Abstract
This Appendix extends, in the sense of providing details and technical information, Chapters 3 and 4.
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Notes
- 1.
The number π is approximately 3. 1459⋯ , where the “⋯ ” indicate the digits continue indefinitely. The first 8 billion digits of π have been calculated — see David Blatner’s 1997 book The Joy of pi published by Walker and Company, NY — [5] in Bibliography. For an example of applying the perimeter formula, note that the perimeter of a circle with a 10-foot radius is 2 × (3. 1459⋯ ) × 10 = 62. 981⋯ feet.
- 2.
The perimeter of a circle of radius r on flat ground will be greater than the perimeter of a circle of radius r measured on the sphere and less than the perimeter of a circle of radius r measured on the saddle surface. For a concise overview of surfaces, see Chapter 8 Riemann’s Legacy of O’Shea’s book, The Poincaré Conjecture — reference [26] in Bibliography.
- 3.
A galaxy is a grouping of billions of stars (our sun is an example of a star but stars in general may be much larger or much smaller than our sun). And of course galaxies may also contain planets and other matter. A galaxy shaped like a gigantic deformed ball may span a hundred thousand light years or more. (A light year is the distance that light travels in one year, and through a vacuum, light travels 186,000 miles per second.)
- 4.
See the article Einstein’s Static Universe: An Idea Whose Time Has Come Back by Aubert Daigneault and Arturo Sangalli published in the January 2001 issue of the Notices of the American Mathematical Society.
- 5.
See footnote 5 above.
- 6.
The proposed “variant of special relativity” appears in Segal’s article A variant of special relativity and long-distance astronomy Published in the Proc. Nat. Acad. Sci., Vol. 71 in 1974, pages 765 to 768.
- 7.
Published by Academic Press, New York, 1976.
- 8.
Within the quote, note the use of Minkowski’s time — in the context of Minkowski’s space which is the Cartesian product R × R 3, Minkowski’s time is the first factor R which is the Real timeline. The second factor R 3 may be viewed as our ordinary 3-dimensional human-vision space.
- 9.
Wolf’s book Parallel Universes was published in 1988 by Simon & Schuster Paperbacks.
- 10.
Franks’ book is published by iUniverse, Inc., 2021 Pine Lake Road, Suite 100, Lincoln, NE, 68512.
- 11.
Ivan Ivanšić and Uroš Milutinović showed that J 3 is sufficient for all of these cases. See Theorem 14 in the The Quest for Universal Spaces in Dimension Theory by Stephen Lipscomb. Also see his article in the AMS Notices, Vol. 56, Number 11, December 2009.
Bibliography
Blatner, David, The Joy of π. Walker and Company, NY, 1997.
O,Shea, Donal, The Poincaré Conjecture, In Search of the Shape of the Universe, (pages 38, 39, and 206), Walker and Company, NY, 2007.
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4-web moving into human vision (54,613 KB)
Hyper-sphere L-4 (50,572 KB)
Hyper-sphere L-5 (124,074 KB)
Hyper-sphere L-6 (143,719 KB)
Hyper-sphere L-7 (234,198 KB)
Hyper-sphere L-8 (278,008 KB)
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Lipscomb, S.L. (2014). Appendix 2: Supplement for Chapters 3 and 4. In: Art Meets Mathematics in the Fourth Dimension. Springer, Cham. https://doi.org/10.1007/978-3-319-06254-9_13
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