If you are wondering what sort of idiot would dispute a math problem with Newton, it was not a dispute at all; it's not even clear that they had a discussion at all: https://hsm.stackexchange.com/questions/5148/how-did-newton-...
Mathematicians discover new way for spheres to 'kiss'
51–60 of 74 posts
Re: Mathematicians discover new way for spheres to 'kiss'
#52Earlier quoted context omitted.
My two cents on this: I've done a lot of math, up to graduate courses in weird stuff like operator algebra. I've also read quite a bit of maths pedagogy. I've come to understand that the key thing that determines success in math is ability to compress concepts . When young children learn arithmetic, some are able to compress addition such that it takes almost zero effort, and then they can play around with the concep…
wow. I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map i…
Just a tangent, but there's a nice trick for 8 x 12.
In algebra, you learn that (a - b)(a + b) = a^2 - b^2. It's not too hard to spot this when it's all variables with a little practice but it's easy to overlook that you can apply this to arithmetic too anywhere that you can rewrite a problem as (a-b)(a+b). This happens when the difference between the two numbers you're trying to multiply is even.
For a, take the halfway point between the two numbers, and for b, take half the difference between the numbers. So a = (8 + 12) / 2 = 10. b = (12 - 8) / 2 = 2.
Here, 8 = 10 - 2 and 12 = 10 + 2. So you can do something like (10 - 2)(10 + 2) = 10^2 - 2^2 = 100 - 4 = 96.
It's kind of a tossup if it's more useful on these smaller problems but it can be pretty fun to apply it to something like 17 x 23 which looks daunting on its own but 17 x 23 = (20-3)(20+3) = 20^2 - 3^2 = 400 - 9 = 391
Re: Mathematicians discover new way for spheres to 'kiss'
#53Earlier quoted context omitted.
wow. I have been posting on hackernews "I have dyscalculia" for years in hopes for a comment like this, basically praying someone like you would reply with the right "thinking framework" for me - THANK YOU! This is the first time I've heard this, thought about this, and I sort of understand what you mean, if you're able to expand on it in any way, that concept, maybe I can think how I do it in other areas I can map i…
Calculating 8x12 in my head relies on a trick / technique - they call it "chunking", I believe, in the Common Core maths curriculum that US parents get so angry about - that (I'm also in my 40s) was never demonstrated in schools when we were kids. (They tried to make me memorize the 12x table, which I couldn't, so I calculated it my way instead; took a little longer, but not so much that anyone caught on that I wasn'…
96, easy. Lets go, real time math tutoring in the hackernews comments, 2025 baby! :D
Re: Mathematicians discover new way for spheres to 'kiss'
#54Earlier quoted context omitted.
The kind of intuition you gain for higher dimension tends not to be visual. It is more that you learn a bunch of tools and these in turn build intuition. For example high dimensional spheres are "pointy" and most of their volume are near their surface. These ideas can be defined rigorously and are important and useful. For medium dimension there are usually specific facts that you exploit. In my own work stuff like "…
I remember learning about the probability of returning to the origin in a 2D random walk versus a 3D random walk when I took stochastic processes. After we proved with probability 1 you return to the origin in a 2D walk (and with probability 0 you return in 3D) my professor said "that's why you hand a drunk man the keys to a car and not an airplane when he leaves the bar". After checking wikipedia it looks like he ri…
That's interesting, about the probability being zero in 3D. Is this on an integer lattice? The source that cannot be cited on HN without loss of karma says that the probability of returning to the origin in Z^3 is approximately 0.34.
I don't see how it could possibly be zero, even for reals, unless you're relying on the idea that the probability of any given real emerging from a uniform RNG is zero. That would seem to apply in 2D as well.
Re: Mathematicians discover new way for spheres to 'kiss'
#55Earlier quoted context omitted.
> For example high dimensional spheres are "pointy" and most of their volume are near their surface I had a visceral reaction to this. In what sense can a sphere be considered pointy? Almost by definition, it is the volume that minimizes surface area, in any number of dimensions. I can see how in higher dimensions e.g. a hypersphere has much lower volume than a hypercube. But that's not because the hypersphere became…
There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke ou…
Yes, but that can be better understood as the hypercube becoming more pointy, not the sphere. And it's true; the cube's vertices get arbitrarily far from the origin, while the centers of its faces stay at ±1.
There are other ways in which a hypersphere can be considered "pointy", though; for example, consider a point lying on the surface being moved some epsilon distance to a random direction. As the dimension increases, the probability that the point ends up inside the sphere approaches zero – the sphere spans a smaller and smaller fraction of the "sky".
