Art of Symmetry of 69

The figure 69 is central symmetrical. In the former section, we classified central symmetry as reflection about a point. In a projective sense, where only incidence of objects is relevant, this would mean that we observe a pencil of lines through a fixed point, the central point, and for each line, its intersections with the figure.

But to introduce symmetry we need more structure. If we add affine structure, ratios and

angles, we can say that the central point is the midpoint of lines whose endpoints are corresponding points on different sides of the figure. After adding metric to our geometry, we say that any two corresponding points have same distance from the central point, and they lie on the same circle whose centre is the central point of the symmetry. The central symmetry can also be described from the viewpoint of the observer standing at the central point. He will see the same image when he turns to the left and when he turns to the right.

Symmetry is not only a description but is also a recipe for the construction of figures. The right part, the 9, can be constructed from 6 via central symmetry, but also through planar rotation. Taking as the centre of rotation the central point defining the central symmetry and rotating for 180° in the positive direction, we get the other part, the 9. This shows that central symmetry also classifies as rotational symmetry.

On the other hand, the figure as a whole also has this same rotational symmetry. If we rotate the whole figure for 180° it remains the same regardless of the choice of the centre of rotation. To describe mirroring, which is reflection symmetry about a plane orthogonal to the plane of the figure, we again projectively look at the pencil of lines through a point, this time an ideal point, a point at infinity which defines the direction of lines in the pencil, so that in the affine sense the lines are all parallel, orthogonal to the intersection line of the two planes. Formerly described regularity of the figure will yield the same mirror image

when reflecting about all 4 sides of a square around the figure. This reversed image is the image the observerwould see if he was looking from the other, back side of the plane of the figure.

The figure 69 has internal metaphysical code of natural duality, the union of two symmetric parts symbolizes the connection of two same but mirrored polarities. The description of the connection is the source of the erotic interpretation of 69 as two lovers engaged in mutual oral stimulation. Imagine them as two spheres, each with two oppositely charged poles. The connection happens at poles as the meet of + and- . For both the same happens, but their mutual relation is connecting to the opposite part of the other. This is also the background of the Yin Yang symbol, where the polarities are additionally emphasized

by the use of colours and the spherical interpretation is more visible. Still, it conforms to the figure 69.

The connection of opposites also leads to symbolism of infinity, the number 8 and the infinity symbol, looking vertically or horizontally, thus indicating that unification of polarities brings oneness, wholeness which is perpetual fusion. With this we have described the content of the symmetry of 69 and context of its interpretations.

VARIATION No 1. ALGEBRA

The 6 and the 9 have a numerical, algebraical regularity reflecting the symmetry of their forms, connected to their greatest and unique common divisor 3.

In the view of two common algebraic operations, adding and multiplying as binary operations, we see that 6 and 9 are obtained as the result of these operations when both

operands are the same number 3.

6 = 3 + 3

9 = 3 * 3

Or, if we pertain to multiplication and its derived operation exponentiation, we can rewrite these equalities,

6 = 3 * 2

9 = 3^2

and see the numbers 6 and 9 as the basic algebraic derivatives of the 3.