How to stay away from each other in a spherical universe

AbstractAristotle’s cosmology divides his finite, spherical universe into two main parts: the sublunary and the supralunary world. The former reaches from the middle of the Earth to the sphere of the Moon and is a domain of constant change, as the four sublunary elements, earth, water, air and fire endlessly change into one another. The latter starts with the sphere of the Moon and ends at the edge of the cosmos, that is, with the sphere of the fixed stars. There is only one element in this realm, the one that Aristotle calls the “first element” – an unchangeable, imperishable principle that guarantees the everlasting motion of heavenly bodies.There is, however, a short passage in the

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How to stay away from each other in a spherical universe

Resonance ◽

10.1007/bf02835977 ◽

1997 ◽

Vol 2 (10) ◽

pp. 38-45

Author(s):

Bhaskar Bagchi

Keyword(s):

Spherical Universe

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Eleatic Questions

The Classical Quarterly ◽

10.1017/s0009838800024423 ◽

1960 ◽

Vol 10 (1-2) ◽

pp. 84-102 ◽

Cited By ~ 25

Author(s):

G. E. L. Owen

Keyword(s):

Physical World ◽

Sensible Qualities ◽

Spherical Universe ◽

Subsequent Introduction

The following suggestions for the interpretation of Parmenides and Melissus can be grouped for convenience about one problem. This is the problem whether, as Aristotle thought and as most commentators still assume, Parmenides wrote his poem in the broad tradition of Ionian and Italian cosmology. The details of Aristotle's interpretation have been challenged over and again, but those who agree with his general assumptions take comfort from some or all of the following major arguments. First, the cosmogony which formed the last part of Parmenides' poem is expressly claimed by the goddess who expounds it to have some measure of truth or reliability in its own right, and indeed the very greatest measure possible for such an attempt. Second, the earlier arguments of the goddess prepare the ground for such a cosmogony in two ways. For in the first place these arguments themselves start from assumptions derived from earlier cosmologists, and are concerned merely to work out the implications of this traditional material. And, in the second place, they end by establishing the existence of a spherical universe: the framework of the physical world can be secured by logic even if the subsequent introduction of sensible qualities or ‘powers’ into this world marks some decline in logical rigour.

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The Lyre and the Cloak

Mapping the Afterlife ◽

10.1093/oso/9780190670481.003.0010 ◽

2020 ◽

pp. 243-276

Author(s):

Emma Gee

Keyword(s):

Ideal World ◽

The Earth ◽

Complete Synthesis ◽

The World ◽

Spherical Universe ◽

The Universe ◽

The Ideal

This chapter studies Plato’s Phaedo. In the Phaedo, the afterlife journey and the synoptic vision of the universe are collapsed into one another. In the myth of the dialogue, we are all, all the time, said to be on an underworld journey, since we live in the “creases” of the earth, not on its surface. At the same time, the True Earth of the Phaedo mirrors in its shape the spherical universe of the vision, as we also see it in the Spindle of Necessity in Plato’s Republic, and in the flight of souls around the universe in Plato’s Phaedrus. The Phaedo is a true geography of soul, in that the fate of the soul is integrated with the shape and motive forces of the earth seen as a whole. What we have in the Phaedo is a complete synthesis of the mythical underworld with the “geographic” earth. Tartarus (Phaedo 111e7–112e3) is the lowest point of the world, but it is also the center of the sphere. The result of Plato’s assimilation of the underworld, the landscape of the soul, with the “scientific” earth, is that earth and soul become analogous. They can be studied in the same way. In the ideal world, the universe itself is our eschatology.

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PRECESSION AND CHAOS IN THE CLASSICAL TWO-BODY PROBLEM IN A SPHERICAL UNIVERSE

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127408020380 ◽

2008 ◽

Vol 18 (02) ◽

pp. 455-464 ◽

Cited By ~ 4

Author(s):

JOHN F. LINDNER ◽

MARTHA I. ROSEBERRY ◽

DANIEL E. SHAI ◽

NICHOLAS J. HARMON ◽

KATHERINE D. OLAKSEN

Keyword(s):

Body Problem ◽

Initial Conditions ◽

Flat Space ◽

Three Body Problem ◽

Two Body Problem ◽

Extreme Sensitivity ◽

Three Body ◽

Spherical Curvature ◽

Spherical Universe ◽

Two Bodies

We generalize the classical two-body problem from flat space to spherical space and realize much of the complexity of the classical three-body problem with only two bodies. We show analytically, by perturbation theory, that small, nearly circular orbits of identical particles in a spherical universe precess at rates proportional to the square root of their initial separations and inversely proportional to the square of the universe's radius. We show computationally, by graphically displaying the outcomes of large open sets of initial conditions, that large orbits can exhibit extreme sensitivity to initial conditions, the signature of chaos. Although the spherical curvature causes nearby geodesics to converge, the compact space enables infinitely many close encounters, which is the mechanism of the chaos.

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