Zeno's paradox exists in several forms, all of them centering around relative movement that can be viewed as a discrete series. Zeno's paradox might be an old philosophical puzzle, but it might hint at the deep structure of the universe and connected to problems of modern science. The above illustration portrays the relative movement of Achilles and a turtle. It is an intuitive notion that the running Achilles reaches and overtakes the turtle in a finite time. Nevertheless, Zeno's logic takes apart the movement into a series of discrete distances, which sections Achilles's motion into an infinite sequence with extension to an earlier position of the turtle. As the turtle moves further on, the shrinking distance between the two is never eliminated. Although Achilles gets infinitesimally closer to the turtle, he can never reach it.

The topology of space between the poles (black and white holes), demonstrated by a Breton hat |

As Einstein has shown, the global picture of the universe is formulated by the spatial field, demonstrated by a Breton hat above. The field curvature changes smoothly from positive curvature near the black holes, to negative curvature near the white holes, whereas the microdimensions form standing waves, therefore form a quantized, step-wise progression between the poles. Constant frequency waves of the microdimensions organize along constant field curvature of space. This way the micro and macrodimensions are mutually exclusive, i. e., orthogonal. The macro- and microdimensions are connected via interaction, or decoherence, which ensures their congruent development and their interconnected, interdependent relationship as shown in Zenon's paradox. However, this is the central quality of wave decoherence in quantum mechanics. The connection between sequences formed of discrete series (quantum mechanics), and a field with smooth gradation (the field of gravity) has been featured in my earlier post and YouTube video. But similar relationships can also be found in many other fields, from physical sciences to social phenomena.

Gaussian (left) and Power law distribution (right) |

The microdimensions maintain a closed minimal surface within each step of the gradual succession of standing wave frequencies that form between the poles; inverse volume changes between macro- and microdimensions by Lorentz transformations enhance field curvature and pressure differences, leading to Gaussian distribution (shown above left). However, the curving field directs the decision of the participant, and the choice of the participant further modifies the field (as observed along the brim and the top of the hat). Hence, in the curving field deterministic, irreversible and self-perpetuating changes give rise to singularities, called poles. The polar singularities stabilize the structure of the cosmos, but allow the center of the field to empty out, leading to a power-law distribution (shown above, right). For example, fracturing resulting from rigid collisions also show power law distribution. The increasing gravitational differences of space, versus the minimal surface of the microdimensions engender the large~~ ~~scale spatial-temporal complexities and cellular structure of the universe and reflect the contradiction of Zeno's paradox: the smooth changes of the field occur parallel to the step-wise changes of energy. These discrete fluctuation in space might give rise to quantum phenomena in material systems, and form the basis of the quantum computer. The microdimensions are reformulated on many scales and lead to submanifolds that form the fractal and cellular structure in the material world, biology and society. Hence, quantum phenomena (correlation between discrete and smoothly changing systems) might be more general across the material world, in biological systems, conscious processing and society.