Simple way to generate the sequence of dimensions used in String Theories.

[pascal2]

Some of you may remember that some time ago I'd claimed (though I can't remember whether I'd posted this here or not) that the sequence of dimensions utilized by String Theories was composed of every other doubled Fibonacci number- that is, 2, 4, 10, 26.... This fits the facts, but remained underivable by simple principles.

Today I realized that I could get these numbers by resorting to the (2,1)-sided Pascal Triangle. As you know, you can generate Fibonacci numbers by summing terms along the shallow diagonals of the Triangle. 

 

But there are infinite numbers of 'generalized' Pascal Triangles with edge values more complex than the simple 1's running down those of the classical Pascal Triangle. 

 

The (2,1)-sided Pascal Triangle has 1's running down one side and 2's down the other (with the identity of the apex term ambivalent between 1 and 2). Summing terms along the shallow diagonal upwards towards the 2's side generates the Lucas numbers, and summing terms upwards towards the 1's side generates the Fib numbers (but upshifted one move, that is, lacking the first 1 of 1,1,2,3,5,8...).  

 

Anyway, the doubled Fibonacci numbers can be gotten by adding Lucas and Fibonacci numbers together. So 2+0=2; 1+1=2; 3+1=4; 4+2=6; 7+3=10; 11+5=16; 18+8=26....

 

Now, the shallow diagonals of any generalized Pascal Triangle aren't all identical. They differ in whether they teminate in an edge term, or not (that is, BETWEEN edge terms). You can generate the doubled Fibonacci numbers (as described above) by INTERSECTING the shallow diagonals that give the Lucas and Fibonacci numbers on the 2's side of the (2,1)-sided generalized Pascal Triangle. The ones that have terminal 1's along the shallow diagonals summing to Fib numbers are the ones that are used by String Theories, and the ones that DON'T have terminal ones are those NOT used.

 

The (2,1)-sided generalized Pascal Triangle also motives terms in equations that give the powers of the Metallic Means (see https://en.wikipedia.org/wiki/Metallic_mean). A decade ago I realized, while working with both these issues at the same time, that the numerical coefficients of the terms in the equations for the powers of the Metallic Means were IDENTICAL to those along the shallow diagonals of the (2,1)-sided generalized Pascal Triangle leading to the Lucas numbers, and furthermore, that the powers that these terms were raise to were IDENTICAL to the dimensional labels of the edge-parallel diagonals that intersected the shallow diagonals at the terms as well. In the classical (1,1)-sided Pascal Triangle, the outer 1's have a 0D label, as they don't change. The natural numbers have label 1D, since they change linearly/monotonically. The triangular numbers are about close-packing of circles in a plane, so 2D. The tetrahedral numbers are motivated by close-packing of spheres in a volume, so 3D, and so on. This works as well for any generalized Pascal Triangle.

 

Jess Tauber