12:01 AM
Is it possible to, if not always then at least for most famous probability distributions, construct a (discrete if appropriate) surface by gluing the PDFs with neighboring parameter values next to each other, and reslice the surface in an orthogonal direction to get another probability distribution? For example, the surface could be the glued Poisson PDFs for each value of $\lambda$, which would then be resliced treating $k$ as the parameter and $\lambda$ as the support variable.
Would this distribution in $\lambda$ adhere to the probability axioms in that case (for each fixed $k$)? Is this basically what Bayesian statistics is doing under the hood by positing a distribution of belief over a parameter?
12:36 AM
@user10478 I'm not sure I understand you well, you're looking at a family of PDFs parametrized by some parameter $\theta$ lets say, $f_\theta$
is that right? And then what do you do?
12:56 AM
is not "difficult" a relative
term
@user85795 yes but the point is that I never meant "difficult" in the first place
"unpleasant or unattractive" things are difficult
in what sense
you avoid them
I don't think thats necessarily the case
usually people avoid unpleasant things, but I don't think the same can be said about unattractive things
1:06 AM
Sure, beauty is in the eye of the beholder.
even if this was something one would avoid, that doesn't necessarily make it difficult, as much as it makes it difficult to deal with
it doesn't make it difficult like how a math problem can be difficult, for example
its more psychological
yes
@Jakobian Yeah, so if for example $\theta$ happened to be discrete, your new probability distribution be $Pr(\theta = 1)$, $Pr(\theta = 2), ... and the old variable$f$used to range over would now be a parameter. 1:32 AM @user10478 So you're treating$\theta$as a random variable. I think I've seen that in a statistics class before. But I'm still not sure what you're asking could you use less words and more symbols/definitions? this might be better to ask among statisticians as well I was just wondering if those probabilities over$\theta$automatically sum to$1$and satisfy sigma additivity, so as to constitute an actual probability distribution. 2:00 AM @user10478 what do you mean? If$\theta$is a random variable then it will naturally have a probability distribution I'm still not understanding Can I treat a$\theta$obtained in that way as a random variable? 2:19 AM In what way though, in what way you're going further but not clarifying what you mean at all I don't think the whole talk with surfaces really made much sense, it didn't to me, so I'm still struggling on what you mean but I get it if you don't want to elaborate because lets say, you don't really know what you want to ask 3:13 AM @Jakobian Sorry, I didn't see latest reply right away. Yeah it might be better if I make a picture in Mathematica at a later time and show what I have in mind. 2 hours later… 4:45 AM @EE18 I thought you were asking about the 1/k^2+k question, so my hint about prime between n and 2n was just about k^2<k^2+k<2k^2, bounding the sum. Nothing to do with prime:) 5:21 AM oh, i dunno. nothing to do with prime? 5:33 AM ðŸ¥¹ 1 hour later… 6:53 AM @copper.hat Hey there. Sorry, I've been extremely busy recently. 2 hours later… 8:33 AM @Jakobian I'm looking at this again and the condition$x\geq 2y$needs to have a lower bound too I believe. When$X=10$, the possible$Y$values are$4$and$5$, but the condition$x\geq 2y$says that$p_{X,Y}(10,y)$is nonzero also for$y=1,2$and$3$. I don't see what the lower bound is right now though. 8:54 AM @Jakobian maybe this is what you were saying here, not sure. Or simply add$x\geq 2y$when$x\leq 7$and$x\geq 2y> 2(x-7)$when$12\geq x>7$, something like that. 9:30 AM I wonder how one can rewrite the function then in a concise way with all them conditions. In this mathscinet review mathscinet.ams.org/mathscinet/article?mr=2044031 the reviewer writes: > this sort of definition is wholly ungeometrical and totally sterile which I find hilarious (just to remark, he is actually praising the book quite a lot in the context of the review, I just like the formulation a lot) 10:32 AM in the proof that "if Ax=0 and Bx=0 have the same solutions, they must be row equivalent" I noticed one thing in the above, when defining the map K, there are m-r! ways of maping deltas to gammas, so that K stays invertible this could mean that there are m-r ! invertible matrices such that KA=B? 2 hours later… 12:13 PM @psie no, I'm not saying this is always non-zero for$x\geq 2y$. Just that its equal to certain probability involving$U_1, U_2$for those$x, y$. Which can and will be zero for most$x, y$@psie split it up because the events are either disjoint or equal depending on if$x= 2y$or not 12:53 PM Oh, heck yes! A Zelda game with playable Zelda! 