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The beauty and harmony of mathematics is the result of strict following simple rules.
Gemstones grow strictly following a simple structure of several atoms.

Preamble

Let's remember basics. 😉
Consider a set of numbers / numerically given function f (x) .
For clarity, let's draw:

Recall that the difference between two neighbor values / numbers fn - fn-1 is the derivative of the function f(x) in (neighborhood) Xn ... up to a factor C = (Xn - Xn-1), which depends on the dimension X (centimeters, inches, kilograms, or whatever):

and the difference between two neighbour differences is the second derivative:

Now let's start. We will look for the "physical meaning" of the well-known
arithmetic mean (average) of two numbers

Let's denote as fn (green in the figure) the new value of the function at point n which we get as the arithmetic mean of its (old) nearest neighbors.

Now let's subtract the old value fn:

On the left we have: "new" minus "old" , that is, the time derivative (up to a factor):

On the right side we have the second derivative with respect to X :

So, we got -- (who remembers the equations of mathematical physics? 😉) -- one-dimensional diffusion equation!

Thus, the arithmetic mean is a solution to the diffusion equation!

So, math physics - it is so easy

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