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Superellipses and Superhyperbolas: Beyond Classical Geometry

2025-03-28

This article introduces superellipses and superhyperbolas, generalizations of ellipses and hyperbolas, respectively. The shapes are controlled by a parameter 'p'. When p=2, they reduce to standard ellipses and hyperbolas. Increasing p makes superellipses more rectangular, but with continuous curvature; superhyperbolas become blunted at the vertices. The article explores why superellipses are far more common than superhyperbolas, speculating on naming conventions and the lack of effective advocacy for the latter.

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(www.johndcook.com)
Misc Superellipse

The Buenos Aires Constant: A Mathematical Puzzle Hiding Primes

2025-02-21

The mysterious number 2.92005097731613..., known as the Buenos Aires constant, generates a sequence of prime numbers when used to initialize a simple Python script. This isn't a coincidence; the constant's definition is intrinsically linked to prime sequences. However, due to computational precision limitations using the IEEE 754 standard, the algorithm fails after generating a certain number of primes. This raises questions about the deeper connection between mathematical constants and primes, highlighting the impact of computational accuracy on mathematical experimental results.

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(www.johndcook.com)
Misc mathematical constant

What *Exactly* Is a Second?

2025-01-01

This article delves into the evolving definition of a second. Initially defined as 1/86400th of a day, the slowing rotation of the Earth rendered this definition imprecise. In 1967, the second was redefined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between two hyperfine levels of the ground state of the cesium-133 atom. Cesium was chosen for its ease of isolation and purification, while the specific number of periods ensured backward compatibility with the length of the tropical year in 1900. Although the astronomical definition has been discarded, the cesium-based definition remains in use today.

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(www.johndcook.com)
Tech Time Measurement Cesium Atomic Clock

Unix Time and a Modest Proposal

2024-12-27

This article delves into the discrepancy between Unix time (the number of seconds since January 1, 1970) and the actual time due to the Earth's slowing rotation and slight variations in its orbit. Leap seconds were introduced to address this, but their complexities lead to a plan to discontinue them by 2035. The author proposes a novel solution: periodically adjusting Earth's orbit to maintain synchronization between the solar year and the average Gregorian calendar year, thereby eliminating the need for leap seconds.

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(www.johndcook.com)
Tech Unix time Earth's orbit