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Comment by 3pt14159 | Hacker News Reader

I know right? I also loved the times where I got to the math and all that internal intuition lined up completely. Torsional deformation for example. I almost blamed myself for not inventing the math myself it so obviously matched my intuitions. It's kinda fun no matter which way it goes.

Black body radiation never quite sat right with me. One of the few subject areas where I just resigned myself to memorizing the formulas and moving on with life. Same with non-integer dimensional spaces for the most advanced partial differential course I took. I can visualize 2 million dimensional spaces just fine professor. But one and a half? What does this even mean?

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It just means that the degrees of freedom aren't used fully. This is usually seen in fractals, where there is a level of redundancy that is respective of the fractional dimension missing.

Think about it like this. If I have a 2d field (x,y) and I enforce every point's x value to be 0, I pretty much just made the x degree of freedom redundant, and can now call the field a 1d field. If instead, I enforce every 3rd point's x value to be 0, I've now got a 1+2/3 dimensional space. Because there is some redundancy, I no longer get the full entropy that 2 dimensions provide.

That is a great, simple explanation, thanks.

Never encountered this idea before, does it have a formal name? Most fractional-dimension spaces encountered refer to either the Minkowski or Hausdorff Dimension.

It's equivalent to the measures you mentioned.

Here's a paper which discusses fractional degrees of freedom: Effective degrees of freedom of a random walk on a fractal by AS Balankin · 2015 · Cited by 42 — This allows us to define the fractional dimensional space allied ... number of effective dynamical degrees of freedom on the fractal

https://pubmed.ncbi.nlm.nih.gov/26764671/

To hammer the relationship home, consider the holographic principle, which is based on observation of black holes, and states that our reality only needs 2 spatial dimensions instead of 3. Both Hawking and Susskind have eluded to this being the case solely because of the symmetries in the laws of physics. The symmetries cause a massive redundancy/pattern in the field values over the 3d space, such that we should theoretically be able to predict the state of the entire field given only the values at two-thirds of the volume.

Therefore, you can imagine a 2d surface which contains the state of our universe, and some kind of computational (possibly geometric/algebraic) projector, which understands the redundancies, reads the 2d surface, and renders a sparse 3d volume. In the case of our universe, the projection operation might be extraordinary complex, requiring a deep understanding of the laws of physics and the redundancies they induce into the underlying state that they operate on.

https://en.wikipedia.org/wiki/Holographic_principle

I looked at you information, and your quote interesting.

If you happen to come back, what do you mean by this partial quote,

"Hardy couldn't have been more wrong about the innocence of pure mathematics."

I'm just getting more interested in math the older I get.

Thanks in advance.

Math is more interesting, purely as math, as I get older. I found the same is true for history as well. The trick is to find good history authors for the particular time you are interested in. Same with math.

I graded very well in math through college, but later in life, I went back and explored more how all the concepts relate. In college, you are sort of fed calculus through a fire hose, and you just have to 'accept it' and move on. And you are left wondering, how were these ideas, these conclusions, reached? Until you go back through and see the long history of infinite series and see the various attempts to codify solutions. The problem is, as a student, you cannot possibly spend that much time deriving the whole solution from scratch and still hope to finish a degree in four years. As Carl Sagan said “If you wish to make an apple pie from scratch, you must first invent the universe.”

Which is why you can never stop learning. I am in my mid 60s, and I still learn something new regularly. Something big at least once a year, something smaller at least once a month. Never stop learning.

I meant that mathematics has been used for nuclear bombs, breaking cryptography, and other nefarious purposes. Physics is a sort of applied mathematics, where reality is the testing ground for models created on paper.

It was originally thought that Einstein's theory of special relativity was just a cool mathematical idea. But it was much much more. Mathematics has consequences that are so deep that even now it is rearing its head in AI..differentiable loss functions.

Isn't black body radiation a problem that drove scientists crazy for many years before Einstein found an explanation that got him a Nobel prize. That explanation is what started the huge mess that is quantum physics.

So I don't think anyone can be blamed for not getting it intuitively.

Yep. It was literally a catastrophe for the traditionalists in physics.

And literally referred to as the Ultraviolet Catastrophe

https://en.m.wikipedia.org/wiki/Ultraviolet_catastrophe

I'm not 100% sure whether or not you're attributing the right physicist here ;).

Max Planck wrote about quantized energy emissions from black bodies first in 1900. With this assumption, the spectrum of black body radiation could be derived successfully. That won him the Nobel Price in 1919. Albert Einstein postulated that light itself was quantized in one of his famous series of papers in 1905. This paper won him the Nobel Price in 1922.

[Side note: confusingly, Max Planck was awarded the 1918 Nobel Price and Albert Einstein was awarded the 1921 Nobel Price. This happened because the committee decided in 1918 and again in 1921 that none of the candidates met their standards and withheld the price for later.]

Ha! I didn't know that. Well, I feel a bit better now. Normally the old physics engine in my head could handle stuff, but that one was impossible.

Thanks for sharing.

I have seen a couple ways to assign meaning to the words "one and a half dimensional" and neither of them are actual spaces (i.e. have a basis, are full of vectors).

Compare https://math.stackexchange.com/questions/1466820/vector-spac... and perhaps https://faculty.math.illinois.edu/~reznick/CantorSetArithmet...

https://www.princeton.edu/~fhs/fhspapers/fhspaper91.pdf might come closer?

I don't understand the first one, the second one isn't about continuous spaces (cantor dust has gaps), but the last one seems pretty interesting and I'm working through it. Thanks for the links!

Edit: The last paper only describes fractional dimensions in a limited sense, because the way the author constructs the coordinate system still involves a positive integer number of coordinates.

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