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I know it sounds like a fudge factor that can be made to do anything, but the truth is that particle models of weakly interacting dark matter can only be made t

I know it sounds like a fudge factor that can be made to do anything, but the truth is that particle models of weakly interacting dark matter can only be made to do a few things. If any distribution of matter imaginable could be created through dark matter, you're right, that would be an empirical fudge factor, but only a few distributions of matter can be created this way, and it's noteworthy that one of them matches the one in real life. An example of a limitless fudge factor is epicycles. They could be made to do anything - now through our understanding of the Fourier transform, we can make them draw elephants. That's not what CDM is like because it can't be made to draw elephants.

denton-scratch 5y ago
Thank you. I didn't know that epicycles had the creative expressiveness to generate elephants, I'm impressed, and I'll have to dig into that - sounds interesting. I thought that Ptolemy's epicyle ideas were defeated precisely because they were not able to explain observations.
- whatshisface 5y ago
  This isn't an elephant but it demonstrates the principle: https://www.youtube.com/watch?v=QVuU2YCwHjw https://www.youtube.com/watch?v=QVuU2YCwHjw Epicycles were originally considered a good idea because it takes exactly two circles to make an ellipse, and Kepler's laws were satisfied by such an ellipse. They were "falsified,", in the two-cycle form, when closer observations revealed that Kepler's laws were not obeyed exactly. At that point, the choice was to either add more epicycles (working your way up to an elephant), or for Newton to discover universal gravitation.
- stordoff 5y ago
  Drawing an elephant: https://i.stack.imgur.com/tOU5r.gif https://i.stack.imgur.com/tOU5r.gif Taken from this StackExchange post: https://mathematica.stackexchange.com/questions/171755/how-can-i-draw-a-homer-with-epicycloids https://mathematica.stackexchange.com/questions/171755/how-c...
  - denton-scratch 5y ago
    That's magnificent! Thanks! So it looks as if you can draw anything with epicycles. But is that appearance correct? I mean, has anyone ever looked into it? Can it be proved that you can draw anything with epicycles? Relatedly: Is that elephant-drawing machine capable of drawing anything? And relatedly-relatedly, is the idea of a Turing-complete spirograph totally off-the-wall? (Don't worry, I already take loads of medication!)
    - vecter 5y ago
      Fourier Series [0] (infinite sums of sines and cosines) can be made to equal any "well-behaved" function. This is the same idea in more dimensions. [0] https://en.wikipedia.org/wiki/Fourier_series https://en.wikipedia.org/wiki/Fourier_series
      - denton-scratch 5y ago
        OK, I get it. Thanks. [Edit] Hang on - so why can't you use epicycles to draw the actual orbits of planets? (modulo the n-body problem) If epicyles is equivalent to fourier, and can model any "ordinary" function, then I'm surprised someone hansn't made an epicycle model that's equivalent to the Newtonian model. Just wondering.
        tene 5y ago
        You can use epicycles to draw the actual orbits of planets. That's what "epicycles" are, or were: a fairly accurate model of the observed orbits of planets relative to Earth built through a composition of circular motions. One way to think about it is that you start by modeling the motion as a simple circular orbit. You then look at how the observed data differs from your circular model and add an epicycle whose parameters minimize the difference between your new model and your observations. Repeat this process until you get as close as you want to get. The success of heliocentric elliptic-orbit model isn't that it's able to represent something impossible with epicycles, but that it's dramatically simpler. Kepler's three laws are: 1) The orbit of a planet is an ellipse with the Sun at one of the two foci. 2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3) The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. In contrast, the epicycle models had 40-80 cycles with parameters tuned to match the observed data. I really like this video on how Fourier series work: https://www.youtube.com/watch?v=r6sGWTCMz2k https://www.youtube.com/watch?v=r6sGWTCMz2k

denton-scratch 5y ago

Thank you. I didn't know that epicycles had the creative expressiveness to generate elephants, I'm impressed, and I'll have to dig into that - sounds interesting. I thought that Ptolemy's epicyle ideas were defeated precisely because they were not able to explain observations.

whatshisface 5y ago
This isn't an elephant but it demonstrates the principle: https://www.youtube.com/watch?v=QVuU2YCwHjw https://www.youtube.com/watch?v=QVuU2YCwHjw Epicycles were originally considered a good idea because it takes exactly two circles to make an ellipse, and Kepler's laws were satisfied by such an ellipse. They were "falsified,", in the two-cycle form, when closer observations revealed that Kepler's laws were not obeyed exactly. At that point, the choice was to either add more epicycles (working your way up to an elephant), or for Newton to discover universal gravitation.
stordoff 5y ago
Drawing an elephant: https://i.stack.imgur.com/tOU5r.gif https://i.stack.imgur.com/tOU5r.gif Taken from this StackExchange post: https://mathematica.stackexchange.com/questions/171755/how-can-i-draw-a-homer-with-epicycloids https://mathematica.stackexchange.com/questions/171755/how-c...
- denton-scratch 5y ago
  That's magnificent! Thanks! So it looks as if you can draw anything with epicycles. But is that appearance correct? I mean, has anyone ever looked into it? Can it be proved that you can draw anything with epicycles? Relatedly: Is that elephant-drawing machine capable of drawing anything? And relatedly-relatedly, is the idea of a Turing-complete spirograph totally off-the-wall? (Don't worry, I already take loads of medication!)
  - vecter 5y ago
    Fourier Series [0] (infinite sums of sines and cosines) can be made to equal any "well-behaved" function. This is the same idea in more dimensions. [0] https://en.wikipedia.org/wiki/Fourier_series https://en.wikipedia.org/wiki/Fourier_series
    - denton-scratch 5y ago
      OK, I get it. Thanks. [Edit] Hang on - so why can't you use epicycles to draw the actual orbits of planets? (modulo the n-body problem) If epicyles is equivalent to fourier, and can model any "ordinary" function, then I'm surprised someone hansn't made an epicycle model that's equivalent to the Newtonian model. Just wondering.
      - tene 5y ago
        You can use epicycles to draw the actual orbits of planets. That's what "epicycles" are, or were: a fairly accurate model of the observed orbits of planets relative to Earth built through a composition of circular motions. One way to think about it is that you start by modeling the motion as a simple circular orbit. You then look at how the observed data differs from your circular model and add an epicycle whose parameters minimize the difference between your new model and your observations. Repeat this process until you get as close as you want to get. The success of heliocentric elliptic-orbit model isn't that it's able to represent something impossible with epicycles, but that it's dramatically simpler. Kepler's three laws are: 1) The orbit of a planet is an ellipse with the Sun at one of the two foci. 2) A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. 3) The square of a planet's orbital period is proportional to the cube of the length of the semi-major axis of its orbit. In contrast, the epicycle models had 40-80 cycles with parameters tuned to match the observed data. I really like this video on how Fourier series work: https://www.youtube.com/watch?v=r6sGWTCMz2k https://www.youtube.com/watch?v=r6sGWTCMz2k