Experimental Mathematics: Finding Number Patterns

[Natus Videre]

Let's see which geometric shape encapsulates this relationship.
At a given iteration, every new prime number needs to be connected to all the primes preceding it.

From left to right, we obtain:

• a point (1 vertex, no edge),
• a line segment (2 vertices, 1 edge),
• a triangle (3 vertices, 3 edges),
• a tetrahedron (4 vertices, 6 edges),
• a 5-cell (5 vertices, 10 edges)

Going a bit further, a way to geometrically map primes would be to associate each prime to a vertex using barycentric coordinates.

In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or barycenter) of these masses.

• Prime 1: (1)
• Primes 1,2: (1,0), (0,1)
• Primes 1,2,3: (1,0,0), (0,1,0), (0,0,1)
• Primes 1,2,3,5: (1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)
• Primes 1,2,3,5,7: (1,0,0,0,0), (0,1,0,0,0), (0,0,1,0,0), (0,0,0,1,0), (0,0,0,0,1)
• ...and so on.

source (image): simplex in nLab

Finding the coordinate of the next prime is very easy, because the next prime occupies a new dimension and embeds the preceding primes into that new dimension. For example, the prime 11 (the 6th prime) would have the coordinate (0,0,0,0,0,1) in six-dimensional space.

But given a coordinate, how would we deduce the numerical value of the prime? The geometry of a simplex seems to put each vertex on equal footing with the other vertices. Remember, when we drew the graph, the edges were directional in order to show the dependence between each vertex. As you can see, the prime 2 doesn't depend on the prime 5, but the prime 5 depends on the prime 2.

But we could also say that the prime 1 "gives birth" to the prime 2, which, in turn, "gives birth" to the prime 3, and so on. And so, by transitivity, the prime 1 "gives birth" to all the other primes.

If we add both graphs together, we obtain an undirected (no arrows) graph, because the direction of travel is unrestricted, i.e. we can travel back and forth from each point to all the other points.

Q: Now, the other night, in front of the psychomantium, I did not exactly have a vision, but something came into my head, and the idea was that prime numbers are important because, the principle that they are only divisible by themselves and by one is indicative of the fact that they are direct links, channels, or conduits to seventh density, or first density, or something...

A: How about all densities?

Q: Okay, that is sort of what I mean, that they are, in a sense, gateways - would that be a good term?

A: Close.

"All is one and one is all," as the C's said.