(For the curious motorcyclist, I recommend "Honda Common Service Manual" as a starting point.)
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?
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.
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.
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.
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.
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.
So I don't think anyone can be blamed for not getting it intuitively.
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.]
Thanks for sharing.
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.
One of my favorite lines from an early CB750 manual was "Pleased to be applying the 26mm spanner to the castellated nut. Thank you."
How can you not follow instructions when they are worded like that?
I learned about train wheels decades ago, when I briefly ran through train history as an interest. What I found interesting at the time was how thoroughly this was understood right at the advent of the age of steam. Rail engineers understood this from the beginning.
Another thing is that since the wheels ride an a narrow portion, going straight most of the time, wheel wear distorts the conical section, which leads to the wheels rubbing (instead of rolling) in corners, which accelerates the wear. So rail wheel wear is non-linear. Once the wear reaches a point wear it causes slipping/rubbing, it leads to positive feedback on the rate of wear. A failure mode that isn't obvious until you thing about it.
Turns out designing a new system to fit requirements is orders of magnitudes harder than fixing a system somebody else designed.
You see this in software all the time. Anyone can follow a tutorial. But can you start from scratch and build something novel? Can you build it such that others can maintain long after you’re gone? That’s hard.
Same with cooking. Anyone can follow a recipe. But can you design a recipe?
Making it scale economically is where all the math gets involved.