while a mere toddler. Indeed, he made his first ground-breaking mathematical discoveries while still a teenager. He completed Disquisitiones Arithmeticae, his magnum opus, at the age of twenty-one (1798), though it would not be published until 1801. This work was fundamental in consolidating number theory as a discipline and has shaped the field to the present day (excerpt from a Wikipedia article). The history of Mathematics is a very interesting subject taught by Dr M.W. Green .
A cross-sectional image of a
Nautilus sea shell shows the spiral curve of the shell and the internal chambers that the animal steadily adds as it grows. The chambers provide
buoyancy in the water. The spiral formed by the chambers is a logarithmic spiral or approximately a Fibonacci spiral
as constructed in the second image. Fibonacci numbers form a sequence defined by the recurrence relation on the right.
That is, after two starting values, each subsequent number in the series is given by the sum of the two preceding numbers. The first Fibonacci
numbers, also denoted as n = 0, 1, are: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... The Fibonacci numbers are named after Leonardo of Pisa, known as
Fibonacci, although they
had been described earlier in India.
A Fibonacci spiral is a spiral created by
drawing arcs connecting
the opposite corners of squares in the Fibonacci tiling, which in
turn is a tiling with squares whose sides are successive Fibonacci numbers in length. This is an example of a subject in Mathematical Biology, a
field our Mathematical Biology Group works on.
An X-ray image of the surface and lower atmosphere of the Sun. It shows magnetic loops emerging from the solar surface, and exemplifies the complex structure of the Sun's magnetic field. The image actually shows not magnetic fields, which are invisible, but the hot, radiating plasma which is closely tied to magnetic field lines and thereby traces out those field lines. In order to get an impression of the size of these magnetic loops, an image of the Earth has been added to provide a length scale (image made by Transition Region and Coronal Explorer (TRACE) spacecraft). The structure and dynamics of these phenomena are part of the field of research of the Magnetohydrodynamics Group. See here for larger version of the image. Approximation of a step function by a Fourier series. Using an increasing number of terms of the Fourier series, the quality of the approximation improves. However, wiggles left and right of the discontinuity occur which do not get smaller using more terms of the series. This 'overshoot' has a constant height (=18% of the discontinuity) and moves towards the jump as the number of terms increases. This effect is called Gibbs' phenomenon, named after the American physicist J. Willard Gibbs. The overshoot is a consequence of trying to approximate a discontinuous function with a partial (i.e. finite) sum of continuous functions. A finite sum of continuous functions is, by definition, continuous, and therefore cannot approximate the discontinuity (and the region "near" it) to within any arbitrarily chosen accuracy. This is an example of a problem in Numerical Analysis, a field of research covered by our Numerical Analysis and Scientific Computing Group. Some happy graduates from the 2007 Graduation. Click here for a larger version of the image. |