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Oh I don't know about that. The solution involves trigonometric functions. Are they not geometric in nature? Said differently: periodic things in some sense ex

• mnw21cam 5y ago
  No. Trig functions can be defined independently of geometry. One way is by considering the differential equation above. Another is through the Taylor expansion of the sin(x), which is periodic as a completely emergent property. Likewise, it's hard to imagine what would happen to that very famous equation e^iπ + 1 = 0 if π were something different.
  • gspr 5y ago
    I know. But they do also reflect geometry.
• HPsquared 5y ago
  In a 1-dimensional spring-mass system there are no circles, but π emerges nonetheless. Edit: natural frequency ω = Sqrt(k/m). ω comes out in rad/s, to convert to more natural units (cycles per second) π is required.
  • gspr 5y ago
    Well, you might say that the circle emerges from the dynamics of that system. Two such spring-masses out of phase draw a circle.
  • BoiledCabbage 5y ago
    To some degree, isn't that essentially just "adding one more turtle"? Now instead of space being Euclidean, you're saying space-time is? And effectively if space-time were no longer Euclidean, then the conversion to cycles/sec would use the "alternate pi" of this non-euclidean space time wouldn't it?
• kortex 5y ago
  You can derive pi/4 from summing alternating reciprocals of odd numbers. No geometry. You do need limits (since pi is transcendental, there are no purely algebraic solutions. You can use special functions, but those have limits or other transcendentals "in them"). https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80 https://en.wikipedia.org/wiki/Leibniz_formula_for_%CF%80
  • gspr 5y ago
    Indeed, good point.
• rssoconnor 5y ago
  I'd say periodic things express motions on a loop, in the topological sense of looping back to itself. But because topology is only defined up to deformations, thus you cannot really say that loops are circles. However, there is indeed a connection to circles. One way of seeing this by looking at the solutions to this differential equation over the complex "plane" and looking at the exp(i x) and exp(-i x) solutions, and seeing how these functions wrap the real number line around in a circle. The complex plane fundamentally has a Euclidean geometry to it. So much so that if you grew up in a world with a strong non-euclidean geometry, or even grew up as some sort of digital being with no real notion of space at all, so long as you are able perform moderately sophisticated mathematics you are going to wind up discovering the complex numbers, and things like their absolute value and multiplication and the exponential function and how they map values around. Perhaps you never mentally arrange the complex numbers into a "plane", but none the less, all your basic geometric concepts are embedded into the algebraic operations on these complex numbers, and you will wind up effectively doing Euclidean geometry, even if you never identify it as such.