Start with the real interval from 0 to 1. Then chop it into two pieces, and add the product of their lengths to your score. Repeat on each piece, then on all four, and so on. What is the maximum score you can obtain?
It has a completely different beautiful solution. Think geometrically!
I guess I must be more mathy than I thought because I've always found math fascinating and beautiful. I've written posts about Euler's Identity ("a math sonnet"), and I've been a big fan of the Mandelbrot for decades.
(I'm currently watch a set of videos by a guy who is going for a world record by zooming down to beyond 10^-20,000. Incredible how the Mandelbrot behaves, especially given such a simple definition.)
I’ll definitely have to do an article about the Mandelbrot set eventually. I think next semester I’ll get to take a special topics class all about the Mandelbrot set and related things — the keyword here would be “complex dynamics”. Perhaps once I’m in that class I can do some articles on it
I look forward to those posts! I posted some Notes about it a while ago. They’re in my backtrail if interested.
Are familiar with British mathematician Dr Holly Krieger? She’s appeared in a number of Numberphile videos about the Mandelbrot. Lot of fascinating territory there.
Maybe you can answer a question I’ve had for a long time. The M-set consists of the main cardioid, its infinite circular bays and infinite “filaments” branching away. Along those filaments are an infinite number of “mini brots”. My question is whether those filaments are indeed one-dimensional curves through the complex plane?
Dr. Krieger is excellent! I grew up on Numberphile during middle and high school, and I still love watching Numberphile nowadays.
I don't know if they're 1-D or not. My gut instinct is that they would be 2-d, and it's just that the thickness is *incredibly* miniscule, but I could be wrong. One thing we do know is that the fractal dimension of the boundary is 2 - that is, the boundary of the Mandelbrot set is so "curvy" that it can completely fill up 2-d space. I'm not sure how that might play a part here, but it seems potentially relevant.
It's a wild guess on my part, but I do wonder if they're 1D. My thought is that along the filaments, a line normal to any point along the filament crosses the filament at only one point. What would the criteria be for thickness? What could define the measure of points along the normal crossing the filament? It seems simpler to imagine 1D filaments and a single point. But totally a WAG.
I read once about something called (IIRC) "rays" in connection with the M-set. Because the M-set is simply connected, no matter how deep and twisty a zoom gets, for any point outside, there is always a continuous path to the outer area. Rays, as I recall, are the putative minimum paths from any outside point deep in. By minimal they meant as distant as possible from all surrounding M-set areas. I got the impression rays extend all the way to "just outside" the filaments. Which I think is like trying to calculate a number progressively closer to zero. Zeno's paradox comes true. That planted the notion of 1D just as 0.0 is one distinct point in ℝ.
Anyway, just something I've wondered about. And meta-wondered if it was something mathematicians would be, like, "Oh, sure, that. Common knowledge, dude!"
I've been immersed in a project (writing a virtual CPU) and am far behind myself now. But catching up. I have some morning errands, but you and one other post are first on my TODO list tomorrow. (If I get in a writing mode, it might not be until afternoon.)
I'm slightly more visually oriented than text, and I like diagrams, so I have a bias towards the second solution. The first seems like a more general tool. I can recall running into inductive reasoning like that more often that "hey, presto" arguments like the second. Okay, back to the post...
Big fan of the first solution: while the second solution definitely has the same vibe as other solutions that mathematicians like to call elegant (i.e., calling in a seemingly unrelated topic to solve the problem), I'd hesitate to say it explains why the result is true nearly as well as the first, largely because it (seemingly) doesn't generalize to other similar games with different scoring methods-- this makes it seem like the second solution is a fluke or lucky coincidence, rather than something fundamental to the kind of thing that the problem is.
Definitely more of a fan of the first solution. You said induction feels more powerful than elegant, in that it doesn't really give insight to the underlying parts of the problem, but I'd definitely disagree. You're computing S(n) by computing S(m) and S(k) for smaller m and k, so it makes all the sense in the world to try and do induction (or descent - same thing)
The graph theory solution seems like it's telling me more about graphs than about splitting piles, which makes it delightful in its own right, but a less natural solution to the problem at hand.
The second problem reminds me of this one from Zach Wissner-Gross from https://thefiddler.substack.com/:
Start with the real interval from 0 to 1. Then chop it into two pieces, and add the product of their lengths to your score. Repeat on each piece, then on all four, and so on. What is the maximum score you can obtain?
