I understand that honeycomb conjecture assumes you use one shape. If you are allowed to use multiple different regular polygons, as long as the length of the side is the same, what pattern would produce the maximum honey to wax ratio?

I seem to have come up with an example tiling that has a greater honey to wax ratio than hexagons. I detailed it here:

https://veniamin-ilmer.github.io/math/honeycomb-conjecture/

Has anyone has written about this before? Am I right?

Let there be the following interval:

I = (0; 𝜀], where 𝜀 ∈ ℝ and 0 < 𝜀

Now |I| := 1 (equal by definition)

Because 0 ∉ I and 𝜀 ∈ I that means the only element of I is 𝜀 and there is no other real number between these 2.

That means 𝜀 is the immediate real positive number following 0.

So I was playing with this idea and it came to me. What real number, when squared, is the number + 𝜀. I also know there is actually a real solution for x^2 = x + 1, that is the golden ratio, but what I want to find out what is the solution for x^2 = x + 𝜀. After plugging the quadratic formula I came to this conclusion:

x = (1 ± sqrt(1 + 4𝜀)) / 2

Now I want to prove that sqrt(1 + 4𝜀) ∉ ℚ and I did like this:

First I assumed that there is a real number that is will be noted q that is equal to sqrt(1 + 4𝜀).

So,

sqrt(1 + 4𝜀) = q

1 + 4𝜀 = q^2

4𝜀 = q^2 - 1

4𝜀 = (q - 1)(q + 1)

This is where I am stumped and I can't really say I am satisfied with x but I can certainly say that it is a number between 1 and 2. Also I defined 𝛾 to be 1 / 𝜀 and so you can say that 𝜀𝛾 = 1, which is nice. All of this is not something established in maths, but really just me trying to discover something interesting. Maybe 𝜀 is irrational, who knows? I think you can pull out so many interesting ideas from this.

say you have

f : R -> R

f(x) = ax + b

this kind of functions are named first degree equations or linear equations. and I was wondering how are we really sure that the function f actually determines a straight line when drawn on an euclidean plane? is there a real proof?

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