Yes, I'd love to, although I'm not sure when! It has indeed been disproven this summer, check it out: www.quantamagazine.org/two-students-unravel-a-widely-believed-math-conjecture-20230810/ arxiv.org/abs/2307.02749
This is such a brilliant perspective -- I would LOVE to learn more about the symmetries/connections of the Apollonian group, and this problem's traditional connection to geometric inversion. I found your channel from #SoME3, and I have absolutely fallen in love with everything I've seen!! Please please please keep it up!!
The title of this video doesn't do justice to how utterly fascinating it is! 😮🤯😅 Indeed, I wish it were longer and went into some more detail, although I wonder if -- like the Apollonian circles themselves -- the topic could just go deeper and deeper into a rabbit-hole of endless detail! 😄 Oooh! Nice shiny links in the description, too! You are so generous, thank you! Just an idea for possible future video(s?) sparked by this one: I really liked your video showing the elastic band trick for visualizing continued fraction convergents and you also connected that to Farey ... (was it 'divisions' or 'neighbourhoods' or something like that? I'm familiar with the term Farey Sequences, but your presentation of it seemed more general and more interesting). I've also seen Farey Sequences being connected to Ford Circles, and the general representation of Rational Numbers as successive mediants (which, as I've discovered on my own (though I'm sure it's not new!), can themselves be thought of as simply vectors (numerator, denominator) and simple vector addition as the 'mediant' operation). The Ford Circles corresponding to specific rationals say n/d apparently have either radius or circumference (can't remember which) as 1/d^2, apparently. This seems *_SUSPICIOUSLY_* 😉 coincidental with this video's remarkable illustration of the integer curvatures of Apollonian circles!!! Could it be that Ford Circles (and hence Farey sequences and mediants and rational numbers in general) have some sort of intimate connection to the Apollonian Circle problem(s)??? I once spent some time investigating (I'm just a hobbyist in maths) the connection between Ford Circles and the shear operations that perform individual mediant steps on rationals (e.g. the 2x2 matrix [1, 1; 0, 1] as an example), and I noticed that the Ford circles representing the integers, along the number line, can either get 'squeezed' into the packing between two other circles, or, alternatively, depending on which shear you use (e.g. the inverse matrix of a previous shear), one of the integer Ford circles ends up *expanding* to 'infinite' size to form a new straight line (e.g. y=1 or maybe it was y=1/2, I can't remember; I think it was y=1 in the case I was looking at). And, in another video I watched a while back, a similar circle-tangents problem was explained by similarly 'expanding' either one or two of the circles to show that the they became two infinite straight lines, and all the previously smaller-and-smaller packed circles (which were previously packed between the two larger circles) became identically sized circles stacked in an infinite row (or column if you prefer) each one tangent to both of the new straight lines and to its two neighbours. Like a long ribbon or strip of -- well -- Ford circles (representing integers I suppose, like in the previous paragraph)! So, in the case of these Apollonian circles you showed here, It seems to me like you could do a similar transformation with your largest 'containing' circle, and perhaps the second-largest circle (being contained) and transform them into two straight lines, and all their immediate-tangent packed circles would then form 'the integers' as Ford circles, and then I guess all the other secondary, tertiary, etc. circles would then become the Ford circles representing the rationals (and their mediants, and hence connecting with Farey sequences). (Hmm, maybe you could actually pick any two tangent circles to transform into straight lines, and all the other circles would somehow transform into.... 🤔💭❓) Am I on the right track here??? *_Here's the suggestion (finally!):_* Would that make for a good video topic for you? (Or something in that flavour, anyway?) All these originally-seemingly-very-different sub-topics in maths now seem so intimately related to me! Why don't they teach this stuff in high school or even start teaching some of it in grade school??? It's so amazing to me!
I haven't even watched the video yet, but I've been a big fan of these kind of fractals since I was young and always wanted to write a program to generate them. I just wrote a program that finds the circles very well for a very specific example. I'm excited to learn more about three circles tangent to each other lol.
Terrific stuff - I had no idea that this mystery still existed, and the visuals are engaging and helpful. Also let me say that this works very well without music. One of your other videos had jazzy piano music in the background which I found very painful and distracting - here is one vote to say no music please
Nice question! I haven't tried that, but we do is similar: we look at the curvatures modulo n. That's actually the last digit of the base n representation.
@@ProofofConceptMath Do you also check for repeated numbers? I mean, to see not just which numbers appear, but how many times they appear (i.e. circles of the exact same size).
@@robertlozyniak3661 Yes, absolutely! We can count "with multiplicity" or "without multiplicity". They are different questions, and both very interesting. Tell you what, I'll add a few references in the video description to good papers on this topic.
