Ciro Santilli @cirosantilli 40

If a product is called "Big Company Catchy Name Does Something", then it came from an acquisition, and they wanted to keep the name due to its prestige and to not confuse users.

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Model of quantum computing Updated 2025-07-16

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Open source video game asset Updated 2025-07-16

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The beauty of mathematics Updated 2025-07-16

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Ciro Santilli intends to move his beauty list here little by little: github.com/cirosantilli/mathematics/blob/master/beauty.md

The most beautiful things in mathematics are results that are:

simple to state but hard to prove:
- Fermat's Last Theorem
- number of unknown rationality, e.g. is $e + π$ rational?
- transcendental number conjectures, e.g. is $e + π$ transcendental?
- basically any conjecture involving prime numbers:
  - twin prime conjecture
- many combinatorial game questions, e.g.:
  - first-move advantage in chess
surprising results: we had intuitive reasons to believe something as possible or not, but a theorem shatters that conviction and brings us on our knees, sometimes via pathological counter-examples. General surprise themes include:
- classification of potentially infinite sets like: compact manifolds, etc.
  - classification of finite simple groups
  - classification of regular polytopes
  - classification of closed surfaces, and more generalized Poincaré conjectures
  - classification of wallpaper groups and space groups
- problems that are more complicated in low dimensions than high like:
  - generalized Poincaré conjectures. It is also fun to see how in many cases complexity peaks out at 4 dimensions.
  - classification of regular polytopes
- unpredictable magic constants:
  - why is the lowest dimension for an exotic sphere 7?
  - why is 4 the largest degree of an equation with explicit solution? Abel-Ruffini theorem
- undecidable problems, especially simple to state ones:
  - mortal matrix problem
  - sharp frontiers between solvable and unsolvable are also cool:
    attempts at determining specific values of the Busy beaver function for Turing machines with a given number of states and symbols
    related to Diophantine equations:
    decidability of Hilbert's tenth problem of a given degree and number of variables
    Hilbert's tenth problem over other rings
Lists:
- math.stackexchange.com/questions/139699/what-are-some-examples-of-a-mathematical-result-being-counterintuitive
- math.stackexchange.com/questions/2040811/what-are-some-counter-intuitive-results-in-mathematics-that-involve-only-finite/2055458#2055458
applications: make life easier and/or modeling some phenomena well, e.g. in physics. See also: explain how to make money with the lesson

Good lists of such problems Lists of mathematical problems.

Whenever Ciro Santilli learns a bit of mathematics, he always wonders to himself:

Am I achieving insight, or am I just memorizing definitions?

Unfortunately, due to how man books are written, it is not really possible to reach insight without first doing a bit of memorization. The better the book, the more insight is spread out, and less you have to learn before reaching each insight.

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Voltage transformer Updated 2025-07-16

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Waring-Goldbach problem Updated 2025-07-16

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It is exactly what you'd expect from the name, Waring was watching Netflix with Goldbach, when they suddenly came up with this.

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Water Updated 2025-07-16

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"Water" is the name for both: