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Years ago, in elementary school, as my mind wandered I would often think of random things. I thought of how people could all perceive different colors (like in RGB, instead someone sees GBR) but never realize others are seeing differently. (which I later found out Einstein talked about in his thoughts on relativity) Er... On the playground years ago I thought about the rate of increase on a line graph. (I'll explain how this applies to life later) Imagine multiple levels of increase and decrease, where higher levels directly effect the levels below them. The bottom level is the number you use. So, if each level is either increase or decrease, you could represent each by either a plus or a minus. The best comparison (which isn't very good) is where each level is an increase rate. If the second level has a positive increase rate, then every step, it makes the level below it have a greater increase rate.

So...

+ would be like y = x.
- would be like y = -x.
++ would be like y = x^2.
+- would be like y = sqrt(x).

Once you get to the third level it becomes very hard to explain in human words.
++- would be where the growth rate is exponential, but after a while the rate at which the growth rate increases nears zero.

After about a half an hour of thought on the subject during school I realized the Game of Life would be a fantastic way of explaining my thought experiment. If you used Life, x would be the generations, and y would be the population.

For +, imagine a puffer.
For -, imagine a clean fuse.
For ++, imagine a breeder.
For +-, imagine a pattern that grows like a logarithm, at a slower and slower rate.
For ++-, imagine a breeder, that for example, creates switch engines at a rate that is a logarithm. So, it might make a switch engine at generation 1000, then 2000, then 4000, then 8000, until it takes a really long time before another switch engine is created. So, the population increase rate's rate of increase slows down over time.


I'm sure this kind of utterly confused probably half of you. But I've thought about this for years and never could create a method of explanation that even came close to making sense.

So guys:
Tell me if this makes sense at all to you! I want to know if you get what I'm talking about or not.

Tell me if some concept like this already exists! I've found that some of the random thoughts I come up with often are already known, and if this exists in some manner, I want to know.

Could a pattern that has a ++- growth rate be created? I imagine a 180 degree reflector paired with a Cordership-based contraption that, whenever it gets a glider, sends one back to the reflector and creates a switch engine to the side.

Thanks, and I hope no one thinks I'm stupid or weird for this.

That "++-" population graph that you suggested is known as a logistic function, which is represented by the formula y = 1/(1+a^x), where a is between 0 and 1.

Just had a thought: Perhaps levels further down are just the derivatives of the levels directly above them.
Another thought: Essentially, when time goes to the right (+) or left (-), it causes a growth increase (+) or decrease (-), which in turn triggers another growth change. Based on this assumption, this is what the graphs would look like:
+++: y = x^3
++-: see above equation
+-+: the middle portion of y = x^3
+--: y = x^(1/3)

Awesomeness wrote:Tell me if this makes sense at all to you!

Yep :3

Can you describe the permutations you haven't yet (-+, --, +++, +-+, +--, -++, -+-, --+, ---)?

I imagine +++ would be exponential, +-- would be log(x), but I'm not sure about the others.

Er... On the playground years ago I thought about the rate of increase on a line graph. (I'll explain how this applies to life later) Imagine multiple levels of increase and decrease, where higher levels directly effect the levels below them. The bottom level is the number you use.

It sounds like you've re-discovered the concept of calculus. If you did that independently, then that puts you in the same boat as Leibniz, who did the same thing 10 years after Newton!

The individual 'levels' of which you speak are formally known as derivatives. For example, consider the following function:

function: x^2
first derivative: 2x
second derivative: 2
third derivative: 0
fourth derivative: 0
...

The first derivative of a function is its rate of change. The second derivative is the rate of change of its first derivative, and so on.

Once you get to the third level it becomes very hard to explain in human words.

But easy to describe mathematically, as I(I(I(0))), where I() is the indefinite integral (or antiderivative) of the function. (You should actually use the integral sign here, but Nathaniel hasn't integrated LaTeX support!)

By the way, you're wrong to assume that exponentiation is the third antiderivative of y = 0 (+++ in your unfathomable notation). The exponential function is its own derivative!

Just a moment ... did you say you realised that in elementary school?!


They say that mathematicians produce their greatest work before the age of 40, but that just takes the cake...

I imagine a 180 degree reflector paired with a Cordership-based contraption that, whenever it gets a glider, sends one back to the reflector and creates a switch engine to the side.

That's O(t log t) growth, by the way.

Thanks for the replies, guys!

Wow. I thought I explained it awfully but people seem to get it! I did think about this independently, and I had no idea it was related to calculus.

Wow, you thought of this in Elementary school?


Son, you need to take advanced math courses, if you're already thinking such things.

Wow!
I learned about calculus in elementary school...
but I never imagined a young kid figuring it out themselves!!