'Bout damn tme. I wonder how one discovered an object like compact sets Im basically done with compactness section of rudins basic topology chapter, and, aside from knowing its cool properties, I dont rly have a "feel" for this concept much more than the definition @nickbros123 Being kind of hard on Platonism, myself, I don't think about "compactness" as having been "discovered", so much as "invented". The idea of compactness is kind of natural in real analysis---think about how many theorems rely on a function being defined on some closed, bounded interval. The extreme value theorem for continuous functions, the mean value theorem, the most basic definition of an integral, etc. In this context, "closed and bounded" turns out to be a nice property. So it is reasonable to ask how much that concept can be extended. Do the theorems work for any closed and bounded set, or just intervals? What happens if you drop one of the conditions? How about in higher dimensions? Or in more radical spaces? Once you start asking those kinds of questions in the context of more general topological spaces, the idea of compactness is more-or-less natural. It gives you some idea of closed and bounded, in spaces where "bounded" doesn't make sense (e.g. because there is no metric). that makes sense I can see how "every open cover has finite subcover" kinda simulates a "bounded" criteria 1:16 PM @nickbros123 See Heine-Borel theorem it was motivated by problems in analysis There is some discussion of the intuition behind compactness on a thread from MSE. I find the idea of compactness being a topological generalisation of finiteness very compelling and useful. That being said, I don't think it is an easy concept to "discover" â€“ the history of compactness would also seem to suggest this. @Jakobian very interesting!! 1:27 PM @nickbros123 the reason for this is that you are reading Rudin @Thorgott what else would you rather I read @nickbros123 the modern definition of compactness is due to Vietoris, see his paper: Vietoris, L., Stetige Mengen, Monatsh. f. Math. u. Phys. 31 (1921),173-204. I believe the translation of the title of the paper would be "continuous sets" that's where he provides various equivalent formulations for compactness, still using countable covers though correction: apparently Vietoris didn't formulate it in terms of open covers @nickbros123: You might want to try Analysis I and Analysis II by Terence Tao. I think it's better suited for self-study of analysis than Rudin. It includes more motivation, at least Vietoris defined compactness using filter bases/nets 1:43 PM @Joe will have a look. IDK if its some kind of stockholm syndrome or inertia from having spent many a day working through Rudin, but I kinda like it well, general way in which it went is that, as far as I understand it, people were first thinking of compactness in terms of sequences, then tried to move away from it, although countability was still persistent, and then finally moved to arbitrary open covers it wasn't like a smooth process @nickbros123: I definitely think it is a very elegant book which is also a decent reference. But it is impossible for a mathematics book to be everything and anything. In the case of Rudin, I think what he gains in succinctness and elegance he loses in motivation @Jakobian is that a paper by bourbaki or is it in their analysis book? @nickbros123 their topology book 1:53 PM What is meant by [0,1] compact spaces? the book from analysis you're referring to is from 1949 I think so it means every compact Hausdorff space is a closed subspace of$[0, 1]^\kappa$for some$\kappa$there is this characterization that any Tychonoff space can be embedded into$[0, 1]^\kappa$for some$\kappa$by using continuous maps$f:X\to [0, 1]$see the property that the Tychonoff cube is universal for the space of compact spaces @Jakobian general topology, not algebraic topology 2:12 PM as another example, realcompact spaces are$\mathbb{R}$-compact there is also$\mathbb{N}$-compact spaces in literature that I know of and although this is definitely an interesting concept, I think it more natural to embedd$X$into$\mathbb{R}^\kappa$using all bounded continuous$f:X\to \mathbb{R}$, rather than just$f:X\to [0, 1]$, since the former has the benefit of making the connection between$\beta X$and the ring$C^*(X)$more explicit Perhaps rather than$\mathbb{N}$-compact spaces, there should be considered$\mathbb{Z}$-compact spaces as well, so that there might be some connection drawn between them and the ring$C(X, \mathbb{Z})$. Thanks! 2:39 PM @nickbros123 depends on what you're trying to learn @Joe As much as I like Tao's book, I would actually recommend against trying to work from both Rudin and Tao at the same time (I would recommend against trying to learn out of Rudin at all, probably). Once you pick a text, I think that it is generally best to work through it as a text, until you feel that you have gotten out of it what you want to get out of it. Switching between books can often leave gaps, as the approaches may be different. And I think that there is value in working through a single text, doing the exercises, and figuring things out on your own. If all you care about is finding some proof, the internet has millions. The goal should be to work out those proofs without looking them up somewhere else. I'd say that it is a good idea to focus on one main text and use others only as reference when struggling, but