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.
Archive for January, 2023
The Cosmology of the Divine Comedy
Published January 19, 2023 Irish Times ClosedTags: Astronomy, Cosmology, Topology
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 irishtimes.com].
Stereographic projection between 
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”.
Summing the Fibonacci Sequence
Published January 5, 2023 Occasional ClosedTags: Arithmetic, Number Theory
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,
and two initial values, 
ThatsMaths 