Physical 3x3x3x3 hypercube and other physical 4d twisty puzzles [video]
youtube.com
youtube.com
But it drives home the reason why I parted ways with advanced math. There just came a point when the math no longer resonated with the natural world that I was comfortable living in.
Imaginary (and complex) numbers were the first real schism where my mind repelled the very idea.
Nonetheless, decades later, I would be talking with a friend who is much smarter than me. We were talking about the lossy nature of converting audio from the time domain to the frequency domain and back. I understood that phase information was lost in the translation (I also thought it was curious that the human brain seemed to be phase-agnostic and so could not perceive the difference when audio was round-tripped through the frequency domain).
"Oh," he said, "when you do a Fast Fourier transform to move audio into the frequency domain the phase information is represented by the imaginary part of a complex number."
Wait, what? So complex numbers have a real-world analog in constructing an FFT — a "real" algorithm that does real work?
I see now that the really smart people just understand the natural world much better than I do — or perhaps see it differently than I do.
https://youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFit...
This is also a great series of videos about how, like it says on the tin, the imaginary numbers are real:
https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
If you're interested, here are some videos about Fourier analysis:
So many of the names of mathematical objects are arbitrary and I think that can be harmful. "Real" numbers are just as imaginary as "imaginary" ones! They both have use in physics as models for natural phenomena, but neither are real like an atom. You may still argue that "natural" numbers are real because we really can have n of something, but general real numbers can't be written down or computed or used to divide physical space and time as we understand them now. The names are mostly historical artefacts that serve as mnemonics, nothing more.
You'd invent them, too, if you spent a few weeks building a computer program to output the values of `t` that make `t^3+pt+q=0` (when `4p^3 + 27q^2 < 0`, there are 3 real solutions, but they can't be expressed in general without involving complex numbers).[2]
[1] https://en.wikipedia.org/wiki/Complex_number#History [2] https://en.wikipedia.org/wiki/Cubic_equation#Cardano's_formu...
If extending the number line bothers you, you should also have a problem with negative numbers. And, in fact, many ancient mathematicians did, so you're in good company :)
No. Complex numbers are "simply" neat construct to represent 2D coordinates algebraically - i.e. without matrices.
Their existence is a natural consequence of the rules of the algebraic operations (addition, subtraction, multiplication, division, powers and logarithms); they aren't incidental, they are integral. Without imaginary numbers, the algebraic operations aren't closed.
(This being a brief summary of the thesis of the videos I linked in a sibling comment.)
That being said, they are useful as coordinates, as well.
The puzzle can be assembled to invalid states, so yes it does come to the user to ensure that only allowed or "canonical" moves are used.
Thank you! It means a lot! Especially because I am just a high schooler...
Yes, it is a lot more complicated this is because of the nature of what the gyro is doing. The gyros are effectively a 4 dimension rotation of the puzzle that actually doesn't change the state of the puzzle, but just the orientation. This doesn't work out to be very pretty in 3d.