generally speaking analysis books are self-sufficient @Joe Motivation and any kind of exposition. Also, I hate his proof of the Mean Value Theorem. And the wheels kind of fall of the book in the last couple of chapters. Yeah a lot of people don't like how he treats functions of several variables. I've never read the chapter, so I can't really comment. However, I did find it useful to learn other areas of mathematics before learning any kind of rigorous calculus on$\mathbb R^n$, i.e. I went wider rather than deeper. That way, I was able to read books on analysis that could lean on tools from topology and algebra. By contrast, if you just read from Rudin, you would encounter Stokes' Theorem before even seeing a precise definition of "vector space", which is ... weird 2 hours later… 4:46 PM I've been playing with the gyroid, "an infinitely connected triply periodic minimal surface discovered by NASA scientist Alan Schoen in 1970". The true gyroid is parameterised by elliptic integrals (of the first kind), but there's a close approximation in terms of circular functions:$\cos(x)\sin(y) + \cos(y)\sin(z) + \cos(z)\sin(x) = 0$Here's the approximation: sagecell.sagemath.org/… Here's the true gyroid, which I adapted from code by Parcly Taxel gitlab.com/parclytaxel/Malibu/-/blob/master/malibu/minimal/… 1 hour later… 5:55 PM The Jewish position of "God is responsible for both good and evil" seems a lot better than the Christian one In particular there is no problem with "God is everything that's good" or whatever Christians say, there is no problem with God being good yet not stopping evil from happening also interesting how Christians just merged all those "evil" beings into one @GratefulDisciple Better in the sense of being more believable 6:16 PM @Jakobian It is better , yes , much harder to be refuted. However , there are other good arguments against any god , no matter whether it is an always good god. there's always Pastafarianism if you want a tasty alternative @robjohn Hope busy means some fun. @Peter definitely. I believe the concept of "there has to be a creator" is just your usual personification that humans do. its one of those things with which religious people try to convince you we like to see ourselves in things other than ourselves, that is nature 6:39 PM @Jakobian Correct. Since money is essential for us , it is also essential for god although god does not need money. Exactly what you say , they project all their wishes and needs to a hypothetical being. And I think , people only believe in a paradise because they wish it would exist. But why then the belief in the hell and the wish that other people arrive there ? This is what I cannot understand. Even if you think about creation, why does the world need to be created? I think even the physical theories like Big Bang don't talk about creation of the universe, only of its expansion I.e. I think Big Bang theory permits the universe to have always existed @Thorgott ...Analysis ? @nickbros123 why the question mark? Your original question was about topology Thorgott's question was reasonable @Jakobian The big bang theory was invented by a priest , our bad luck is that some observations seemed to back it up. When observations against the big bang theory occured , instead of giving it up , the dark matter was invented :( In reality , the universe was not "born" and won't "die". I don't know. A lot of physicist are for it, it might be reasonable. It won't matter to me if its true or not anyway its true that the one who came up with the theory was religious and so he had his own reasons to advocate for it but this doesn't discredit it 7:06 PM @Jakobian Big Bang theory says that space & time as we know it began ~13.7 billion years ago. BB theory talks about what happens at times t>0, but it doesn't talk about t=0 itself, because in pure General Relativity you get an unremovable singularitt at t=0. Perhaps a quantum gravity theory will avoid that problem. There's some info at physics.stackexchange.com/a/136861/123208 & the various linked questions. One option is that there's a discontinuity, and there is no actual t=0. (If you use logarithmic time, t=0 corresponds to negative infinity). A related notion is that the very earliest instants of time may not be well-orderable (due to quantum uncertainty), so for times on the order of the Planck time, it's not possible to define causal sequences. @Jakobian Indeed. OTOH, it did slow down the initial acceptance of BB theory (to an extent), by people who thought BB theory was too motivated by religious belief in a creation event. At the time, the dominant paradigm was an eternal steady-state universe. @PM2Ring very interesting, so if BB did happen, there might have been some weird shenanigans due to quantum mechanics at small times$t$could it be then said that ~13.7 billion years ago was, most likely the origin of space and time, the universe as we know it, but not necessarily the universe itself? Right. But even if we do create a consistent quantum gravity theory, it will be really hard to empirically verify it. Standard GR works superbly in regimes we can