The discussion centers on the connection between Wallis' formula for π and quantum mechanics, highlighting the formula's geometric proof and its applications in quantization structures. Wallis' formula is expressed as π/2 = (2/1) * (2/3) * (4/3) * (4/5) ... A geometric proof can be found in a linked PDF, while the user also mentions exploring zeta and log functions in relation to Wallis' product. The conversation emphasizes the potential for these mathematical concepts to intersect with quantum mechanics and artistic applications.
PREREQUISITES
Understanding of Wallis' formula for π
Familiarity with geometric proofs in mathematics
Knowledge of zeta functions and their properties
Basic concepts of quantum mechanics
NEXT STEPS
Research geometric proofs of Wallis' formula
Explore the properties of zeta functions and their applications
Study the relationship between quantum mechanics and mathematical frameworks
Investigate the use of quantization structures in art
USEFUL FOR
Mathematicians, physicists, artists exploring mathematical concepts, and anyone interested in the intersection of geometry, quantum mechanics, and artistic expression.
Does anybody know what the connection is between Wallis' formula for ##\pi## and quantum mechanics? There was an article about it: https://www.eurekalert.org/pub_releases/2015-11/aiop-ndo110915.php
but like all articles for the lay public, all the details were left out.
I've always thought there should be a geometric proof of this, say approximating a semicircle by rectangles of some sort, but the proofs I've seen are analytic.
3Blue1Brown has a video about a geometric proof of the Wallis product:
#4
Matt Benesi
134
7
3blue1brown rocks. Not sure about dropping the 0 though. Didn't get all the way through.
A while back (didn't finish it yet), I started working on a zeta/log chart, with a few switching formulas between them (and I was thinking about using it as a quantization structure for some art). The Wallis Product pops up (a few thingies down):
Since, log(\frac{x}{y}) =\sum\limits_{n=1}^{\infty} \, (\frac{x-y}{x})^n \cdot \frac{1}{n}, if you set y=x-1...
<br />
\begin{array}{|c|c|c|c|}<br />
\hline & \frac{\zeta(1)}{1} & \frac{\zeta(2)}{2}& \frac{\zeta(3)}{3}& \frac{\zeta(4)}{4} & \frac{\zeta(5)}{5}&\dots\\<br />
\hline \lim\limits_{x\to 1^+} log(\frac{x}{x-1}) & \frac{1}{1 \cdot 1^1} & \frac{1}{2 \cdot 1^2} & \frac{1}{3 \cdot 1^3}& \frac{1}{4 \cdot 1^4}& \frac{1}{5 \cdot 1^5}&\dots\\<br />
\hline log (\frac{2}{1}) & \frac{1}{1 \cdot 2^1} & \frac{1}{2 \cdot 2^2} & \frac{1}{3 \cdot 2^3}& \frac{1}{4 \cdot 2^4}& \frac{1}{5 \cdot 2^5}&\dots\\<br />
\hline log (\frac{3}{2}) & \frac{1}{1 \cdot 3^1} & \frac{1}{2 \cdot 3 ^2} & \frac{1}{3 \cdot 3^3}& \frac{1}{4 \cdot 3^4}& \frac{1}{5 \cdot 3^5} &\dots\\<br />
\hline log (\frac{4}{3}) & \frac{1}{1 \cdot 4^1} & \frac{1}{2 \cdot 4^2} & \frac{1}{3 \cdot 4^3}& \frac{1}{4 \cdot 4^4}& \frac{1}{5 \cdot 4^5} &\dots\\<br />
\hline log (\frac{5}{4}) & \frac{1}{1 \cdot 5^1} & \frac{1}{2 \cdot 5^2} & \frac{1}{3 \cdot 5^3}& \frac{1}{4 \cdot 5^4}& \frac{1}{5 \cdot 5^5} &\dots\\<br />
\hline \dots &\dots & \dots&\dots &\dots&\dots&\dots\\<br />
\hline<br />
\end{array}<br />
A couple of functions for smooth movement between quantized log and zetas:
Log shift function, with x ? N a ? R : [0,1].. note that it's 2a/ 2n, so I drop the 2:
From the above, from the Wallis product for pi:
\sum\limits_{n=1}^\infty \,-1^{n+1} \, \cdot \frac{\eta{(n)}}{n} \, = \, log(\frac{\pi}{2}) I kept on going, because I thought that perhaps I could make a quantized framework for some art- connections between adjacent points in space would be mapped x to zeta, y to log, z to zeta/eta depth (determined by 1-2^{1-s})... not finished, because I have to do real work too.
When we do a zeta to eta conversion, we lose log(2), and are adding different logs together to get log (pi/4):
I got to the point where subsequent zeta/eta conversions add depth, have log charts, etc. so they can be stacked. It's written on paper though... and needs to be completed. But I ended up working on something else because this computer was a gift from Keith Anderson in the fractal entertainment industry (Fractaled Visions), so I feel like I should work on fractals. A bit.