# Applications of integration applied to Rubik's cubes



## goodatthis (Nov 15, 2016)

In my calculus II course we have to do a project where we apply integration to some aspect of our lives. We've learned a whole bunch of different applications of integration, including area, volume, work, averages, centroids, moments, arc length, surface area, laminar flow, cardiac output, probability, and also some of those applications applied to parametric and polar equations. Does anyone have any ideas on how I could somehow apply integration to the rubik's cube? 

Also, I'm completely fine with learning a few more advanced topics such as matrix theory if that means a better project.


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## xyzzy (Nov 15, 2016)

Twisty puzzles are inherently discrete while calculus largely deals with the continuous, so I doubt you can make an interesting project out of the "theoretical" aspects of twisty puzzles. (I definitely would like to be proven wrong, though!)

More on the practical side, maybe you could look into cube designs and how calculus factors into that. (Moments seem particularly relevant here.)


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## Rcuber123 (Nov 15, 2016)

U can look at the steps of all the methods and calculate the probability of getting a skip for each step.

Edit: take a look at the probability thread


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## shadowslice e (Nov 15, 2016)

I once mapped the 3x3x3 cube onto the complex plane by considering stereographic projection (in a way similar to the extended complex plane through the riemann sphere just to see if I could find anything useful/interesting) but I didn't pursue it very far. Maybe you could try doing contour integration or similar though honestly, I think you might be better off trying to create a different system of notation instead of integration such as something similar to kociemba coordinates though more in depth.

If you want to continue you could also look at stuff like quaternions (maybe the integration extended to that system could prove interesting) or similar.


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## AlphaSheep (Nov 16, 2016)

Averages can be calculated by the integral from a to b of a function divided by the length of the interval. It's not useful for calculating averages for solves though, because it reduces to just the sum divided by the number of solves, so it's kind of pointless.

Any time you have a rate, you can integrate to get a forecast. If you track your times for a while you can fit a curve to the rate at which you improve, then integrate those curves to forecast what your times will be like at specific points in the future. It's easier and better just to fit a curve to the times though, but that doesn't involve integration. 

The other option is to go to classical mechanics. Speed and acceleration are always easy candidates for integration. How far do the pieces travel during a typical solve? 

Can't think of much else though. The cube is inherently discrete, so for most things you can (and should) just sum instead of integrating. Integration is the generalisation of summing for continuous systems, so while you can always apply integration to discrete systems (using numerical methods, etc) , it's hard to do so without making things unnecessarily complicated and getting meaningless rubbish out, unless there's a meaningful continuous phenomenon behind the discrete measurements.


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## Dash Lambda (Nov 17, 2016)

Hmm...
You could use integration to calculate the surface area of the irregularly shaped pieces of some modern speedcubes and look for a correlation with relative friction (which could also be affected by the type of plastic and the nature of the shape, so a correlation is not guaranteed, hence looking for it).

... That's all I could think of at the moment.


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