investigate. We need quantum gravity for stuff we can never observe directly, like the core of a black hole, and the state of the universe within a few Planck times of t=0. We're pretty confident about BB theory for times after a picosecond or so after t=0, because we can investigate those energies in our particle accelerators. But even a particle collider the size of the galaxy can't get to the energies involved at the Planck time. 7:23 PM The big bang theory is , as much more of the cosmology , so-called "mainstream physics" , not reasonable , it is a linear regression to an assumed event billion years ago with no evidence that it happened. This is like claiming that the USA will once have$10^{12}$people. Even if the universe was more compact in the past (not the space , but the matter contained in it) , this does not mean that it was concentrated in a point. We know the early universe was quite uniform, to ~1 part in 100,000. That's the scale of the inhomogeneity in the CMB (cosmic microwave background). That radiation was released when the universe became transparent, ~370,000 years after the BB. Presumably, there's also a CNB, the cosmic neutrino background, which was released around t=1 second (IIRC). But we can't detect it: our best neutrino detectors need neutrinos with kinetic energy ~1 million times greater, and even then they only detect ~1 neutrino per billion, or so. @Jakobian why would I learn topology from rudin @nickbros123 to learn analysis what were you trying to ask me because I don't understand you were learning topology from Rudin so its normal to think that you might have wanted to learn more topology Right, yeah thats possible But I was intending to learn analysis @Peter There's plenty of evidence for the BB. Eg, the observed spectrum of the CMB, and the isotope ratios of light elements are consistent with the theoretical predictions. But you're certainly entitled to maintain some scepticism. Plenty of cosmology is rather speculative, without a very firm foundation. @Peter The universe is probably infinite in its spatial extent. And if that's the case, then it's always been infinite. However, there is a density singularity at t=0. But that doesn't mean you can claim that the universe was a point at t=0. But as I said earlier, BB theory doesn't try to talk about t=0. :) 7:43 PM @Jakobian Tbf, I kinda already knew Rudin's topology is bare minimum and can't learn actual topology from that, A) considering (even in metric space context) I have a proprietary metric space book (which deals metric space results topologically rather than with epsilon delta) which itself goes for 300+ pages B) it's metric spaces, not the general thing I'm just wondering but, given we can make a recursive sum function into a closed form expression like: n * (n + 1) // 2 how come we can't do the same for recursive multiplication? I mean, without using iteration like with the closed form version of recursive sum @NordineLotfi I don't understand your question. @XanderHenderson sorry, let me make it clearer: given a recursive sum function (let's say in Python, that would be: pastebin.com/raw/pJjfds6m) and the closed form version would be n * (n + 1) // 2, I'm wondering if an equivalent expression exist but for recursive multiplication (same as the Python version but using *= instead of += ) (here I uploaded an iterative version since it's easier to read then the recursive version which use list slicing) I read math, not python. I am not going to try to parse your code. Can you please ask the mathematical question you are interested in, in the language of mathematics? That being said,$n!$can be expressed in terms of the Gamma function. Which means that you can express things like$\prod_{j=m}^{n} j$in terms of Gamma. ah, yeah already knew about the gamma function. I can approximate it using gamma(N+1), N being the natural number. But this is just an approximation I think, isn't there a similar expression like with the recursive sum example I provided? 7:53 PM No, it is not an approximation.$\Gamma(n)$is precisely equal to$(n-1)!$. @nickbros123 compact and connected sets are all that matters for most people that's how I look at it. Everything else is optional After having read good and convincing articles about the issues with modern cosmology , I am very careful with all this mainstream physics. Cemented ideas can still be utter wrong. But I think, this is not the right place to discuss this. This site is about math. @XanderHenderson got you, might have misunderstood then :) Thank you @Peter well, "Associated with Math.SE; for both general discussion & math questions alike." as per channels description. This chat is for math, but not only or math. Discussion about, say, types of bread, is actually welcome here @NordineLotfi this is not very clear question, what is and what isn't a closed form is a subjective question. Usually we can try and define what it means. For example, for roots of polynomials we say that generally speaking, polynomials of$5$th degree are unsolvable, but that means they are unsolvable by radicals. So this sort of "closed expression" is provably impossible for all such polynomials. But you can still find a closed expression in terms of elliptic integrals... Here Xander showed you that$n!