Environment, The legislator

The math is the theoretical label for the environment, In other words, We are using math to show what is and happens in the environment, And as well when there are impossible things to do in the environment, there are impossible things to do in math! We are using some symbols for impossible things in math or some symbols to show "change is required in the environment" to make math more usable in future calculations, But we can avoid these mistakes by using some tricks discussed in this article. In this article, The `pi` number, `∞` and imaginary numbers are discussed

Regular polygons

In Euclidean geometry, a regular polygon is a polygon that is equilateral (all sides have the same length). The first of them is regular triangle with 3 sides with same length. In this part `n` is the count of sides and `S` is the area of any polygon.

The area of a triangle is `1/2 r a` and `a` as base and `r` as height of the triangle.

Regular polygons are contained with triangles and their area can be calculated by counting triangles area. For example, there are 4 triangles in a square with `theta` insider angle.


`S = 4(1/2 r a) = 2 r a`

`a = 2 r rarr S = a^2`

Or the area of a hexagonal:

`S = 6(1/2 r a) = 3 r a`

`r = sqrt{3}/2 a rarr S = (3 sqrt{3} a^2)/2`

For `n` sided polygons:

`S = (nra)/2, a = 2r tan((pi)/n)`

`rarr S = nr^2 tan((pi)/n)`

With adding triangles to the polygon it's going to be a circle. So that we have:

`S = lim_{n->∞} (nra)/2 `

`S = lim_{n->∞} nr^2 tan((pi)/n) = pi r^2`

By how many triangles we can put in the polygon `pi` number is going to be accurate.

`lim_{n->∞} (na)/2r = lim_{n->∞} n tan((pi)/n) = pi`

Testing `pi`

In every regular polygon with n sides there is a `pi` number that i name it `pi_{n}`. For example `pi` of a square is 4 and area of a square is `S = 4 r^2`.

`S = pi_{n} r^2`, `pi_{n} = n tan((pi)/n)`

`n` `n tan((pi)/n)` `pi_{n}`
`3` `3 tan(pi/3)` `5.1961524227066`...
`4` `4 tan(pi/4)` `4.0000000000000`...
`6` `6 tan(pi/6)` `3.4641016151378`...
`2^100` `2^100 tan(pi/2^100)` `3.1415926535898`...

In reality, in fact, By growing `n` to infinity, `a` is going to be zero that there is not any zero length for sides of a polygon, So this rule is used to prove that there is no any infinitive number and obscure number for `pi` and it is an environmental number that based on conditions is an accurate number in that environment. 

How about `0, 1, 2` for `n`? The environment does not let the triangle to have 1, 2, or 0 sides and tells us that something is impossible or should be changed!

Environmental theoretical numbers

The environment is the lawyer, It decides what can happen and what is illegal. An experiment is prepared to show what is environmental numbers and environmental theoretical numbers.

In this experiment, there is a lamp in a box that has a hole, And a wall that the lamp is lightning to the wall. The distance of box from the wall is assumed `10`.


This document has not been completed yet and is being edited.

Environment, The legislator - 2018 Apr 24