Wikipedia:Reference desk/Mathematics#Area of a .22square.22 on the surface of a sphere

:File:IIHF World Junior Championship.png Sagittarian Milky Way (talk) 16:44, 7 May 2025 (UTC)

:All conventional projections of a sphere to the plane have a circular outline. The outline in the logo does not have a constant curvature; it is some artistic fantasy projection. One can therefore only guess at the latitudes of the parallels. If the angular distance between successive parallels is a constant $\backslash delta$, and the next one, not shown, would be the equator at $0^\backslash circ,$ those visible are at $\backslash delta,\; 2\backslash delta,\; 3\backslash delta.$ Then one might guess (but it remains a guess) that $4\backslash delta=90^\backslash circ.$  ​‑‑Lambiam 23:13, 7 May 2025 (UTC)

::At least the left of the horizon seems to be hidden by the white band. If it's x degrees of lat each line y of long of a sphere/Earth at some distance finite (General Perspective projection) or infinite (orthographic projection) then this should be answerable by painstakingly comparing image plane coordinates of intersections with image plane coordinates of candidate combos of latitude/longitude/nadir coordinates/viewing distances etc made with the method I embarrassingly couldn't figure out till you told me (i.e. convert to Cartesian (0N0E nadir), roll image plane coordinates to nadir 0N/desired longitude with easy trig, roll that to desired nadir with easy trig) Sagittarian Milky Way (talk) 00:14, 14 May 2025 (UTC)

= May 9 =

We know about parity. However, has anyone ever proposed a term like tri-arity that means the classification of a number n by whether n, n-1, or n+1 is a multiple of 3?? (Every integer belongs in exactly one of these 3 categories; they have properties that parallel being even and odd; the only difference is that they relate to 3 the way even and odd numbers relate to 2.) Georgia guy (talk) 17:28, 9 May 2025 (UTC)

:The parity of a number is (in one-one correspondence with) its residue modulo 2, so the question can be rephrased as, is there a snappy term for "residue modulo 3"? I have never seen one. For the ternary analogon of a parity check, we find the terms "modulo-3 residue check" and "residue modulo-3 check" in the literature.  ​‑‑Lambiam 20:04, 9 May 2025 (UTC)

:This question reminds me a bit of a sort of inquiry that might have been popular in the 17th–19th centuries, among figures like Henry More and Charles Saunders Peirce, who were fond of making up names for things that it wasn't clear needed names, or were even well-specified things. Of course the residue of an integer mod 3 is a well-specified thing, but it isn't clear to me that it needs a name. --Trovatore (talk) 20:33, 9 May 2025 (UTC)

::It may be worth mentioning that not all residue systems are created equal; residue modulo 2 is used much more frequently and in a wider variety of ways than other residue systems. For example permutations can be assigned "even" or "odd" parity, with the rules of parity being preserved under composition, in other words even*even=even, even*odd=odd, odd*even=odd, odd*odd=even. There is no mod 3 way of doing this, nor is there for any higher order modulus. So "parity" gets its own special name due simply to the frequency of situations in which it appears. If triarity appeared as frequently then it might get official status as well. --RDBury (talk) 07:50, 10 May 2025 (UTC)

• RDBury, assuming the operation is addition; yes there is. Let's call the 3 kinds of numbers black (divisible by 3,) red (one more than a multiple of 3,) and green (one less than a multiple of 3.) So the answer can be:

• Black+black=black
• Black+red=red
• Black+green=green
• Red+black=red
• Red+red=green
• Red+green=black
• Green+black=green
• Green+red=black
• Green+green=red

(Please note that I'm just using these names for convenience; any response you have must hold regardless of what names I'm using for the 3 kinds of numbers.) Georgia guy (talk) 12:36, 10 May 2025 (UTC)

:I was confused at first by RDBury's response as well, but look closer; he's talking specifically about permutations. There's a theorem that every (finite) permutation can be decomposed into a composition of permutations where you swap two elements at a time. The decomposition is not unique, but the number of pair-swaps in it always has the same parity. --Trovatore (talk) 17:57, 10 May 2025 (UTC)

:More information: Permutation#Parity of a permutation. --Trovatore (talk) 17:59, 10 May 2025 (UTC) Oh actually we have a whole article: Parity of a permutation. --Trovatore (talk) 18:10, 10 May 2025 (UTC)

:Here is a further exposition. Let $S\_n$ stand for the set of permutations on a finite set of size $n,$ and consider a function $c:S\_n\backslash to\backslash \{\backslash text\{black\},\backslash text\{red\},\backslash text\{green\}\backslash \}$ – that is, a way of assigning one of these three colours to each of these permutations – such that the identity $c(\backslash sigma\backslash tau)=c(\backslash sigma)+c(\backslash tau)$ is satisfied (using the addition table above). We can then prove that it assigns the colour $\backslash text\{black\}$ to all permutations.

