Mathematical models of reality and site swap mathematics

Laboratory 15 has seen a number of different prefaces. Originally I started lab fifteen with "the mathematics stack" starting with counting and moving up through addition, algebra, and into calculus. I then note that there are other fields of mathematics that are not "in the traditional stack" such as statistics, fuzzy math, chaos theory, and site swap notation.


 For a couple terms I shifted to presenting site swap notation abstractly on the board, deliberately obfuscating the connection to juggling. I then ask the class if they understand. Someone usually says yes, although everything I have said is devoid of any possible comprehensible meaning.

Regina Pelep

On April 10 I showed Morgan Freeman's Through the Wormhole Is Reality Real?  This was followed by Neil deGrasse Tyson's Science in America and the episode on the Higg's boson Is There a God Particle? on April 24. On April 26 I ran Is Gravity an Illusion?, in keeping with the theme that reality may not be real.


This put the course on track to return to the quotes of Galileo and Dyson which began the laboratories the first week of classes.


I led off by reading the Galileo quote from the first laboratory.

[Science] is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language,... – Galileo Galileo


At 8:00 I opted not to then read the Dyson quote, but that quote probably should be resurfaced at this point. I went ahead and shared that quote in the 11:00 section.

For a physicist mathematics is not just a tool by means of which phenomena can be calculated, it is the main source of concepts and principles by means of which new theories can be created... ...equations are quite miraculous in a certain way. I mean, the fact that nature talks mathematics, I find it miraculous. I mean, I spent my early days calculating very, very precisely how electrons ought to behave. Well, then somebody went into the laboratory and the electron knew the answer. The electron somehow knew it had to resonate at that frequency which I calculated. So that, to me, is something at the basic level we don't understand. Why is nature mathematical? But there's no doubt it's true. And, of course, that was the basis of Einstein's faith. I mean, Einstein talked that mathematical language and found out that nature obeyed his equations, too. – Physicist Freeman Dyson

Magen with site swap equations on the board behind her


Then I read an abridged version of the article on Max Tegmark's book, Is the Universe Made of Math?  Is the universe a reality or a mathematical illusion, a construction. I noted too that while this may sound strange, there are religions with teachings to the effect that, "This is not your home, when you pass away you go home," in other words, this reality that you take to be real is not real.

Pelida. Moesha, Tiffany, and Joana are in the background

I then segue from the mathematical nature of the universe and the possibility of the unreality of reality back to the idea that whether or not the universe is math, mathematical models do inform us about physical systems.

Vanessa Saret in front of a 3 pattern with a 342 swap (324) shown

With that I turned back to the world of mathematical modeling and wrote out site swap notation, not obfuscated, on the board. I used red, green, and blue balls. I diagrammed a 3 cascade using black and purple dashes for the sites.

Then I added in a red, green, and blue ball, showing that the red ball could only land three sites down range in time.

Johnner, Dannia (Tanya)

Then I diagramed a 51 shower. I first put up a red, green, and blue 1 shuttle left to right in the first six sites. Then I showed that is this set-up is taken as given, the red lead ball cannot land until five sites later. 


I then returned to the 3 pattern, swapped a red and green ball mid-sequence, and showed how this led to a swapping of the sites and the ball order down stream. 

Tedrick, Yoma

After this I asked if anyone understood the diagrams and the system, or knew what physical system was being modeled. Although one student in the morning section said he thought he understood, the other students conveyed to me only their confusion. In the afternoon section some students nodded, but when pressed could not produce an explanation for what system was being modeled.


Kimsky, however, realized that the system was juggling. Three balls and two hands. He saw through from the abstract to the concrete. 

I explained that math without a physical system, math done abstractly on a board, is not an optimal way to learn and understand mathematics. Perhaps not a way mathematics can be learned. A few math geniuses might be able to work purely abstractly, but most students need physical examples to help guide their understanding. 


I then juggled the 3 and went on to do 51 and 342 as demonstrations.

Dannia demonstrates bounce juggling

The class wrapped up with the students attempting to either learn to juggle or to learn new site swap patterns.

Dannia, Pelida. Aimina Wilson, background left.

This laboratory also has the intent of wrapping up the term, expanding the students conception of what constitutes a mathematical model of a physical system, and having some fun. Daniel Kahneman in Thinking, Fast and Slow noted that the remembering mind rates experiences using a peak-end rule. We remember the best moment and the final moments of an experience when we reflect back on that experience. Although I had not known this particular fact when I designed laboratory 15 five years ago, I had always shared George M. Cohan's belief that one should "always leave them laughing when you say goodbye. I want the students to have positive affective domain responses to science, and ensuring the end experience is fun has an impact on those perceptions.


The students start off being exposed to a mathematical system that is puzzling and confusing, but end the session having fun juggling. The balls, are becoming threadbare after a decade of use, worn, cracked in some cases, and rather dingy.


Dannia bounce juggles

An instructor who once taught journalism at the college noted that if there is no picture, then it didn't happen. The photograph is everything. My son now teaches me that if there is no video, then it didn't happen. Photographs are too easy to fake, to "photoshop" and alter. Video is harder to photoshop. 


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