Square roots turn out to have a strange relationship to the fundaments of the world. The square root of 2, found as the diagonal of a unit square, and the square root of 3 of the diametric across a cube; these are the simplest expressions of two and three dimensions, in area and volume. This can be shown graphically as:
three-dimensional space.
The number 1 is, by definition, the smallest number, the unit which additively creates the successive ordinal numbers through adding one. It is unchanged by any number of multiplications or divisions by itself, making it invulnerable to any powers of other numbers. That is, (1/1n) = 1 = 1n for n>0.
Negative numbers were problematical until recent centuries, when it was seen that they could be created by the positive numbers, times the square root of -1, which was called the imaginary number i, because no square has an area of minus one, area being defined as positive. However, i has been found to be a dimensional operator representing orthogonal movement at right angles by 90 degrees or π/2 radians. This can therefore represent our modern vectors, using complex numbers which are right triangles defined in two ways: either, (a) having a length (of hypotenuse) at an angle (the vertex angle) which is consistent with (b) a complex number of x (= the base) + i × y (= the opposite “imaginary” side of a right triangle, that “opens up” two-dimensional space).
For a vector, the hypotenuse length is called the radius R of a circle and the vertex angle is θ (theta) in “r-theta” notation and the dimensional alternative is the complex number notation (the tangent function). This gave birth to our trigonometry; whose functional form is the right triangle but whose archetype is the circle and its radii.
The unit square and unit cube (above), in geometrically establishing 2 and 3 dimensionality, can be geometrically developed as their square roots, since each new diagonal creates a new diagonal length which can be arced down to extend the rectangle (below).
generating the successive roots of the ordinals.
This form of development can be seen in the design of the King’s Chamber of the Great Pyramid where, by increasing the height to half the floor’s diagonal length (√5/2) , the end wall diagonals become cubits of 3/2 so that the length of the floor is 4-halves forms a rectangle with 3 halves, whose diagonal must then be 5 halves; and this was obviously intentional, a design criteria, and demonstration of the “first” {3 4 5} triangle plas a lesson in manipulating roots inside volumes.
In arriving at 5 and its root something new was harmonizing the three dimensions, seen as the root of 5 in two dimensions but then expressing the first Pythagorean triangle {3 4 5} where the squares of the sides sum in a Pythagorean way (32 + 42 = 9 + 16 = 25 = 52). The ratios between its sides express the musical intervals 4/3 (the fourth), 5/4 (major third), 3/5 x 2 = 6/5 (minor third). There are four such triangles (two shown), in two angled planes.
4 is the first square number and octave (which is 2:4) with 3/2 (the fifth) and 4/3 (the fourth) within it, and the number of sides of the square.
6 is the first cross-multiple (2 x 3 = 6), the number of faces of the cube.
5 is the addition of the first two numbers, (2 + 3 = 5), and these start the Fibonacci series expressing the golden meanThe Golden Mean is that unique ratio {1.618034}, relative to ONE {1}, in which its square and reciprocal share the same fractional part {.618034}. It is associated with the synodic period of the planet Venus, which is 8/5 {1.6} of the practical year {365 days}, by approximation. It is a key proportion found in Greco-Roman and later "classical" architecture, and commonly encountered in the forms living bodies take. through both growth and recurrence, through living processes. The number 5 is a vector that fulfills the three dimensional reality created upon the surface of the earth as a thing-in-itself which is dynamic and transient.
The two themes, of harmony and life, are reflected quite distinctly in the geocentricThe ancient view of the planetary world in which the inner planets orbit the sun and the sun and planets orbit Earth (the geo part). The Chaldean order was Earth - Moon - (Mercury - Venus -Sun) - Mars - Jupiter - Saturn. solar system, as shown in my two books** in the numbers found in traditional knowledge and, at the inception of this understanding of numbers as creational due to megalithic astronomy.
**see my Harmonic Origins of the World
and Sacred Geometry: Language of the Angels;