The number pi has an evil twin
A math explainer on pi’s “evil twin” introduces ϖ, a constant related to the perimeter of the lemniscate of Bernoulli, an ∞‑shaped curve defined by distances to two focal points rather than one as in a circle. Commenters explore how this fits into a broader family of lemniscate- and ellipse-related constants, generalizations to curves defined by products of distances from more points, and connections to elliptic integrals and dynamical systems. Along the way, they branch into etymology of mathematical terms, historical alphabets and notation, and other notable mathematical constants that arise in similar geometric or analytic contexts.
Language and Misnegation
- A side thread unpacks the phrase “never fails to disappoint.”
- Some argue it’s only correct as a sarcastic insult (meaning “always disappoints”), and that using it as praise is a misnegation like “could care less.”
- Others note that many native speakers now use it colloquially to mean “never disappoints,” demonstrating linguistic drift.
- Logical nitpicks appear (distinguishing “never failed” vs “never fails,” empty-set/vacuous-truth readings), with pushback that this is not how people actually use the phrase.
Lemniscate Spelling and Etymology
- Multiple commenters notice inconsistent spellings (“leminscate,” “lemniscate”) and clarify that “lemniscate” is standard.
- This leads to discussion of classical-language borrowing: Greek vs Latin roots, with observations that math and science freely mix Greek, Latin, Arabic, and even Sanskrit origins (e.g., sine/cosine history).
- Examples of Arabic and Greek loanwords in math, science, and everyday language are listed and briefly debated.
Greek Letters and Notation
- The symbol ϖ is discussed; some would misread it as omega with a bar.
- Commenters note variant and archaic Greek letters (digamma, heta, koppa, etc.) and that ϖ is a “variant of pi.”
- A Greek-speaker explains that the handwritten π resembling ϖ is standard in modern Greek schooling.
Geometry of Lemniscates and Generalizations
- The lemniscate of Bernoulli and the associated constant ϖ are related to curves defined by the product of distances to two foci.
- Commenters explore generalizations:
- Cassini ovals and polynomial lemniscates for more than two foci.
- “Trilemniscate” style shapes from three points; claims that areas can remain constant while perimeters diverge as the number of points increases.
- Other figure‑eight curves (Gerono lemniscate, Lissajous curves) are compared visually and parametrically.
- There is interest in whether analogous transforms built from “lemniscate sines/cosines” could parallel Fourier transforms.
Constants, Primes, and “Evil Twins”
- Speculation about “evil twins” of prime numbers yields candidates such as even numbers, highly composite/anti-primes, and “lucky numbers,” with no consensus on a best analogue.
- Broader discussion lists other notable mathematical constants (Euler–Mascheroni, Feigenbaum, Khinchin, etc.) and where they arise.
- A side calculation examines using arithmetic vs harmonic means to approximate the Gauss constant, trading square roots for more iterations.
Speculation, Culture, and Miscellany
- Several commenters imagine civilizations or sci‑fi settings where lemniscate geometry is more fundamental than circles or where alien math is non-integer-based or “logarithmic.”
- Others share curiosity-driven links: unusual world map projections, Penrose-like diagrams, and various curve visualizations.
- Minor complaints appear about Mastodon’s keyboard navigation, with userscripts offered as a workaround.