$can be written in terms of Gamma function, which has definition as an integral, so that might be somewhat satisfactory for you. The question about if it has a certain, well-defined type of expression in case that isn't satisfactory, is most likely very hard whatever you decide on to call "a closed expression" similarly how its hard to show that only polynomials of degree$\leq 4$are solvable by radicals i.e. for the question to be objectively answerable, you would need to provide a proper definition of what it means to be a closed expression, that is, a definition not subject to your or other people opinions of course that would be a definition only for the purposes of the question itself 8:09 PM Anyone watching Euro 2024? Problem: Take the number line and randomly select a point to "zoom into". As an example, zooming into 0 takes ...(-3)(-2)(-1)(0)(1)(2)(3)... to (-3)(-2)(-1)(-1/2)(0)(1/2)(1)(2)(3)... so that the distance between 0 and 1/2 becomes 1. This is still a valid metric space, but the question is to quantify how different it is to the regular number line for this, zooming in gives the number halfway between the number you're zooming into and the numbers adjacent to it disastrous showing for Ireland in UEFA 2024 @nnabahi if I understood you well, this should be a uniformly equivalent metric, so the properties are basically unchanged say, you have a function$f:\mathbb{R}\to\mathbb{R}$defined by$f(x) = 2x$on$x\in [-1, 1]$and linearly extended to be continuous and linear with slope$1$on$[1, \infty)$and$(-\infty, -1]$with$d(x, y) = |f(x)-f(y)|$believe this is what you're trying to do 2 Motivation. The Euro 2024 soccer football championship is in full swing, and the male part of my family are avid watchers. Right now the championship is in the group stage where every group member plays against every other member exactly once. I got the impression that the games in the groups get... @SoumikMukherjee there was a lot of discussion about whether Ireland's best strategy was to lose against the NL 8:22 PM @Jakobian you're right, sorry for the misunderstanding. I'm just not really versed with the math notation or specific terminology as you can see :) @Jakobian This seems right. I'm a bit confused since wikipedia says "[uniform] equivalence requires that there is a single set of constants that holds for every pair of points in X." So at every blow up, there's a new set of constants defined for it? also, as far as I tried, the solution with gamma presented earlier is actually not always exact, e.g.: after some specific size of integer, it start to be just an approximation, with a specific error rate. @NordineLotfi No, the formula$\Gamma(n+1) = n!$is true in the mathematical sense, the two are literally equal. Your computer might be giving wrong value for$\Gamma(t)$depending on how it calculates it, but its not because math is wrong, its because computers work this way most likely the program you're using, language, whatever, package, doesn't give an exact value to$\Gamma(n+1)$@Jakobian right, from a theoretical point of view, I do agree, it is 100% exact. But on any CPU, GPU, basically any hardware or software stack, unless you can afford infinite floating precision (which will use more memory and computation), it's going to always be an approximation at some point when using it on large integer. @Jakobian I can decrease the error rate, but only through increasing the floating precision but that's only temporary if I'm working with very large number all very fuzzy 8:28 PM that's outside of the scope of what mathematician usually cares about I wonder how computer would do the gamma integral. does it just give value from the stirling approx? @nnabahi what exactly are you asking me the constants should be the same as long as the "zoom" is always by the same amount. So they work uniformly for all "blow up points" 8:32 PM in fact it shouldn't be hard to find what those constants are explicitly @Jakobian (-2)(-1)(0)(1)(2) ->on 0 (-2)(-1)(-1/2)(0)(1/2)(1)(2) -> on 1 (-2)(-1)(-1/2)(0)(1/2)(3/4)(1)(1 1/2)(2), to say that the zoom on 1 blows the right side up by a different amount than the left sweet god @Jakobian that's fair, didn't think about it from that perspective :) @nickbros123 this is gorgeous @nickbros123 gamma(N+1) is actually much more accurate than Stirling approx. Also faster depending on the implementation of gamma I tried it yesterday with a bunch of different other alternative and implementation 8:36 PM @nnabahi I think for any discussion to be had here, you need to stop using numbers and begin using actual formulas and symbols that make whatever you are trying to do well-defined and precise so many things evaporate when formalised like doubts about what is being talked about the number of convex questions had dwindled, is ai/ml losing popularity? if i listen to music with mandarin words on youtube, then all the ads i get are in mandarin. was hoping for more subtlety with prediction my overall life experience