The purpose of many sciences is to find the most accurate mathematical formula that can be found in the experimental value of fine-structure constant. Attempts to find a mathematical basis for this dimensionless constant have continued up to the present time. However,no numerological explanation has ever been accepted by the physics community. In this paper we will present the exact expressions for the fine-structure constant. A simple and absolutely accurate expression for the fine-structure constant in terms of the Archimedes constant. The equivalent expressions for the fine-structure constant from the madelung constant. Also a exact expression for the fine-structure constant in terms of the golden angle,the relativity factor and the fifth power of the golden mean. Also other expressions for the fine-structure constant. Finally we will present the continued fractions for the fine-structure constant.

 In this paper in an elegant way will be present the simple unification of the fundamental interactions. We will calculate the unity formulas that connect the coupling constants of the fundamental forces. We will find the formulas for the Gravitational constant. It will be presented that the gravitational fine-structure constant is a simple analogy between atomic physics and cosmology. We will find the expression that connects the gravitational fine-structure constant with the four coupling constants. Perhaps the gravitational fine-structure constant is the coupling constant for the fifth force. Also will be presented the simple unification of atomic physics and cosmology. We will find the formulas for the cosmological constant and we will propose a possible solution for the cosmological parameters. Perhaps the shape of the universe is Poincaré dodecahedral space. Finally will be presented the law of the gravitational fine-structure constant followed by ratios of maximum and minimum theoretical values for natural quantities. This article will be followed by the energy wave theory and the fractal space-time theory.

 We propose the exact formula for the fine-structure constant α in terms of the golden angle,relativity factor and the fifth power of the golden mean. Also we will present the continued fraction for the fine-structure constant.

 In the first chapter we will present the definitions,history,relations and arithmetics approximations of Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 220 expressions. In the second chapter we will present exact expressions between Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 127 expressions. In the third chapter we will present approximations expressions between Archimedes' constant π,the golden ratio φ,the number Euler e and the imaginary number i. A total of 112 expressions.

 In this short work we will study the simple and absolutely accurate expression for the fine-structure constant with the madelung constant b2(2). This simple expression for the fine-structure constant in terms of the Archimedes constant π is. Also we will present the continued fraction for the fine-structure constant.

  We present 26 exact mathematical expressions of the proton to electron mass ratio with exact value. We propose an exact mathematical expression using Fibonacci and Lucas numbers. Also we present the exact mathematical expressions between the proton to electron mass ratio with the mathematical constants. Finally we propose the exact mathematical expressions between the proton to electron mass ratio with the fine-structure constant.

 In this paper will be presented the unity formulas that connect the fine-structure constant and the proton to electron mass ratio. The equations are simple,elegant and symmetrical in a great physical meaning. At the beginning we will make a review of the last works. We will suggest the exact formula for fine-structure constant in terms of the golden angle,the relativity factor and the fifth power of the golden mean. Also we propose a simple and absolutely accurate expression for the fine-structure constant in terms of the Archimedes constant. After we propose two exact mathematical expressions for the proton to electron mass ratio. It will be presented  the mathematical formulas that connects the proton to electron mass ratio μ,the fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro's number NA,the gravitational coupling constant αG of the electron and the gravitational coupling constant of the proton αG(p). Ιt will be explained that the product of the proton to electron mass ratio with the reciprocal of the fine-structure constant is one of the roots of a trigonometric equation. It will also be shown another way to show this trigonometric equation with the exponential form of the equation. The beautiful formula that connects the fine-structure constant,the proton to electron mass ratio and the fifth power of the golden mean. Also the formula that connects the fine-structure constant,the proton to electron mass ratio and mathematical constants π,φ,e and i. Finally we will show that the speed of light and the maximum speed of sound depends only from the fine-structure constant and not from the proton to electron mass ratio.

   In this paper we will present a new exact formula for the fine-structure constant in terms of the golden angle,the relativity factor and the fifth power of the golden mean. A new interpretation and a very accurate value of the fine-structure constant has been discovered in terms of the golden radio. We propose the exact equivalent mathematical expression for the proton to electron mass ratio using Fibonacci and Lucas numbers and two other exact mathematical expressions for the proton to electron mass ratio. We present the exact mathematical expressions that connect the proton to electron mass ratio and the fine-structure constant. Also we will find a new formula for the Planck length and a new formula for the Avogadro's number. First we will find the beautiful unity formula that connects the gravitational coupling constant and the fine-structure constant. This equation is the simple unification of the  gravitational and the electromagnetic interactions. We will present the exact mathematical formulas that connect six dimensionless physical constants. The six dimensionless physical constants are the proton to electron mass ratio μ,the fine-structure constant α,the ratio Ν1 of the electric force to the gravitational force between electron and proton,the Avogadro's number NA,the gravitational coupling constant αG of the electron and the gravitational coupling constant αG(p) of the proton. Also we will find a new formula for the gravitational constant G. Finally we will be presented the formula for the cosmological constant Λ. This unity formula is a simple analogy between atomic physics and cosmology.  All these equations are simple,elegant and symmetrical in a great physical meaning.

 In this paper we will study the strong coupling constant. The recommended theoretical value for the strong coupling constant is αs=Euler's number/Gelfond's constant. It will turn out that this value is the key that solves many problems of Physics. We will find a lot of formulas for the strong coupling constant. All these equations prove that the value of the strong coupling constant depends on the energy scale. First we will find the beautiful unity formulas that connect the strong coupling constant and the fine-structure constant. These equations are the simple unification of the strong nuclear and the electromagnetic interactions. It will be presented the mathematical formulas that connect the dimensionless physical constants with the strong coupling constant. From these equations we reached the formula of the unification of the strong nuclear,the electromagnetic and the gravitational interactions. Also we will find the formula for the Gravitational constant. Finally we will be presented the formula for the cosmological constant. This unity formula is a simple analogy between atomic physics and cosmology.  All these equations are simple,elegant and symmetrical in a great physical meaning.