﻿ Spaghetti curves

# Spaghetti curves: the hidden math of a polar coordinate integration problem

Villanova University Pi Mu Epsilon Induction Talk, May 2, 2014, 9:30am Mendel Rm 115
by Robert T Jantzen

r = sin2(2.4 θ)  + cos4(2.4 θ)

Inspiration: Stewart 7e-8e Section 10.3 (removed from edition 9e):
Parametric equations and polar coordinates Figs 16, 17. [How did this talk come about?].
"Just a figure" but it seemed interesting to push the idea of arclength integration in section 10.4 while demonstrating the power of Maple (bob is curious to push math ideas). [This Figure 16 was removed from later versions of the textbook.]

Scratching on the problem it dawned on bob (eventually, bob was a bit slow here)  that there was some interesting stuff to play with here.

 Disclaimer: bob has past experience with pasta geometry. Radial oscillations of the circular cross-section of tube pasta make them "rigati" = ridged, "big" cylinder plus ridges: "rigatoni", ridged corkscrew pasta: "cavatappi rigati"] but that is another story...

First, why does this seemingly complicated expression for the radial oscillation look very much like a pure sinusoidal oscillation about the central circle at the average radius? Answering this question opens a mathematical can of worms that touches a number of important idea whose natural extension leads to the recent exciting discovery of the effects of primodial gravitational waves on the anisotropy of the cosmic microwave background.
[see our own Camille Carlisle's VU '08 astro major discussion in Sky and Telescope:

Maple scratch worksheet spaghetticircle-short.mw [full worksheet below]

Power reduction formulas: Wiki: trig power reduction formulas [inverse of double, triple, n times angle formulas]

## Part Two: Stewart's "bumpy sphere" (another intriguing figure from his calc book)

Fourier "harmonic" analysis on the circle generalizes directly to "spherical harmonic" analysis on the 2-sphere,
for example, used to analyze the celestial sphere image of the 10^(-5) perturbations of the extremely isotropic 2.725 degree Kelvin cosmic microwave background (CMB) radiation recently in the news for apparent confirmation of the inflationary model of the universe:
Maple worksheet: bumpyspheres.mw improved to bumpyspheres2.mw
[Wiki: Spherical Harmonics, CMB images, CMB power spectrum, CMB review]