A regular shape has borders that have the same length of distance from the center to the sides or in other words with a regular length distribution in it like a circle or a sphere.

In fact, there is no circle or sphere in reality and all of circles or spheres we can see are regular polygons that contained with triangles or pyramids.

Regular rule is about environmental numbers like `pi` number that is changing based on environmental conditions.

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`

`theta = (2pi)/n`

`alpha = pi - theta`

`tan((pi)/n) = a / (2r)`

`rarr a = 2r 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`

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`... |

`8` | `8 tan(pi/8)` | `3.3137084989848`... |

`2^10` | `2^10 tan(pi/2^10)` | `3.1416025102568`... |

`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!

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.

Regular rule - 2018 Apr 24