I worked on motorcycles for years using DIY guides and YouTube tutorials. Opening up real engineering books was an eye-opening and humbling experience that made me a better mechanic, driver/rider, homeowner and software engineer.
(For the curious motorcyclist, I recommend "Honda Common Service Manual" as a starting point.)
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?
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.
Never encountered this idea before, does it have a formal name? Most fractional-dimension spaces encountered refer to either the Minkowski or Hausdorff Dimension.
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
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.
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.
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.]
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).
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.
I worked professionally as a motorcycle mechanic from mid 1970s through the early 1980s. Honda's service manuals were very good, but my favorites were the early manuals from the late 1960s to mid 1970s, mostly for their Japanese to English translations.
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.
I wonder if a similar phenomenon is the reason my final drive chain and sprocket seems to wear a little after install, be stable for 10,000+ miles and then suddenly wear a lot.
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?
This brings to mind my amusement when friends would bash Ikea cabinets as too flimsy. I was struck that they were just strong enough to do what they claimed. No more.
Everyone loves to shit on value engineering but the reality is that it's a lot harder and takes a lot more craftiness than designing things under less price pressure. When you start talking about revisions to complex things where you have to both work fast and avoid revising too much of the thing (to keep costs down) it gets really crazy.
You do know that the architects hand off their work to structural engineers on anything more complex than a single pavement building before it gets built, do you?
It has a resonance mode at 2Hz, now successfully damped. Bridge designers know to look out for resonances at those frequencies, because pedestrians tend to identify and accentuate them by walking, especially in large numbers immediately after opening. The resonance mode wasn't identified during design because unusually it is in the horizontal plane - most bridge designs have very slow resonances in that plane (sway) but the millennium bridge is a side mounted suspension bridge, with the pylons alongside the roadway.
(For the curious motorcyclist, I recommend "Honda Common Service Manual" as a starting point.)