Mathematicians discover new way for spheres to 'kiss' • Hacker News Board

Re: Mathematicians discover new way for spheres to 'kiss'

#71

Post by eszed » Fri, Jan 17, 2025, 6:27 AM UTC

neom wrote: post #69

eszed wrote: post #68

Earlier quoted context omitted.

:-) 12 is made up of a 10 and a 2. What's 8 x 10? 80. What's 8 x 2? 16. Add 'em up? 96, baby! They teach you to do math on paper from right to left (ones column -> tens column, etc), I find chunking works best if you approach from left to right. Like, multiply the hundreds, then the tens (and add the extra digit to the hundreds-total you already derived), then the ones place (ditto). It's limited by your short-term m…

Seems my math teachers in school...er..didn't. That makes sense, I know how to write math out on paper and solve it, but then my instinct has always been to reach for that method mentally, so I literally draw a pen and paper in my imagination, and look at it and do the math and it takes way too long so I just give up, this seems like I can just learn more rules and then apply them, as long as I have the rules. Thank…

My pleasure! I'm no one's idea of a mathematician, but I enjoy employing arithmetic tricks and shortcuts like this one.

A few years ago I had an in-depth conversation with a (then) sixth-grader of my acquaintance, and came away impressed with the "Common Core" way of teaching maths. His parents were frustrated with it, because it didn't match the paper-based methods of calculation they (and you and I) had been taught, but he'd learned a bunch of these sorts of tricks, and from them had derived a good (probably, if I'm honest, better than mine) intuition for arithmetic relationships.

Re: Mathematicians discover new way for spheres to 'kiss'

#72

Post by iNic » Fri, Jan 17, 2025, 9:08 AM UTC

carltg_ wrote: post #56

iNic wrote: post #42

Earlier 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…

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…

Yes that is true, but there are other ways to see spiky-like behavior.

First, the volume of spheres (or balls rather) in higher dimensions goes to zero as the dimension grows. Said another way, to keep unit volume on a ball you need to grow the radius more and more (which I interpret as spiky).

Second, the volume of spherical caps grows like ~exp(- d h^2 /2), in particular the caps lose volume fast in higher dimensions. To interpret this as "spikyness" I like to visualize it as two balls intersecting (which is just 2x the cap volume). If they are of the same radius, but their centers are just slightly off their intersection volume goes to zero quickly!

Re: Mathematicians discover new way for spheres to 'kiss'

#73

Post by jacobolus » Sat, Jan 18, 2025, 8:18 AM UTC

neom wrote: post #53

eszed wrote: post #48

Earlier quoted context omitted.

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

There are lots of good ways to break down this multiplication problem:

8 × 12 = 8 × (10 + 2) = 8 × 10 + 8 × 2 = 80 + 16 = 96

8 × 12 = (10 − 2) × 12 = 10 × 12 − 2 × 12 = 120 − 24 = 96

8 × 12 = (10 − 2) × (10 + 2) = 10 × 10 − 2 × 2 = 100 − 4 = 96

8 × 12 = (5 + 3) × 12 = 5 × 12 + 3 × 12 = 60 + 36 = 96

8 × 12 = 4 × 24 = 2 × 48 = 96

8 × 12 = 2³ × (2² × 3) = 2⁵ × 3 = 32 × 3 = 96

etc.

Re: Mathematicians discover new way for spheres to 'kiss'

#74

Post by burjui » Mon, Jan 20, 2025, 7:54 AM UTC

nejsjsjsbsb wrote: post #60

anarchonurzox wrote: post #17

Earlier quoted context omitted.

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.

At least, we can assume that after a few weeks, he became not quite that depressed.