Quantum Gravity and String Theory


The Curvature and Dimension of a Closed Surface

Authors: Shawn Halayka

In this short memorandum, the curvature and dimension properties of the $2$-sphere surface of a 3-dimensional ball and the $2.x$-dimensional surface of a 3-dimensional fractal set are considered. Tessellation is used to approximate each surface, primarily because the $2.x$-dimensional surface of a 3-dimensional fractal set is otherwise non-differentiable (having no well-defined surface normals). It is found that the curvature of a closed surface {\it must} lead to fractional dimension. Notes are then given, including how this tessellation model applies to a toy Universe.

Comments: 9 Pages. Clarified the role of the Planck length in terms of scale.

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Submission history

[v1] 2018-12-25 17:37:59
[v2] 2018-12-26 15:49:33
[v3] 2018-12-27 14:44:30
[v4] 2019-01-06 16:35:48
[v5] 2019-01-08 10:44:26
[v6] 2019-01-11 20:21:50
[v7] 2019-09-16 18:42:23
[v8] 2019-09-19 16:36:41
[v9] 2019-09-21 20:55:02
[vA] 2019-09-26 22:35:40
[vB] 2019-10-08 16:31:29
[vC] 2019-10-22 21:39:38
[vD] 2019-10-31 14:20:25
[vE] 2019-11-26 09:08:26
[vF] 2019-12-09 15:04:44

Unique-IP document downloads: 105 times

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