It has a completely different beautiful solution. Think geometrically!
I guess I must be more mathy than I thought because I've always found math fascinating and beautiful. I've written posts about Euler's Identity ("a math sonnet"), and I've been a big fan of the Mandelbrot for decades.
(I'm currently watch a set of videos by a guy who is going for a world record by zooming down to beyond 10^-20,000. Incredible how the Mandelbrot behaves, especially given such a simple definition.)
I’ll definitely have to do an article about the Mandelbrot set eventually. I think next semester I’ll get to take a special topics class all about the Mandelbrot set and related things — the keyword here would be “complex dynamics”. Perhaps once I’m in that class I can do some articles on it
I look forward to those posts! I posted some Notes about it a while ago. They’re in my backtrail if interested.
Are familiar with British mathematician Dr Holly Krieger? She’s appeared in a number of Numberphile videos about the Mandelbrot. Lot of fascinating territory there.
Maybe you can answer a question I’ve had for a long time. The M-set consists of the main cardioid, its infinite circular bays and infinite “filaments” branching away. Along those filaments are an infinite number of “mini brots”. My question is whether those filaments are indeed one-dimensional curves through the complex plane?
Dr. Krieger is excellent! I grew up on Numberphile during middle and high school, and I still love watching Numberphile nowadays.
I don't know if they're 1-D or not. My gut instinct is that they would be 2-d, and it's just that the thickness is *incredibly* miniscule, but I could be wrong. One thing we do know is that the fractal dimension of the boundary is 2 - that is, the boundary of the Mandelbrot set is so "curvy" that it can completely fill up 2-d space. I'm not sure how that might play a part here, but it seems potentially relevant.
It's a wild guess on my part, but I do wonder if they're 1D. My thought is that along the filaments, a line normal to any point along the filament crosses the filament at only one point. What would the criteria be for thickness? What could define the measure of points along the normal crossing the filament? It seems simpler to imagine 1D filaments and a single point. But totally a WAG.
I read once about something called (IIRC) "rays" in connection with the M-set. Because the M-set is simply connected, no matter how deep and twisty a zoom gets, for any point outside, there is always a continuous path to the outer area. Rays, as I recall, are the putative minimum paths from any outside point deep in. By minimal they meant as distant as possible from all surrounding M-set areas. I got the impression rays extend all the way to "just outside" the filaments. Which I think is like trying to calculate a number progressively closer to zero. Zeno's paradox comes true. That planted the notion of 1D just as 0.0 is one distinct point in ℝ.
Anyway, just something I've wondered about. And meta-wondered if it was something mathematicians would be, like, "Oh, sure, that. Common knowledge, dude!"
This is far outside my boundaries of expertise; perhaps I’ll learn more in the spring though and be able to comment on it more intelligently then…
Unrelatedly, looking forward to your input on ITGT7. I think I got it into a form that isn’t half bad, finally
I've been immersed in a project (writing a virtual CPU) and am far behind myself now. But catching up. I have some morning errands, but you and one other post are first on my TODO list tomorrow. (If I get in a writing mode, it might not be until afternoon.)
I'm slightly more visually oriented than text, and I like diagrams, so I have a bias towards the second solution. The first seems like a more general tool. I can recall running into inductive reasoning like that more often that "hey, presto" arguments like the second. Okay, back to the post...
Big fan of the first solution: while the second solution definitely has the same vibe as other solutions that mathematicians like to call elegant (i.e., calling in a seemingly unrelated topic to solve the problem), I'd hesitate to say it explains why the result is true nearly as well as the first, largely because it (seemingly) doesn't generalize to other similar games with different scoring methods-- this makes it seem like the second solution is a fluke or lucky coincidence, rather than something fundamental to the kind of thing that the problem is.
The second solution is more beautiful, because it uses an apparently unrelated idea (graphs) to solve the problem.
Definitely more of a fan of the first solution. You said induction feels more powerful than elegant, in that it doesn't really give insight to the underlying parts of the problem, but I'd definitely disagree. You're computing S(n) by computing S(m) and S(k) for smaller m and k, so it makes all the sense in the world to try and do induction (or descent - same thing)
The graph theory solution seems like it's telling me more about graphs than about splitting piles, which makes it delightful in its own right, but a less natural solution to the problem at hand.