Re: Mathematicians discover new way for spheres to 'kiss'
#56Earlier quoted context omitted.
> For example high dimensional spheres are "pointy" and most of their volume are near their surface I had a visceral reaction to this. In what sense can a sphere be considered pointy? Almost by definition, it is the volume that minimizes surface area, in any number of dimensions. I can see how in higher dimensions e.g. a hypersphere has much lower volume than a hypercube. But that's not because the hypersphere became…
There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke ou…
I hear this point parroted all of the time, but I think it is a misunderstanding and a poor visualization. Consider the same situation, but instead of focusing on the radius of the center sphere, focus on the distance between the spheres on the corners to the origin. For 1-dimension, these 'spheres' are unit intervals and so the distance is 1 (Central radius is 0). For 2-dimensions, these are circles at a distance of root(3) (Central radius is root(2)-1). 3-D: root(3) (Central radius is root(3)-1). Etc. So, it isn't the central circle getting more 'pointy' allowing the central radius to increase, but rather that the corner circles are getting further from the origin, allowing larger N-spheres (increasing proportional to the root of N). Thus, pointy is not the right way to conceptualize these spheres. For the more visual folk, I would recommend drawing this out and you can see this in action.
More clearly, if a sphere became 'spikey' then the distance on the surface of the spike should be further than a neighboring point, which is NOT the case.
Not trying to attack you, I just see this same point over and over and think that this warrants more thought
Re: Mathematicians discover new way for spheres to 'kiss'
#57Earlier quoted context omitted.
> etc. That's handwaving the answer just as you were getting to the crux of the matter. "Are mathematicians really visualising spheres with physical space between them" in higher dimensions than 3 (or maybe 4)? From the experience of some of the bigger minds in mathematics I met during my PhD, they don't actually visualize a practical representation of the sphere in this case since that would be untenable especially…
Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”
> Reportedly, Geoffrey Hinton said: “To deal with a 14-dimensional space, visualize a 3-D space and say 'fourteen' to yourself very loudly. Everyone does it.”
He did. You can see / hear that line in this video from his old Coursera course.
https://youtu.be/TNhgCkYDc8M?list=PLLssT5z_DsK_gyrQ_biidwvPY...
Exactly how seriously he intended this to be taken is a matter of debate, but he definitely said it.
Re: Mathematicians discover new way for spheres to 'kiss'
#58Earlier quoted context omitted.
There is a standard thought experiment where you start with a hypercube of side-length 2, centered at the origin. You then place a radius 1 sphere on each vertex of this hypercube. The question then becomes: what is the largest sphere you can place at the origin so that it is "contained" by the other spheres. As it turns out in like dimension 6 or so the radius of the center sphere exceeds 1. It will actually poke ou…
Yes, but that can be better understood as the hypercube becoming more pointy, not the sphere. And it's true; the cube's vertices get arbitrarily far from the origin, while the centers of its faces stay at ±1. There are other ways in which a hypersphere can be considered "pointy", though; for example, consider a point lying on the surface being moved some epsilon distance to a random direction. As the dimension increa…
Specifically, of course, d = sqrt(N), where N is dimension and d is distance of a vertex of the unit hypercube from the origin.
Re: Mathematicians discover new way for spheres to 'kiss'
#59The interesting ta for me: > Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.
A small anecdote: my dad is a mathematician. For a significant portion of his postdoc/early career (in the 80's/90's) he worked on proving a particular conjecture. Eventually he abandoned it and went to be much more successful in other areas. A few years ago someone found a counterexample. He was quite depressed for a few weeks at the thought of how much of his strongest research years had been devoted to something i…
At least he didn't "prove" a theorem that turned out to be false!
Re: Mathematicians discover new way for spheres to 'kiss'
#60The interesting ta for me: > Had she been one of his graduate students, he would have tried harder to convince her to work on something else. “If they work on something hopeless, it’ll be bad for their career,” he said.
A small anecdote: my dad is a mathematician. For a significant portion of his postdoc/early career (in the 80's/90's) he worked on proving a particular conjecture. Eventually he abandoned it and went to be much more successful in other areas. A few years ago someone found a counterexample. He was quite depressed for a few weeks at the thought of how much of his strongest research years had been devoted to something i…
Thanks that is a good anecdote. Did he get over it and how?
To me such a career is useful for (a) the greater good: you can't make discoveries without dead ends and (b) the maths created along the way! Or if not shares then the skills developed.