Closest packed

(12)    There are six points that can be represented by three integers that add up to zero such that the sum of the absolute values of the three coordinates divided by two equals one.

(13)    The variable six contains the unit vectors that can be added to a point in the plane to step to the closest neighbors of the point with integer coordinates. There are six neighbors with this coordinate system instead of four neighbors as in the omni-perpendicular Cartesian system.

(14)    If you add another dimension you get twelve sets of four coordinates that add up to zero and have a VectorDistance of one and also have integer coordinates. The four coordinates refer to the perpendicular displacement of the four defining planes of a regular tetrahedron. In other words, the surface of the paper with the triangular grid is fixed by the fourth coordinate.

(15)    The unit of perpendicular displacement is the height of a regular tetrahedron of edge length one instead of an equilateral triangle. You can see why if you make a model of the cuboctahedron (Bucky Fuller's Vector Equilibrium (VE)) with 36 tubes: 24 edges of the cuboctahedron and 12 radial tubes to the 12 vertices from the center, and look at the four hexagonal planes (cross sections) of the VE to see them as the four planes that define a tetrahedron when the tetrahedron has an edge length of zero and imagine one or more of the planes moving outward or inward from the center of the tetrahedron to define other regular tetrahedrons of different sizes in different places. The T_2 simplex (line segment), the T_3 simplex (equilateral triangle) and the T_4 simplex (regular tetrahedron) are interlocked in the VE. Some things can only be understood or discovered operationally.

(16)    The variable twelve defined above contains the 12 unit vectors that can be added to a sphere in closest packed sphere space to step to its nearest neighbors.

(17)    The agglomeration of spheres above keeps the same shape when more and more layers of closest packed spheres are added. The shape is called a cuboctahedron. It is the Vector Equilibrium. It is an equilibrium of vectors from equal diameter objects when the centers of the objects are connected to the centers of their nearest neighbors by vectors, and the spheres removed. It has six square faces and eight triangular faces. If you know the formulas for the square numbers () and the triangular numbers () you can compute the shell growth rate of closest packed spheres. Bucky Fuller published the formula in 1941(?) and it can be used to find the number of protein nodes on the outer shell of a virus.

(18)    Objects under consideration can be assumed to act in equal amounts in all directions sometimes, i.e. like spheres. If you know the Avogadro constant that 22.4 liters of ANY gas at one atmosphere at zero degree Celsius contains 6.02252 times molecules, then you can compute the size of a tetrahedron with a volume of 22.4 liters and think of the molecules as being like stacked cannon balls in a court yard. is the formula for the number of balls in a tetrahedron of edge length n-1, because it is the sum of the triangular numbers, .  So, you can figure out the spherical influence of any molecule of gas.

Created by Wolfram Mathematica 6.0  (17 March 2008)