ISSN 2517-5718

New Analogues of the Pascal Triangle and Electronic Clouds in Atom

Alexander Yurkin*

Independant Researcher, Puschino, Moscow, Russian Federation

CitationCitation COPIED

Yurkin A. New Analogues of the Pascal Triangle and Electronic Clouds in Atom. Adv Comput Sci. 2018; 1(3):114


A new flat analogue of Pascal’s triangle based on the consideration of parts of regular polygons (regular hexagon, regular icosagon, etc.) is proposed. The classification of nuclear models of an atom is clearly shown: with circular, elliptical, cloudy and belt orbits of electrons. The belt model of atom consisting of a system of rays, a layer of electrons moving along wavy trajectories can be represented as a “cloud of trajectories.” A comparison is made of various types of wavy trajectories: a broken wavy trajectory, the trajectory made up of parts of a regular polygon and the sinusoidal path. It is shown that many calculations for our system of rays can be performed not only for rays that are inclined at small angles (smallangle paraxial approximation), but also for rays that are inclined at any angles, the results of such calculations will coincide. It has been suggested that it is possible to more fully explain the origin of the splitting of atomic spectral lines and fullerene schemes construction.


Pascal triangle; Electron; Trajectories; Regular Polygon; Fullerene; Geometrization of physics


  1. Statistics and computer science have grown as separate disciplines with little interaction for the past several decades. This however, has changed radically in recent years with the availability of massive and complex datasets in medicine, social media, and physical sciences. The statistical techniques developed for regular datasets simply cannot be scaled to meet the challenges of big data, notably the computational and statistical curses of dimensionality. The dire need to meet the challenges of big data has led to the develop
  2. Geometric models of the atom consist of lines and geometric shapes (Figure 1). The nuclear models of the Bohr atom [1,2] and Somerfield [1,2] are based on the Kepler planetary model [2] with circular (Figure 1a) or elliptical (Figure 1b), respectively, electron orbits rotating around the atomic nucleus. There are nuclear models of atoms consisting of a nucleus and electron clouds [1] of various configurations, for example, in the form of a torus (Figure 1c). The geometric figures of the models shown in Figure 1 contain different, relatively large and relatively small distances:




    Figure 1: Geometric models of the atom consist of lines and geometric shapes

    Where ∆ is the thickness of the layers of the electron orbits or the electron cloud, δ≈2*10- 15 [m] is the thickness of the individual electron orbit (approximately equal to the size of the electron [3]), and RB≈5,3*10-11[m] is Bohr radius. The value of δ in comparison with the values of ∆ and RB is small.

  1. In [4], based on a consideration of the system of rays, a new geometric belt nuclear model of the atom was proposed (Figure 2). In this work, it was shown that this model corresponds to the de Broglie wave and the Heisenberg uncertainty principle. In [5], the connection of this model, containing broken wavy trajectories, with the Schrödinger wave equation and Born’s probabilistic interpretation of finding the electron was shown. In the works [5-7] the connection of this model with the periodic system of chemical elements, the Pauli principle, and the energy spectrum of the atom and the splitting of the energy levels of the atomic spectrum was shown. 

    Figure 2: New geometric belt nuclear model of the atom

    In [8], using our model, laminar and turbulent flow of fluid through pipes was described. In [9,10], the possibility of using our model in biology was shown.

    We assume that in this model, relation (1) is preserved, but instead of relation (2) we use another relation:


    Figure 2a shows a general view of the belt model. Figure 2b shows this model on an enlarged scale, consisting of broken wavy trajectories of electrons stretched along the horizon. Broken wavy trajectories consist of links inclined at small angles multiple of the angle \[\gamma\] to the horizon. That is, we assume:


    The shortest trajectory \[\lambda_{1}\] consists of two links inclined at angles \[\pm\gamma\]. Its length is equal to the wavelength \[\lambda_{d}\] de Broglie:

    \[\lambda_{1}=\lambda_{d}=2\pi R_{b}\].........................(5)

    Length [4 - 7] of longer wavy paths:


    The broken wavy trajectories are placed in a uniform coordinate grid with the height and length of the cells k and 1 respectively (Figure 2b).

  1. The arithmetic triangle of Pascal was known in ancient India and China [11]. The description of this arithmetic triangle is found in the papers of the poet Omar Khayyam [11]. In [12], a new first planar analogue of Pascal’s triangle was presented for the case of small angles.
  2. Figure 3 presents various analogs of the Pascal triangle. The first analogue of the Pascal triangle for the case of small angles 4γ is shown in Figure 3a. This analogue was investigated in [13]. In this case, the rays, passing a certain distance (link) of length ≈a, are symmetrically split in two at each iteration at small angles ≈±2γ.

Figure 3: Various analogs of the pascal triangle

In this paper, we consider the case corr4esponding to our model, but we will assume that the corners γ',3γ',5γ',… are not small that is the expressions (4) for approximate calculations do not hold.

In Figure 3b and Figure 3c two variants of the second analogue of Pascal’s triangle are shown for the case of ordinary angles 4γ'. In this case, the rays, passing a certain distance (link) a, symmetrically split at each iteration in two at the angles of ±2γ' and form parts of a regular polygon

In the example shown in Figures 3b, c: a≈3,4 [sm], γ'=9°, the radius is inscribed in an regular icosagon circumference r≈10,7 [sm].After several iterations, a part of the rays overlap each other, as in the Pascal triangle [11, 14] and the first analogue of Pascal’s triangle Figure 3a. Note that the ray patterns in Figure 3a and Figure 3b are similar. It is also not difficult to draw a diagram of rays (at small angles ≈±2γ ) similar to that shown in Figure 3a, but similar to the diagram in Figure 3c.

It is interesting to note that if γ'=30° then in the sequential construction of an analogue of the Pascal triangle (as in Figure 3b), the entire plane will be densely filled with regular hexagons. In other cases (Figure 3b,c), the arrangement of some hexagons will be loose and they will overlap each other.

Such regular hexagons located on a sphere or cylinder can form fullerenes [11].

In Figure 4, the analog of the Pascal triangle (a) and the ordinary Pascal triangle (b) from [11] are presented in the form of regular hexagons:

Figure 4: (a) The analog of the Pascal triangle, (b) the ordinary Pascal triangle

  1. Figure 5 shows a wave-like trajectory similar to that shown in Figure 2b, but composed of parts of an icosagon (Figure 3b,c). The wave-like trajectory is placed in an uneven coordinate grid with the height and length of the cells k' and l', respectively. The total length of this trajectory: L=14a (Figure 5); an analogue of λ4 in Figure 2b of the “wave” length: λ'4 = l' 1+l' 2+⋯+l' 14 (Figure 5).