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Pappas has taken an axiomatic approach (3, 3) where each space coordinate corresponds to a temporal coordinate, each pair being inde- pendent of the other two-dimensional coordinates, with the produced space being the direct sum of the three two-dimensional coordinates. Lehto argued that in classical, continuous space-time we should be able to obtain infinite energy with an infinite frequency when dx, the small spatial part, nears zero. This was contested by Pauli and Weis- skopf. As a consequence, Planck energy (1.2 x 10’ GeV) should be the largest local energy around. In parallel, the very foundation of quan- tum mechanics (the ultraviolet catastrophe) dictates that space varia- tion should be proportional to time variations, based on light speed. So cyclical time, Lehto felt, had to follow Planck’s law. He replaced the fre- quency by t (cyclical time): E=ht Lehto said that non-linear forced oscillators generate factor 2 sub- harmonics. This phenomenon is called period-doubling. M. I. Feigen- baum extended his research to incorporate all universal behavior. The maximum local energy corresponds to a minimum period. Lehto argued that all the minimum local energy must be obtained by period- doubling. According to him, the existence of the process of period- doubling was underlying stable-particle systems. He thus took ratios in base 2 of the energies and lengths characteristic of different physi- cal phenomena. The periods of a simple period-doubling system then obey the equation: x/y = 2 where M is an integer and x and y can be any system period. This equation can also be used to find out whether a system is a period- doubling system by solving for the exponent M: where A and B are measured values, lengths or energies, for instance. Attentive readers will not be surprised to see the expression of a frac- tal in this equation. M is in fact a logarithmic “binder” (neperian in this case) that increases with powers of 2. Lengths and energies, Lehto said, have their corresponding periods obtained from the relations r=ct and ris E=h/t. 178 The Science of Extraterrestrials: UFOs Explained at Last * Eric Julien M =In(A/B) / In2