is that people skip the basics/axioms. unfortunate. then they become teachers and propagate the vagueness. not everyone, of course. indeed, i think i may have skipped peano, etc. @copper.hat trends.google.com/trends/… dwindling a little, it seems, but stable ohh, that's cute! 8:48 PM Take$\mathbb{R}$and the usual metric$d(x,y) = |x-y|$, and begin by considering only$\mathbb{Z}$. Pick a point to zoom in on, say$x$. This imposes a new distance on$\mathbb{R}$, where$d(x,\frac{x+1}{2}) = 1$and$d(x,\frac{x-1}{2}) = 1$, and otherwise distances are preserved. Both$\frac{x+1}{2}$and$\frac{x-1}{2}$are now part of our set to choose from,$\mathbb{Z} \cup {\frac{x-1}{2},\frac{x+1}{2}}$, and the process is iterated. huh? What can I clarify what do you mean by considering only$\mathbb{Z}x\in\mathbb{Z}$? the starting set of points I want to zoom in on are only from$\mathbb{Z}$, and then points are appended to$\mathbb{Z}$So, you are trying to define a distance on the dyadics? 8:55 PM you can't change only distance from$x$to$\frac{x\pm 1}{2}$because of triangle inequality that won't be a metric then I'll try a picture, otherwise I'll be back later when this is more properly formalized it might be easier if you describe what you are trying to achieve i'm not a fan of reverse engineering something that is not engineered in the first place... I think the point is to iterate this countably many times and see what we get also I think it should be$x\pm \frac{1}{2}$and not$\frac{x\pm 1}{2}$that would make some sense this exemplifies (to me, at least) the difficulties in communicating remotely. 5 mins and a white/black board would converge quickly sorry how do I send pictures here? like nickbros123 9:03 PM i use imgur but there may be more appropriate ways so that the distance between any 2 ticks is 1 there is upload button on the right side next to send so you are defining a distance on the dyadics? I don't know what a dyadic is but if that's what I'm talking about I'll go do my reading the picture doesn't explain anything 9:06 PM n/2^m @nnabahi rational number of the form$k/2^n$oh then yes in the final step of the picture, the distance between 7/8 and 1 is 1, and the distance between 2 and 3 is 1 otherwise again I'll go make it more precise and maybe try later Ah I get it$d(-1, {7 \over 8})$would be? -1/2, 0, 1/2, 3/4, 7/8, so 5 9:10 PM you need the triangle inequality to hold well anyway, what is your question we're discussing some kind of construction, but there is no question as far as I know @copper.hat as I see it, its set so that$d(x, y)$counts the number of points in$[x, y]$minus$1$but this is a distance on$\mathbb{Z}$plus whatever dyadics we have added My question is to find a way and "quantify" how different this space is to the normal number line. For example, in a random walk on the number line with step values of 1, you'd always end up an integer distance away from 0. Here though, if you zoom in on 0 a bunch of times, it would take forever to get to 1, so you've made less progress and so if we iterate this construction what is this supposed to lead us to i may have misunderstood, but i thought$d(0,1) = 1$, etc. excuse my very imprecise wording 9:13 PM @nnabahi how does this give you a number line like i say, formalising is where the work is... @copper.hat that's what I'm currently learning as far as I see it, distance isn't the same even on$\mathbb{Z}$per each step of construction so we can hardly iterate it to define it on dyadics this is a problem Ah, because my random walk question, I'm switching between the "local metric", where d(7/8,1) = 1 and the regular metric on R? In any case this has been a very helpful exercise in being clear with my language.. Thank you guys I don't see how to go on and do all those things you are claiming could be done and you don't know either it seems so I don't see much point, unless you think about it further and actually manage to formalize it I don't think this will be fruitful but who knows 9:20 PM mathematics is a dangerous siren It looks a little like you are trying to (locally) snowflake the real line. We'll see,, I'm not sure about fruitful but I think I can do some of this more rigorously I'm not quite sure what it would mean to "locally snowflake" a space, but I'm not sure what else to call it. 1 I've seen a few places in the literature (in particular in fractal geometry) where we consider a metric space$(X,d)$, and then for$0 < \epsilon < 1$define a new metric space$(X,d^{\epsilon})$where: $$d^{\epsilon}(x,y) = d(x,y)^{\epsilon}$$ This new metric space is called an$\epsilon$-snow... might help to spread the dyads out into$\mathbb{R}^2\$.
@XanderHenderson cool definition, I might use it
9:22 PM
i thought it was a political label...
@XanderHenderson that is beautiful and definitely adjacent to what I was doing
As a quick final thought, if you "zoom in" to every even or odd number, you get the regular number line but with every distance spread out by 2, again very imprecise language give me a few days :)

2 hours later…
11:29 PM
@copper.hat Some fun, but a lot of medical things for my wife and I and also for our dogs.