:(Proof sketch: The identity permutation $\backslash iota$ gives rise to the equation $c(\backslash iota)+c(\backslash iota)=c(\backslash iota\backslash iota)=c(\backslash iota),$ which is only possible if $c(\backslash iota)=\backslash text\{black\}.$ Then each involution $\backslash pi$ gives rise to the equation $c(\backslash pi)+c(\backslash pi)=c(\backslash pi\backslash pi)=c(\backslash iota)=\backslash text\{black\},$ which is only possible if $c(\backslash pi)=\backslash text\{black\}.$ The involutions generate the whole group $S\_n.$)

:On the other hand, if we use:

::$\backslash text\{black\}+\backslash text\{black\}=\backslash text\{black\},$

::$\backslash text\{black\}+\backslash text\{white\}=\backslash text\{white\},$

::$\backslash text\{white\}+\backslash text\{black\}=\backslash text\{white\},$

::$\backslash text\{white\}+\backslash text\{white\}=\backslash text\{black\},$

:an addition-respecting assignment $c:S\_n\backslash to\backslash \{\backslash text\{black\},\backslash text\{white\}\backslash \}$ is possible, using both colours.  ​‑‑Lambiam 18:50, 10 May 2025 (UTC)

: There is a term for it in variant sudoku. One feature of such sudoku is lines with constraints on the numbers on the line. "parity" lines the numbers on them alternate even and odd. "modular" lines any three adjacent numbers are different mod 3. The following page includes these as well as the many other variant rules: https://eev.ee/fyi/variant-sudoku/ .--2A04:4A43:904F:FAD8:81F7:FD12:6AD7:80B3 (talk) 10:24, 18 May 2025 (UTC)

::The description on that page uses "alternating parity" for the first constraint. A constraint generalizing both "alternating parity" and "modular" is "unique-modulo-{{math|n}}": each of the digits in a group of {{math|n}} consecutive digits along an indicated line is unique modulo {{math|n}}. For example, $\backslash textsf\{962354178\}$ is unique-modulo-5 (but not unique-modulo-2, unique-modulo-3 or unique-modulo-4). Then "alternating parity" is unique-modulo-2 and "modular" is unique-modulo-3. The standard constraint of Sudoku is unique-modulo-9. All of this fits under the rubric Modular arithmetic. Reserving "modular" generically for specifically "modulo 3" is not a good idea.  ​‑‑Lambiam 13:28, 18 May 2025 (UTC)

= May 18 =

[https://www.youtube.com/watch?v=sGCmu7YKgPA This video] talks about some of the new mathematical results from Google Deepmind. Before anyone changes their major, it seems worth pointing out that these are all examples of a certain type of optimization problem and that Deepmind only improved on the best known result, it didn't prove or even claim they were the best possible, only that they were an improvement on what human intelligence was able to find in those particular cases. So I wouldn't count on the Collatz conjecture being solved tomorrow. But it does seem to indicate that the way math is done in the future will be significantly different than the way it was done in the 20th century. Thoughts? RDBury (talk) 02:07, 18 May 2025 (UTC)

:The main takeaway of the video is that Matt Parker should take a vacation more often.

:Constructing formal proofs is not an optimization problem, but in a sense it is a combinatorial puzzle, and I expect more artificial help there in the future. Formalizing published proofs (possibly uncovering seemingly obvious but unjustifiable skips) could be a nice start. Another area where an indefatigable agent can prove its worth may be finding interesting conjectures connecting different fields, similar to the discovery of the monstrous moonshine, which started as an observation of a curious coincidence.

:I do not expect the Collatz or Goldbach conjectures to be solved ever, also not with the help of a superintelligent agent. We cannot, with the knowledge we have, exclude the possibility that although true they are not members of the theory of ZFC. (Consider that a simple generalization of the Collatz problem is undecidable, so it is (non-constructively) certain that some Collatz variants are true but don't hold a ZFC club membership card.)  ​‑‑Lambiam 09:32, 18 May 2025 (UTC)

= May 19 =