Zeno's paradox centers on the relative movement that can be viewed as a discrete series. The above illustration portrays the relative movement of Achilles and a turtle. In the paradox, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 meters. Each racer starts running at some constant speed, one faster than the other. When Achilles will have run 100 meters, he is at the tortoise's starting point. During this time, the tortoise has run a much shorter distance, say 2 meters. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles arrives somewhere the tortoise has been, he still has some distance to go before he can even reach the tortoise.

Intuitively, 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' motion into an infinite sequence regarding the turtle's position. Although Achilles gets infinitesimally closer to the turtle, he can never reach it.

The old philosophical puzzle might be representative of the universe's deep structure, the separate entities of space and time. Time represents compact dimensions forming standing waves of discrete energy vibrations. In contrast, space is the smooth spatial topology of the gravity field. The contrasting dynamics of time and space leads to the many contradiction of quantum mechanics and can lead to Zeno’s paradox. Zeno's paradox is an example of a discrete system interacting with a smoothly changing one.

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

The global topology of the universe is formulated by the spatial field, demonstrated by a Breton hat above. The field curvature changes smoothly from the black holes toward expansion near the white holes, whereas the micro dimensions form standing waves, a quantized, step-wise progression between the poles. Constant frequency waves of the micro dimensions organize along the constant field curvature of space. The macro-and micro dimensions are connected via interaction, or decoherence, which ensures their congruent development and interdependent relationship, as shown in Zenon's paradox. This is the quality of wave decoherence in quantum mechanics. The connection between sequences formed of discrete series (quantum mechanics) and a field with smooth gradation (gravity) has been featured in my earlier post and YouTube video. Such orthogonal relationships can also be found in many other areas, from physical sciences to social phenomena.

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

The compact dimensions maintain a minimal closed surface within each standing wave succession between the poles; inverse volume changes by Lorentz transformations enhance field curvature and pressure differences (above left). However, the curving field directs the participant's decision, and the participant's choice further modifies the field (as observed along the brim and the top of the hat). Hence, deterministic, irreversible, and self-perpetuating changes give rise to singularities, called poles in the curving field. The polar singularities stabilize the structure of the cosmos (shown above, right).

The increasing gravitational differences of space versus the compact dimensions' minimal surface 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 fluctuations in space might give rise to quantum phenomena in physical systems and form the basis of the quantum computer. Fractal and cellular structure reappears on many scales in the material world, biology, and society. Hence, quantum phenomena (correlation between discrete and smoothly changing systems) might be more general in the material world, biological systems, conscious processing, and society.

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