Archive for the 'Occasional' Category

Wavelets: Mathematical Microscopes

In the last post, we saw how Yves Meyer won the Abel Prize for his work with wavelets. Wavelets make it easy to analyse, compress and transmit information of all sorts, to eliminate noise and to perform numerical calculations. Let us take a look at how they came to be invented.

Wavelets-CWT-Example-BOTTOM

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Hearing Harmony, Seeing Symmetry

Musical notes that are simply related to each other have a pleasing effect when sounded together. Each tone has a characteristic rate of oscillation, or frequency. For example, Middle C on the piano oscillates 264 times per second or has a frequency of 264 Hz (Hertz). If the frequencies of two notes have a ratio of two small whole numbers, the notes are harmonically related and sound pleasant when played together.

Lissajous-Interval-16-17-SemiTone

Beats from two notes close in pitch.

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A Geometric Sieve for the Prime Numbers

In the time before computers (BC) various ingenious devices were invented for aiding the extensive calculations required in astronomy, navigation and commerce. In addition to calculators and logarithms, several nomograms were devised for specific applications, for example in meteorology and surveying.

MS-Sieve-Detail

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Torricelli’s Trumpet & the Painter’s Paradox

 

Torricelli-03

Torricelli’s Trumpet

 

Evangelista Torricelli, a student of Galileo, is remembered as the inventor of the barometer. He was also a talented mathematician and he discovered the remarkable properties of a simple geometric surface, now often called Torricelli’s Trumpet. It is the surface generated when the curve {y=1/x} for {x\ge1} is rotated in 3-space about the x-axis.

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Treize: A Card-Matching Puzzle

Probability theory is full of surprises. Possibly the best-known paradoxical results are the Monty Hall Problem and the two-envelope problem, but there are many others. Here we consider a simple problem using playing cards, first analysed by Pierre Raymond de Montmort (1678–1719).

SpadesHearts

Shuffle spades in one pile, hearts in another. Place both piles face downwards. Turn over a card from each pile. Do the two cards match?

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Numerical Coincidences

A numerical coincidence is an equality or near-equality between different mathematical quantities which has no known theoretical explanation. Sometimes such equalities remain mysterious and intriguing, and sometimes theory advances to the point where they can be explained and are no longer regarded as surprising.

Cosine of 355 radians is almost exactly equal to -1. Is this a coincidence? Read on!

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Brun’s Constant and the Pentium Bug

Euclid showed by a deliciously simple argument that the number of primes is infinite. In a completely different manner, Euler confirmed the same result. Euler’s conclusion followed from his demonstration that the sum of the reciprocals of the primes diverges:

\displaystyle \sum_{p\in\mathbb{P}} \frac{1}{p} = \infty

Obviously, this could not happen if there were only finitely many primes.

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