Johann Carl Friedrich Gauss: In 1587 the most efficient stacking of cannonballs was addressed by Thomas Harriot and then in 1611 by Johannes Kepler. It took over 200 years before Johann Carl Friedrich Gauss actually started to prove these conjectures and about another 200 years before the conjectures were more formally proven by Thomas Hales (website) and his people (2014). This question about density had become a key mathematical challenge, deemed by David Hilbert in 1900 to be the eighteenth problem; there appears to be no references to the size of the spheres. For example, I would ask, “Is it possible to have a sphere the size of the Planck Length?” Given the ineffable work of pi, I would argue, “Yes,” and begin sphere stacking at the Planck scale.
Fourier series, transform: The goal here is to bring everyday physics and mathematics to bear to grasp the foundations of our universe so there is nothing esoteric or extra-logical about it. How very satisfying it will be if key mathematicians throughout our history, people like J. Kepler, C.F.Gauss, T.C. Hales (cubic-close packing), Poincaré-Feigenbam (period doubling bifurcation), and Fourier-Dirac-Strogatz (Fourier transform), are responsible for the concepts that describe and predict the behaviors of our infinitesimal universe.
Cubic close packing. Our knowledge of cubic close packing goes back to Thomas Harriot (circa 1587), Johannes Kepler (circa 1611), and Johann Carl Friedrich Gauss (circa 1801). More recently, through the work of Thomas Hales (1998, 2014), we learned that these scholars were each proven to have calculated a very good approximation of sphere-packed densities . Also, notably, in the 2010 Wikipedia’s summaries of this discipline inspired a programmer to create a simple, but highly-informative simulation of sphere stacking.1
Path integrals and Gaussian fixed point. See Assaf Shomer’s on page 7: “The derivation of the path integral formula in quantum mechanics of a massive particle involves chopping up the quantum evolution into very short time intervals and inserting complete sets of states between them.”
http://slideplayer.com/slide/4427954/ Physics 213: Gauss
In the letter, Langlands described a way to extend some of Carl Friedrich Gauss’ pioneering work on prime numbers. Number theorists before Gauss had noticed a hidden relationship among primes: that all the primes that can be formulated as the sum of two squares (for instance, 2^2 + 1^2 = 5 or 3^2+2^2 = 13) have a remainder of 1 when divided by 4, but didn’t know if it held true in all cases Quanta magazine reported. Gauss proved this idea in what’s now known as the quadratic reciprocity law.
Langlands took Gauss’ work and showed that the prime numbers that can be expressed as the sum of numbers raised to the third or fourth power (such as 1^3+2^3+4^3=73) can be tied to the distant mathematical realm of harmonic analysis. (This kind of analysis includes Fourier transforms, a mainstay tool used by scientists and engineers to analyze signals that have a periodic nature, such as sound waves or electromagnetic radiation spectra.)
- “As an application of the Law of Large Numbers, let z be a d-dimensional random point whose coordinates are each selected from a zero mean, 1/2π variance Gaussian. We set the variance to 1/2π so the Gaussian probability density equals one at the origin and is bounded below throughout the unit ball by a constant.” Page 13ff
- Properties of the Unit Ball
- Generating Points Uniformly at Random from a Ball Context-free language: Hopcroft & Ullman 1979, p. 100, Theorem 4.7.
- DFA Minimization
- Minimal Supersymmetric Standard Model:
• Joseph L. Walsh, Walsh matrix
November 2018 (update): “Generate n points at random in d-dimensions where each coordinate is a zero mean, unit variance Gaussian.” from Foundations of Data Science, 2.1 page 12