Archive for January, 2023

Curvature and the Osculating Circle

Curvature is of critical importance in numerous contexts. An example is shown in the figure above, a map of the Silverstone Formula 1 racetrack. The sharp bends (high curvature) force drivers to reduct speed drastically.

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The Cosmology of the Divine Comedy

Divina Commedia: Online at Columbia University.

If you think poetry and maths are poles apart, think again. Around the sixth century, Indian poet and mathematician Virahanka codified the structure of Sanskrit poetry, formulating rules for the patterns of long and short syllables. In this process, a sequence emerged in which each term is the sum of the preceding two. This is precisely the sequence studied centuries later by Leonardo Bonacci of Pisa, which we now call the Fibonacci sequence [TM241 or search for “thatsmaths” at].

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Adding a Point to Make a Space Compact

Stereographic projection between {\mathbb{R}} and {S^1}. There is no point on the real line corresponding to the “North Pole” in {S^1}. Can we add another point to {\mathbb{R}}?

The real line is an example of a locally compact Hausdorff space. In a Hausdorff space, two distinct points have disjoint neighbourhoods. As the old joke says, “any two points can be housed off from each other”. We will define local compactness below. The one-point compactification is a way of embedding a locally compact Hausdorff space in a compact space. In particular, it is a way to “make the real line compact”.

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Summing the Fibonacci Sequence

Left: Fibonacci, or Leonardo of Pisa. Right: Italian postage stamp issued on the 850th anniversary of his birth.

The Fibonacci sequence must be familiar to anyone reading this. We define it by means of a second-order recurrence relation,

\displaystyle F_{n+1} = F_{n-1} + F_n \,. \ \ \ \ \ (1)

and two initial values, {F_0 = 0} and {F_1 = 1}. This immediately yields the well-known sequence

\displaystyle \{F_n\} = \{ 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, \dots \} \,.

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