RIght-angled Triangles

Everyone knows that 3 x 3 + 4 x 4 = 5 x 5. The proof of Pythagoras theorem. Less well known, though still in common awareness is the combination 5, 12 and 13.

Before looking further let us consider what we know of the relationship between 5 and 12. In the regular Dodecahedron, the fifth of Plato’s solids of the elements equating with Ether and Cosmos the 12 faces (do-deca – hedron) are composed of regular even sided Pentagons. This is the height of completion. It is the number of the Soul for the Platonic School. Having attained to the Dodecahedron the initiate has achieved that status known as self-mastery. The Soul expresses through the initiate into the world, the personality is dissolved in the greater self.

Some familiar squares

12, as we have seen under the page on the Cube and Sphere, is the number of regular spheres that surround a central one of the same radius. This is the 13th. It is interesting then that 12 + 5 – already so closely identified, should yield 13 when theior squares are added together. It should also be noted that the Solar cycle of the zodiac is 12 full signs of 30 degrees each (360), while the Lunar year is 12 complete lunations plus about half a further one, or ‘somewhere between 12 and 13’.

The human mind likes symmetry and prefers things to be neatly rounded or squared off to create something it recognises as complete. Yet these slight anomalies in nature give the indication that through such an incomplete expression the Universe remains in a state of evolution.

But other close approximations are to be found in the right-angled Isosceles triangles. If we take 5, for example the square of 5 is 25, when this is the square on both sides of the Isosceles triangle it yields a total of 50, the square on the hypotenuse must equal 50, but 7 x 7 = 49. ‘Close enough for jazz’ as a friend would say. Close enough to be considered equal or approximately equal to the other. In this way 7 implies 5 and vice versa. If we follow the series through we find that 7 x 7 when added to the same yields 98, two short of 10 x 10 = 100.

100 + 100 = 200 4 more than 196 which is 14 x 14. 196 + 196 gives 392, 8 short of 400 = 20 x 20.

Some squares contained in Squares

We can predict that if the series is continued the interval will grow by the next number in the duple series.

400 + 400 = 800 which we find is 16 more than 784, the square of 28.

But 25 + 144 = 169 which is elegantly precise in its definition. The whole thing is in focus, as it were.

Elsewhere we have considered the relationship between 22, 11 and 19, with particular reference to the circle and the year.

12 x 12 = 144, and 144 + 144 = 288. But the square of 17 = 289. back to the jazz. However there is a more perfect solution than this and that is that 289 – 225 (15 squared) = 64!

15 is the sum of 5 + 5 + 5, but 125 is the product of 5 x 5 x 5. In a way 15 implies the spherical number and so represents a degree of fullness not found elsewhere. The Sphere represents the world of Spirit. But it is combined with 8 the first Cube and expression of material manifestation. So here in 17 we have the combination of both the Celestial realm of 12, the zodiac and wheel of the stars, and the unification of the Sphere with the Cube. This is bliss.