[HN Gopher] Math That Goes on Forever but Never Repeats
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Math That Goes on Forever but Never Repeats
Author : rbanffy
Score : 71 points
Date : 2023-05-24 11:24 UTC (1 days ago)
(HTM) web link (www.quantamagazine.org)
(TXT) w3m dump (www.quantamagazine.org)
| callamdelaney wrote:
| i = 0
|
| while True: i += 1
|
| Also never repeats..
| zx321 wrote:
| https://en.wikipedia.org/wiki/Champernowne_constant
| Xeoncross wrote:
| except for the forever part
|
| ints(8-128) will either wrap around (repeating) or will
| overflow error in any language I've ever used
| missblit wrote:
| Hey this is a math fight, no computer architecture allowed!
| callamdelaney wrote:
| Yes, if I knew the mathematical notation for the same thing
| I'd have used that. Overflow is not the point..
| miloignis wrote:
| Python uses BigInts, so it will go until it runs out of RAM,
| I believe.
| Xeoncross wrote:
| Imagine a computer lasting long enough to count to a number
| larger than what fits in 8GB of RAM...
| kkylin wrote:
| Interesting topic, terrible (IMO) title.
| gylterud wrote:
| The AB patterns in the article remind me a bit of the sequences
| which occur in "The anti-pattern game" - a game I invented for a
| course I took in modal logic. The game produces sequences where
| no pattern repeats more than twice in a row:
|
| * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
| * * * * * * * * * * * * * * * * *[?]
|
| I put a brief description up on it here:
|
| https://hakon.gylterud.net/antipattern/
| EddieEngineers wrote:
| Wouldn't a simple version of this be * * * * * * * * * * ...
| etc?
| orblivion wrote:
| I was just thinking the same thing, but it only accounts for
| repeating patterns that start from the beginning. If you look
| at your example, you can already see * * * repeats the
| pattern * three times in a row.
| EddieEngineers wrote:
| Oooooh, wow! Good point & makes things much harder! Will
| read OPs post in more depth later then.
|
| Edit: I wonder if you can use property based testing to
| prove this fact...
| [deleted]
| gowld wrote:
| Title should be "Math _Patterns_ That Go On Forever but Never
| Repeat.
|
| The article title is correct, but the article page's HTML title
| is wrong / SEO
| im3w1l wrote:
| So one thing I that this aperiodic monotile had me wondering,
| does it have any non-trivial _non-viable partial tiling_? By
| which I mean you start laying down tiles right, and it looks like
| they fit together fine. Your construct has no holes and is
| "reasonably convex", like nowhere does it curve inwards by a lot.
| But because of some mysterious long-range self interaction it's
| actually impossible to complete the tiling process into something
| encompassing the whole plane.
| causi wrote:
| I have a question for serious math nerds: are there degrees of
| never repeating? Like if you think of repetitiousness like escape
| velocity, as you get less repetitious the number of digits
| between repeats increases until it hits infinity, aka escape
| velocity. Can repetitiousness decrease even beyond that?
| l33t233372 wrote:
| One way of looking at things like this is from the perspective
| of so called Ramsey theory. Notions like syndenticity(the
| property of a set having bounded gaps), upper density(the lim-
| sup of the proportion of numbers in the set), thickness(the
| property of having arbitrarily large successive subsequences),
| etc. all capture different notions of this.
|
| Edit: let's not forget the rich theory involved in examining if
| sets/sequences contain arithmetic (or geometric) progressions!
| The well known Green-Tao theorem about primes containing
| arbitrarily long arithmetic progressions is one such result. In
| fact, the real Green-Tao theorem says that any set which has
| nonzero upper density with respect to the set of primes
| contains arbitrarily long arithmetic progressions.
| readyplayernull wrote:
| True randomness with infinite sequences should generate
| infinitely repeating sequences and infinitely non-repeating
| sequences.
| l33t233372 wrote:
| I'm not sure I agree. Can you be more precise about what you
| mean by repeating? If you mean "contains an infinite
| arithmetic progression" then surely this is not true.
|
| Could you also be more precise about what you mean by true
| randomness?
| [deleted]
| gylterud wrote:
| There is Kolmogorov complexity, which is (relative to a
| programming language) a measure of how easy a sequence is to
| produce algorithmically - i.e. what is the length of the
| shortest program which can produce the sequence. Most
| mathematicians believe that not all sequences can be produced
| algorithmically, so this leaves some sequence too complicated
| to have a finite Kolmogorov complexity. But you can go on to
| the infinite by adding oracles, and start assigning infinite
| ordinals to sequences, depending on how many oracles you need
| for a given sequence. If a program uses an oracle for the
| halting problem, say, you nominally add o to its length.
|
| All the repeating sequences have finite Kolmogorov complexity.
| So, if you want, you can say that the sequences with infinite
| Kolmogorov complexity have even lower repetitiousness than the
| ones we can produce algorithmically. And this will go on and on
| through the countable ordinals.
| karpierz wrote:
| > Most mathematicians believe that not all sequences can be
| produced algorithmically, so this leaves some sequence too
| complicated to have a finite Kolmogorov complexity.
|
| Isn't this just true?
|
| The set of algorithms is countable, and the set of sequences
| of real numbers is uncountable.
| gylterud wrote:
| It depends on believing in classical logic. Constructive
| mathematicians can instead choose to adopt intuitionistic
| logic and postulate Church thesis, which states that all
| functions, and thus sequences, are computable.
| kmod wrote:
| IANAMathematician, but it's "true" in the sense that we
| primarily use a set of axioms (ZF/ZFC) which produce this
| result. But in a sense one of the axioms is that these
| things exist (powerset axiom), so this result is a bit less
| illuminating than one might hope.
|
| My understanding is that we can reason about these sorts of
| non-constructible sequences but we can't really know if
| they "really exist" because it's not clear what that even
| means if their existence has no implications for physical
| reality (which is possible but not known). So now we're
| into the territory of whether there is such a thing as
| abstract truth independent of reality.
|
| I wrote a blog post in the past about my reading on this
| topic, and I can't claim that's it's accurate but at least
| I tried to talk about this subject:
| https://blog.kevmod.com/2022/04/09/do-the-real-numbers-
| exist...
| [deleted]
| fsckboy wrote:
| I'm not clear on the definition of sequence such that you
| could have one without having a description of how to
| produce it? Can you have a sequence of random numbers?
| kmod wrote:
| They are necessarily non-constructive existence proofs,
| for sure. You're not alone in some potential unease with
| that: https://en.wikipedia.org/wiki/Constructivism_(philo
| sophy_of_...
| rbanffy wrote:
| Any representation of any irrational number will be an
| endless and never repeating sequence of seemingly random
| digits.
| kmod wrote:
| I think the term of interest here is non-computable:
| irrational numbers can have finite kolmogorov complexity.
|
| You're kind of "begging the question" here, where you're
| assuming that non-computable numbers exist and then using
| that to show that some numbers are non-computable. You
| can definitely show that these things exist, but that
| relies on "believing" the set of axioms that you used to
| prove it.
| _a_a_a_ wrote:
| Cantor liked this post.
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