The first groups of cyclic six are the circular functions: sine, cosine, tangent, cosecant, secant, cotangent function. The second groups of hyper six are the hyperbolic sine, hyperbolic cosine, hyperbolic tangent, hyperbolic cosecant, hyperbolic secant, and hyperbolic cotangent function. Respectively, their inverses are the arcsine, arccosine, arctangent, arccosecant, arcsecant, arccotangent, archyperbolic sine, archyperbolic cosine, archyperbolic tangent, archyperbolic cosecant, archyperbolic secant, and archyperbolic cotangent function. The first group relies its existence on the unit circle: x+y=1 while the second relies its existence on the unit hyperbola: x-y=1. Solving these two equations simultaneously give the solutions as the ordered pairs: (1,0) and (-1,0). These are the points of tangency on a Cartesian coordinate system between the unit circle and the unit hyperbola where and when they are in contact and touch each other only at these two points. However, for both to become functions the y-range must be nonnegative such that 0≤y<∞. Once the lower halves of the circle and of the hyperbola are removed the graph looks like a curvy “W” or the frontal one dimensional silhouette of a bat with extended wings on the left and on the right. Since the given asymptotes are y=x and y=-x the wings of the 1D bat figure extend to positive infinity on the right and to negative infinity on the left. In its totality this 1D batman configuration represents completely the whole physical reality of the inner microuniverse and the outer macrouniverse.

The intervals of all x’s of the semicircle is the microcodomain and the intervals of all x’s of the two disjointed partial branches of the hyperbola are the macrocodomain respectively given by the sets: [-1,1], (-∞,-1], and [1,∞). This is partially equivalent to the rational functional reality that for every two given rational numbers a and b, ab=a-bis always true and whose hidden topology is the equivalent of the one dimensional Hopf-Möbius topology connecting zero to unity to infinity and from infinity back to unity to zero.