Archive for the 'Occasional' Category

Listing the Rational Numbers II: The Stern-Brocot Tree

The rational numbers are countable: they can be put into one-to-one correspondence with the natural numbers. But it is not obvious how to construct a list that is sure to contain every rational number precisely once. In a previous post we described the Farey Sequences. Here we examine another, related, approach.

Mediant-red Continue reading ‘Listing the Rational Numbers II: The Stern-Brocot Tree’

Listing the Rational Numbers: I. Farey Sequences

We know, thanks to Georg Cantor, that the rational numbers — ratios of integers — are countable: they can be put into one-to-one correspondence with the natural numbers.

Rational-Numbers-Small

Continue reading ‘Listing the Rational Numbers: I. Farey Sequences’

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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A Zero-Order Front

DosTemp18

Sharp gradients known as fronts form in the atmosphere when variations in the wind field bring warm and cold air into close proximity. Much of our interesting weather is associated with the fronts that form in extratropical depressions.

Below, we describe a simple mechanistic model of frontogenesis, the process by which fronts are formed.

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The Flight of the Bumble Bee

Alice and Bob, initially a distance l apart, walk towards each other, each at a speed w. A bumble bee flies from the nose of one to the nose of the other and back again, repeating this zig-zag flight at speed f until Alice and Bob meet. How far does the bumble bee fly?

Flight-of-BumbleBee-Music

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Euler’s “Degree of Agreeableness” for Musical Chords

Euler-10_Swiss_Franc_banknoteThe links between music and mathematics stretch back to Pythagoras and many leading mathematicians have studied the theory of music. Music and mathematics were pillars of the Quadrivium, the four-fold way that formed the basis of higher education for thousands of years. Music was a central theme for Johannes Kepler in his Harmonices Mundi – Harmony of the World, and René Descartes’ first work was a compendium of music.

Continue reading ‘Euler’s “Degree of Agreeableness” for Musical Chords’

Grandi’s Series: A Second Look

Grandis-Series
In an earlier post, we discussed Grandi’s series, originally studied by the Italian monk Dom Guido Grandi around 1703. It is the series

\displaystyle G = 1 - 1 + 1 - 1 + 1 - 1 + \dots

This is a divergent series: the sequence of partial sums is {\{ 1, 0, 1, 0, 1, 0, \dots \}}, which obviously does not converge, but alternates between {0} and {1}.

Continue reading ‘Grandi’s Series: A Second Look’


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