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In 3-D Euclidean space, the formula for a sphere of radius a located at (0,0,0) has the form:x ^{2} + y^{2} + z^{2} = a^{2}with the metric ds ^{2} = dx^{2} + dy^{2} + dz^{2} = a^{2} (d^{2} + sin^{2} d^{2}) ---------- (20g)in spherical coordinates (r, , ), where the radius r = a is constant (Figure 10p). The sphere has only two degrees of freedom. Further simplification is possible by considering only those circles around the z-axis, or z = a cos = constant, which implies = constant. In this case | |

## Figure 10p 2-D Sphere | ds^{2} = a^{2} sin^{2} d^{2}. The length of the circumference is 2 a sin, which varies from 0 to a maximum of 2a and falling back to 0 as varies from 0 to /2 and then to . The point at = 0 or (sin = 0 at the North and South poles) is not a singularity as it can always be removed by re-defining the origin of reference. |

In the four-dimensional Euclidean space, the 3-D hypersphere is defined by the formula:

x

and ds

where d

The formulation can be generalized further to 5-D Euclidean space with the 4-D hypersphere metric:

ds

where the 3-D d

The no boundary proposal makes use of Eq.(20i) and suggests that at the beginning the universe was running with as the time component from = 0 to /2 in the Euclidean space. At /2 the character of the time component underwent a transformation, e.g., /2 + ict/a. Since then the 4-D hypersphere has changed into a Minkowski hyperboloid described by the metric:ds ^{2} = -c^{2}dt^{2} + a^{2} [cosh^{2}(ct/a) d_{3}^{2}]where the 3-D hypersphere is expanding as "a cosh(ct/a)", and approaches a de Sitter universe in the form (a/2)e ^{ct/a} as ct/a +.
Such a hybrid universe is shown in Figure 10q. Thus, there is no singularity at the beginning of time. That particular region becomes as smooth as the North
| |

## Figure 10q Hybrid Uni-verse [view large image] | (South) pole on Earth. It is suggested that the (red) region in Euclidean time represents the moment of nucleation via tunneling through some sort of energy barrier from nothing (in the form of vacuum fluctuation for example). |

ds

where d

This 5-D AdS space has been used to apply the holographic principle to physical theories or objects (such as superstring theory or black hole) by encoding them from such 5-D space to a 4-D hypersphere.

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