The number pi has an evil twin

Original Article ↗ Hacker News Discussion ↗

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.