Ray trajectories, binomial of a new type, and the binary system; on binomial distribution of the second (nonlinear) type for big binomial power

The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be binomial coefficients of a new type. An analogy has been drawn with the binomial coefficients calculated with the use of the Pascal triangle. The paper provides a geometrical interpretation of binomials of different types in considering the branching systems of rays. Results of numerical calculations of binomial distribution of the second (nonlinear) type for big power of a binomial are given. Difference of geometrical properties of linear and nonlinear arithmetic triangles and envelopes of binomial distributions of the first and second types is drawn. The empirical formula for half-sums of binomial coefficients of the second (nonlinear) type is offered. Comparison of envelopes of binomial coefficients sums is carried out. It is shown that at big degrees of a binomial a form of envelops of these sums are close.