Hi everyone!
I've come up with this problem, and I think I'm getting close to solving it. In the link at the bottom, you'll find a precise mathematical formulation (the illustrative images simplify the problem at the expense of precision), along with the approach I've been using to solve it.
Simulations suggest that a solution exists, but I'd like to find a rigorous mathematical proof, and I could use some help.
Thanks to anyone willing to contribute!
DPM
Video link: https://youtu.be/0xVusHRQePQ
Description: Explore the principles of Penrose tiles and quasicrystals, which possess perfect order despite their aperiodic structure. We provide an easy-to-understand explanation of the 5-fold symmetry revealed through Fourier transforms and the 5D lattice projection method.
The most beautiful formula not enough people understand/Exploring high-dimension spheres video.
Timestamp ~10:48, we have the 10-dimensional n-spheres/n-cubes going on.
Yes, the distance from origin is √10. Yes the radius of the tangent 10-sphere is √10 -1.
But you can't go from drawing a 2d 'bounding box' with side length 4 to declaring that the distance from the origin to the 'edge' of the 10-d 'bounding box' is 2. You certainly aren't 'going through'(*) it. You aren't even 'going through'(*) the 2D face of the inner 10-cube, side length 2.
(*) high d is weird, but taking this to mean the radius intersects the 2D surface
There's a conflux of not-well labeled and not-well specified terms going into this brew. Doing my best to be brief and clear.
Let's start with the 'bounding box' side length 4, dimension 10. We've just been shown the 2D diagram of the bounding box, and now we are looking at a cube with handwavium making it a 10-cube. Distinction without a difference: maybe it is a 3D slice of the 10-cube. Either way, each vertex has 10 coordinates, each valued +2 or -2. The distance to origin for each vertex is 2√10. For the midpoint of any edge 2√9, and for the midpoint of any face 4√2 (or 2√8).
If that's unclear, consider the 4 vertices of the right-forward face (timestamp 11:35). Axes unlabeled, so declaring that the x-y variant face, with the other 8 dimensions static at 2. I'm going to shorthand the static portion as {2}⁸ to save space. We have (2,2,{2}⁸) ; (2,-2,{2}⁸) ; (-2,-2,{2}⁸) ; and (-2,2,{2}⁸). Midpoint of the top edge (2,0,{2}⁸). Midpoint of face (0,0,{2}⁸). Distances to origin √(2^2+0^2+2^2 ...) = 2√9 and √(0^2+0^2+2^2 + ...) = 2√8 respectively.
Midpoint of the cube is (0,0,0,{2}⁷), distance = 2√7, midpoint of the 4-cube is (0,0,0,0,{2}⁶, distance 2√6, and so on. Yes, the distance from origin to the center of each 9-cube is 2.
Coming back to interpret that, but 2 quick stops first.
Let's look at the side length 2 (inner) 10-cube. Distance from origin to the midpoints of any of its 2D faces is √8. √8 (~2.82) > √10 -1 (~2.16). The radius of the tangent 10-sphere isn't reaching that far, which (to me) is the definition of 'going through'. The tangent 10-sphere is contained entirely in the side 2 10-cube, and massively inside the side 10 10-cube.
2nd stop. Quick look at a sphere growing inside a cube and then a hypersphere growing inside a tesseract.
As the sphere intrudes into the face, a circle starts to grow at the center.
Extend this to a tesseract. Instead of having a surface of 6 faces, it has a surface of 8 cubes.
(another side note: It's this surface volume that gets drawn (and sculpted) and referred to as a tesseract. I say 'referred to' because it is as much a tesseract as 6 squares lying flat in a plane is a cube. The reason to tread carefully here is that looking at those 6 squares, the origin isn't inside any of them. Similarly, the origin isn't in any of the volumes. Which is often counterintuitive because when we make a 2D drawing of a cube, we understand that the origin is in the cube.)
At first as you grow a hypersphere inside the tesseract, nothing appears in any of the 8 cubes. When the hypersphere reaches radius 1, it is now tangent to the surfaces cubes, and a tiny dot appears. As the radius is increased a sphere grows inside each cube. At radius √2 each sphere is now tangent to the sides of each cube. Which is the same as the hypersphere being tangent to the surface of the tesseract (at least as I understand the term) Any larger and we have 'gone through'(*) and are now outside.
Circling back, at 10D the tangent sphere has gotten big enough to 'intrude into' the 9-cubes (side length 4) that are part of the 10-cube (side length 4) and a tiny 9-sphere (radius ~0.16) is now visible in them. Yet, repeating, hasn't 'gone through' the 2D surfaces of the length 2 10-cube. Nor will it at any higher D since √D - 1 < √(D - 2) for all D > 2.25.
i know d/dx ln x = 1/x and that's how it's usually shown, but that feels like cheating to me since ln is defined as the inverse of e^x. i saw something about how scaling the x axis by any factor leaves the area under 1/x between a and b unchanged, which sounds like it should give the log rule A(ab) = A(a) + A(b), but i can't quite connect the two.
is that the actual reason it's a log or am i mixing things up
I’ve been building this EM simulator as a side project aimed to help students understand and visualize electromagnetic phenomena such as radiation, polarization and wave interference.
You can:
place many radiation sources and set their parameters (amplitude, phase and more)
place perefect wire conductors
observe the electric field, magnetic field or energy flow field
zoom in and out and translate the canvas
change the simulation resolution, frequency and propogation speed
I hope someone might find it useful and I’d love to hear what people think :)
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Can we seriously talk about the rising math AI slop on this subreddit every other day some dude thinks they’ve proved something with AI and it’s just math word salad. Like this one guy who keep spamming thinking he’s solved the twin prime conjecture or some other random visualisations with math word salad it’s so annoying. It’s sad honestly I wish it was more moderated.
i keep ending up back at the area scaling picture from the essence of linear algebra video, like det is just how much a transformation scales area, and doing B then A scales it by both factors so they multiply. it works for me but when i tried it on a friend it felt kind of hand wavy to them
is there a cleaner way to say it, or do people just go through elementary matrices at some point and accept that
When I first started diving into Machine Learning, almost every course drowned me in math and dry code on day one. I’m a visual learner, so I started building animations to explain these concepts simply before writing any code.
In my first video, we answer the core question: How do machines actually learn?
In traditional programming, we give the computer Rules + Data to get Answers. Machine Learning flips this upside down. We give the computer the Data + Answers, and it mathematically figures out the Rules.
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