Simple math questions about Conway's Game of Life

[Tikax]

Hi everyone,

I'm a French student, and this year I'm graduating from high school. As part of my graduation requirements, I have an oral exam on a science subject (either mathematics or physics) that I've chosen. I need to prepare something for both disciplines and have decided to discuss the Game of Life for my math question. However, to adhere to the exam rules, I must incorporate mathematical concepts related to the topics we've covered in class. While I know there's a lot of math in Conway's Game of Life, I'm struggling to find suitable approaches related to specific notions.

The math portion of my presentation should last at least 3 minutes and must be related to one of the following topics:

Combinatorics and counting
Vectorial geometry, including lines and planes in space
Planes, scalar product, orthogonality, and distance in space
Sequences and recurrence
Limits of functions
Continuity
Differentiation and convexity
Natural logarithm function
Trigonometric functions
Antiderivatives
Differential equations
Integral calculus
Sequences of independent events and binomial distribution
Sum of two variables
Law of large numbers

Could someone please help me find something in the Game of Life that can be related to one of these topics? It could be a problem, a particular way certain structures function, or anything that can be developed for at least 3 minutes. If I can't find a suitable topic, I'll have to choose another math subject for the exam, but I'm really keen on making an interesting presentation about the Game of Life because I'm fascinated by it every day when I play.

Thanks in advance for your responses; even a small suggestion would be greatly appreciated!

[hotdogPi]

The number of 1xn patterns with no dots or dominoes (i.e. that don't die immediately) has the ratio of f(n):f(n-1) approach phi.

The law of large numbers can be seen in "is A more common than B given a specific sample size" as the sample size tends toward infinity.

There are quadratic growth patterns. You can take the derivative to see the increase per generation.

[dvgrn]

How about the "Limits of functions" item? It would take about three minutes to go through the math to check and verify, for example, one of Dean Hickerson's "unusual growth rate" patterns -- like the "Life computes pi" one, for example.

Or condense a presentation like that into a minute or so, and then build a new pattern with some metric that converges on some other interesting constant, and go through the math for that. Three minutes will get used up very quickly explaining things along these lines.

[Haycat2009]

I do not think any CA can do natural log, just an approximation at best. Reason being that time in Cgol is discrete, and as you know, e cannot be represented as a fraction.

[confocaloid]

I do not think any CA can do natural log, just an approximation at best. Reason being that time in Cgol is discrete, and as you know, e cannot be represented as a fraction.

Nothing prevents implementing symbolic manipulation inside a Life universe. Expressions involving logarithms could be transformed/simplified with appropriate data structures and algorithms implemented using only arbitrary-precision integer arithmetic. Terribly slow and impractical (compared to installing and running an actual computer algebra system directly rather than in a cellular automaton), but possible.

[dvgrn]

Yup.

For example, it would be an ambitious but not impossible project to use the tools described in the Universal Computation chapter of the Life textbook (Chapter 9) to write a program that calculates the value of e and prints it out in the Life universe.

There's plenty of other math -- combinatorics and so on -- in that textbook, so flipping through it might bring up some other good ideas. Just pick a topic, run it past whoever will be judging the graduation requirements to make sure there won't be any last-minute objections, and things should be good.

[Tikax]

Hello,
I hadn't expected so many responses from you. This just shows me how much people continue to be passionate about this wonderful Game of Life with such a great community.

Thank you all for your ideas and advice. I will delve into all of them to create the best presentation I can.
I hope this community will stay like this forever.

Goodbye !

[tommyaweosme]

have fun at graduating! meanwhile im off to 6th grade...

i wonder what that will be like...

remember to pay taxes!

edit: you put a space between the final word of the sentence and the "!". that reminds me of someone i know, but i cant pinpoint who