Talk:Minkowski's question-mark function: Difference between revisions

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== About the derivatives ==

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I have a copy [[On Numbers and Games]], and the box function is clearly the Minkowski question mark function itself, not its inverse. I suggest removing the whole "Inverse" section from the article. [[User:Reyk|<b style="color: Maroon;">Reyk</b>]] _{[[User talk:Reyk|<b style="color: Blue;">YO!</b>]]} 07:05, 11 September 2020 (UTC)

*{{tick}}- since there were no objections I removed it. [[User:Reyk|<b style="color: Maroon;">Reyk</b>]] _{[[User talk:Reyk|<b style="color: Blue;">YO!</b>]]} 16:48, 10 October 2020 (UTC)

== ?(2/3) ==

Either the continued-fraction—binary definition is wrong, or I'm misunderstanding something.

The continued fraction representation of 2/3 is [0;1,2]. This run-length-encodes the binary number 0.011₂, which equals 3/8.

However, by the finite-sum equation given ([https://www.wolframalpha.com/input?i=minkowski+question+mark+function+%282%2F3%29 and confirmed by WolframAlpha]), ?(2/3) = 3/4 = 0.11₂. This would appear to correspond to either [0;0,2]=2 or [0;2]=1/2.

(Am I doing the whole process backwards? 2/3 = 0.1010101...₂ ⇒ [0;1,1,1,1,...] = [[Golden ratio#Continued fraction and square root|1/φ]]. That's not right either.)

What's going on?

— [[User:Sonata Green|Sonata Green]]^{([[User_talk:Sonata_Green|talk]])} 01:07, 13 April 2023 (UTC)

::@[[User:Sonata Green|Sonata Green]] There's an extra factor of 2 in front of the sum. The correct procedure is as follows:

::#Convert x to its continued fraction representation. Example: 2/3 = [0; 1, 2].

::#Interpret the numbers in the fractional part as sequences of alternating 0's and 1's, starting with 0 and ending with an infinite sequence of either zeros or ones: 011000... (1 zero and 2 ones).

::#Add a dot after the first zero and interpret the result as binary: 0.11₂ = 3/4.

::Here's another example: x = 3/5 = [0; 1, 1, 2] = → 0100111... ⇒ ?(3/5) = 0.100111...₂ = 0.101₂ = 5/8.

:[[User:Hugo Spinelli|Hugo Spinelli]] ([[User talk:Hugo Spinelli|talk]]) 10:10, 17 October 2023 (UTC)

::Then why does [3;2,1,2,1,4,5,...] generate 11.001001000011111... rather than 11.01001000011111...? — [[User:Sonata Green|Sonata Green]]^{([[User_talk:Sonata_Green|talk]])} 21:34, 17 October 2023 (UTC)

:::@[[User:Sonata Green|Sonata Green]] The example is wrong. It should be <math>[3; 3, 1, 2, 1, 4, 5, ...]</math>. See [https://www.wolframalpha.com/input?i=minkowski+question+mark+function+%28FromContinuedFraction%5B%7B3%2C+3%2C+1%2C+2%2C+1%2C+4%2C+5%7D%5D%29 WolframAlpha], for instance. One of the references given (Pytheas Fogg, 2002) appears to make the same mistake. The other one (Finch, 2003) is very explicit that the first sequence of zeros is <math>(a_1 - 1)</math>-long. —[[User:Hugo Spinelli|Hugo Spinelli]] <small>([[User talk:Hugo Spinelli|talk]])</small> 17:33, 20 October 2023 (UTC)

== Where do the matrices come from in the "Periodic orbits as continued fractions" section? ==

I tried implementing this myself and ran into problems with the matrices given. Deriving them on my own, I got different matrices, and I updated the article with the ones I derived. I can't actually find where they appear in the cited reference, though. — [[User:Flipdisk|Flipdisk]] ([[User talk:Flipdisk|talk]]) 08:30, 27 March 2024 (UTC)

Actually, now that I read the earlier parts of the article more carefully, this section does seem to make sense in context, so I reverted my changes. But I still don't see where it appears in the source material. [[User:Flipdisk|Flipdisk]] ([[User talk:Flipdisk|talk]]) 08:32, 27 March 2024 (UTC)