Art Meets Mathematics in the Fourth Dimension
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To see objects that live in the fourth dimension we humans would need to add a fourth dimension to our three-dimensional vision. An example of such an object that lives in the fourth dimension is a hyper-sphere or “3-sphere.” The quest to imagine the elusive 3-sphere has deep historical roots: medieval poet Dante Alighieri used a 3-sphere to convey his allegorical vision of the Christian afterlife in his Divine Comedy. In 1917, Albert Einstein visualized the universe as a 3-sphere, describing this imagery as “the place where the reader’s imagination boggles. Nobody can imagine this thing.” Over time, however, understanding of the concept of a dimension evolved. By 2003, a researcher had successfully rendered into human vision the structure of a 4-web (think of an ever increasingly-dense spider’s web). In this text, Stephen Lipscomb takes his innovative dimension theory research a step further, using the 4-web to reveal a new partial image of a 3-sphere. Illustrations support the reader’s understanding of the mathematics behind this process. Lipscomb describes a computer program that can produce partial images of a 3-sphere and suggests methods of discerning other fourth-dimensional objects that may serve as the basis for future artwork.
each t × S 1 (Figure 4.2). It turns out that the entire Cartesian Product R × S 1 with each of its “rows” t × S 1 may be pictured as a hollow pipe. Fig. 4.2The Cartesian product R × S 1 as a hollow pipe of infinite length. The hollow-pipe picture of R × S 1 is a faithful picture because each point in the picture corresponds to one, and only one, point in R × S 1. And we clearly see that distinct times t and t′ induce distinct non-touching universes t × S 1 and t′ × S 1.3 Now let us consider
beginning: it is infinite in both directions, and so the universe has always existed and will always exist. Hence EU may be presented as the Cartesian product R × S 3 where R is the whole real timeline. … �22 Comments Again, the goal here is not to explain modern cosmology. Rather, the goal in this chapter is the same as it has been in the previous chapters — show relevance of the 3-sphere.5 Footnotes 1The article (subtitle A tribute to Irving Ezra Segal (1918–1998)) is authored by Aubert
Image?” art on the cover is centered at c = (. 25, . 25, . 25, . 25) and has radius r = . 4. 2The derivation of the transformation from hyper-space into human visual space may be found in reference  and also in the author’s book . 3Version 3.1g.watcom.win 32[Pentium ll optimized], copyright �1991-1999 by the POV-Team. � Springer International Publishing Switzerland 2014 Stephen Leon LipscombArt Meets Mathematics in the Fourth Dimension10.1007/978-3-319-06254-9_9 9. Prelude to
point of observation.” Simply put, Einstein’s cosmic time t is the “real” one, whereas Minkowski’s time is only an approximation of t. �A16 Final paragraph of quote The remainder of the article that tracks Einstein’s Universe R × S 3 concerns the chronometric cosmology theory (CC) and how it addresses other cosmological theories. To close the quotes, so to speak, we include the final paragraph of the article: Hence we may conclude on a rather speculative but perhaps conciliatory note. Assuming
universes creates many possibilities for speculation. Indeed, in his 2003 book The Universe and Multiple Reality M. R. Franks speaks of a superuniverse.12 Franks’ superuniverse is a universe itself, a universe that contains all of the “parallel universes”. He defines energy states (static universes) as either contiguous — two such static universes differ by just one quantum transition — or noncontiguous — two such static universes differ by more than one quantum transition. For an analogy, he