Decagon

As is known, a number of irrational (incommensurable) numbers is infinite. However some of them take the special place in the history of mathematics, moreover in the history of material and spiritual culture of mankind. Their significance consists of the fact that they express some proportions, relations having universal nature and found out in the most unexpected places.

The first of them is the irrational number equal to the ratio of the diagonal to the side of the square. Discovering so-called "incommensurable line segments" and the history of the most dramatic period in antique mathematics are connected to this number resulted to development of the theory of irrationalities and irrational numbers and in the final analysis to creation of modern "continuous" mathematics.

The next two irrationals are the number of p expressing a ratio of a length of circumference to its diameter, and the "Napierian number" of e expressing some relevant geometrical ratios in a hyperbola. A significance of these two major mathematical constants in a calculus consists of the fact that they "generate" the main classes of "elementary functions" (trigonometric functions generated by the number of p, exponential function e^x, logarithmic function log_ex, at last, hyperbolic functions generated by the number of e). Between p and e, that is between "two numbers dominating above the analysis", there exist a refined relation:

where is an imaginary unit, one more exotic creation of mathematical thought.

The golden proportion of t also falls into category of the fundamental mathematical constants. But then there is a question: whether there is any connection between these mathematical constants, for example between numbers p and t? The analysis of the regular decahedron called "decagon" (Fig.1) gives the answer this question.

Figure 1. Regular decahedron ("decagon").

Let's consider a circumference of radius R together with the "decagon" inscribed in it (Fig.1). From geometry it is known, that the side of the "decagon" a₁₀ is connected to radius R by the following ratio:

If to execute some trigonometric transformations using the formulas well known for us from school trigonometry, we get the following outcomes:

1. The side of the regular decahedron inscribed to the circle of radius R is equal to the larger part of the radius R divided by the golden section, that is
   
   

   a10 = R/t.

   
   
2. The golden proportion is connected to the number of p with the following ratio:

t = 2 cos 36° = 2 cos.

(2)

This formula obtained as a result of the analysis of geometrical proportions of the "decagon" is one more testimony of fundamentality of the golden proportion, which alongside with the number of p rightfully can be ranked to category of the major mathematical constants.