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2.3.2 Structural Curvature

A 'straight' line drawn on the surface of a sphere will project around the sphere and join up to form a circle. Such a line is as straight as it can be since it is required to adhere to the surface of the sphere. Analogously, a 'straight' line projected through the curved volume of a hyper-sphere will return to its origin to form a circle. Again, this line is as straight as it can be since it must remain within the curved volume. On the sphere of planet Earth, such a circle is known as a 'great circle'. By analogy, the circle described by a 'straight' line in the curved volume of the ideal universe may be called a 'universal circle'.

In both cases, 'straight' lines form circles with radius, \(R\). Hence curvature of such a line is given as the reciprocal of radius, \(1/R\). In the case of a sphere, at each point, curvature has two components, one for each of the two directions of independent movement on its two dimensional surface. For a perfect sphere these are equal, so that the total curvature at each point is given by \(1/R+1/R = 2/R\). Analogously, in the three dimensional volume of a perfect hyper-sphere, there are three directions of independent movement, so that total curvature at each point is \(3/R\).

In the perfect hyper-sphere which represents the ideal energy continuum, this structural curvature has the same value at each point. However, in the real universe this is not so since the radius vectors are not of equal length. This is impossible to visualize in curved volume, but it easily understood in the two dimensional analog of an ordinary sphere. Bumps and dips in the surface of a sphere obviously cause deviations from its normally smooth surface.

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