I've been reviewing the teachings of Pythagoras, which I've not looked at in any detail since the late 1980s--before I discovered Dewey Larson and the Reciprocal System of theory. With that RS background, I can see what Pythagoras was trying to describe in his structure of the Universe, with the tetractys, his fascination with triangles (everything was triangles) and his tripartite worldly receptacles.
Here's where the "magic" comes from... it starts with the Monad, which is unity (Larson does the same). That Monad becomes a Dyad (dichotomy) that is symbolically represented as the triangle. The Monad is the apex, the "divine origin" that casts its shadow down to the dyad (base line with 2 points) below. The triangle, being composed of 3 points and 3 lines, is the geometric reciprocal of itself. So what you have is two, joined coordinate systems--one of points (yang, 3D space, called the "superior") and another of interconnecting lines (yin, 3D time, the "inferior"). The triangle combining the three systems in yet another triple set of point-line associations creates three dimensional ratios--Larson's scalar dimensions, where the triangle represents "motion."
The Greeks only had 4 "elements," which are not atomic elements--they are actually symbolic of the states of matter (of which conventional science only recognizes three... with the possible exception of a 4th "plasma" state, which really isn't a "state," but an electric ionization). Larson only dealt with the three, known states, but Prof. KVK Nehru, when reviewing his "Liquid State" papers, noticed the dimensional structure indicated there were 4 states of matter, including a "vapor" state that is intermediate between a liquid and gas (clouds, for example--not a liquid and not a gas, because gasses have no cohesion). Knowing we have these four states, instead of three, the Greek "elements" make more sense. Pythagoras also associated each element with one of the regular solids:
Earth, solid, cube
Water, liquid, icosahedron
Fire, vapor, tetrahedron
Air, gas, octahedron
Since 1 (Monad) and 2 (Dyad) were the building blocks of the Universe and "sacred," the Pythagoreans actually started their numbering system for the mundane world with 3 (triangle) and 4 (square), which were the shape of the faces of the regular solids describing the four elements: cube has square faces, whereas the icosahedron, tetrahedron and octagon all have triangular faces. The "four" of earth matched up with the four cardinal directions, etc., basically explaining all the "material" quaternities and being the most stable form, as the "triangle" based elements tended to flow (water, fire, air).
So the numbering system went like this:
Sacred: 1 (Monad) + 2 (Dyad) = 3 (Harmonia)
Mundane: 3 (Triangle) + 4 (Square) = 7 (7-fold structure of creation)
Sacred (3) + Mundane (7) = 10 (the Universe)
That is why "10" indicates a completion of cycles or a new beginning--it is the Monad (1) shifted into another dimension (1<--0), where "0" is the placeholder of the old dimension.
You can also see why 1, 2, 3, 4 and 7 play their symbolic roles, with 1 & 2 being the original Unity and the dichotomy of space and time as "scalar dimensions" of motion, with 3 & 4 as the manifestation in a coordinate system (in the RS, the "time region," the region of atomic rotation that makes the "earth" elements solid, is 4-dimensional (square), whereas the stuff outside the "earth" solids, water, fire and air, are 3-dimensional (triangular).
Ah, "but what about the dodecahedron?" you may ask... "that's a regular solid, and omitted from the Pythagorean teachings!" Well, actually it wasn't omitted, but was part of those secret teachings that only the advanced students got to learn about. The Greeks had a 5th element, the aether, which was associated with the 12-sided dodecahedron, a structure that had a 5-sided pentagon for a face, that didn't fit in anywhere in the mundane world of triangles and squares (somehow, I suspect that Edwin A. Abbott had known about the Pythagorean numbers, when he wrote Flatland).
In the RS, the aether is the realm of 3D coordinate time, the cosmic sector, which cannot be directly observed (as we only observe spatial relationships)--but can be detected by how time influences space, as motion. So the dodecahedron, and it's 5-sided pentagon (or pentagram), became the base, magical symbol to represent the effect that this aether had on the mundane world of triangles and squares. The 12 faces of this solid matched up with the 12 Olympian gods (the Annunaki leaders), the signs of the Zodiac, and others. 3 x 4 = 12 -- all the mundane elements triangles and squares combined, gave 12 "faces" of the dodecahedron, the shadow of the mystical, invisible aetheric influence on the mundane world.
Though I've not found anything in publicly-available writings of Pythagoras, the man was on the same path of deductive thinking that Larson was, so he must have realized that the aether had dimensions of its own, and that the dodecahedron was the shadow of the aetheric triangles and squares on the mundane world. Knowledge of these aetheric dimensions formed the basis of magical law.
So that is why "10" is considered an important number, in ancient times.
Keeper of the Troth of Ásgarðr, Moriar prius quam dedecorer.