Art Meets Mathematics in the Fourth Dimension
Format: PDF / Kindle (mobi) / ePub
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.
previous chapter was to demonstrate, using Dante’s Divine Comedy, that the 3-sphere played a role in world literature and Christian faith. And these examples, however, are not exhaustive. Within mathematics the 3-sphere has a rich history. The reader who desires a challenge may consider the following article by R. H. Bing (1914–1986)6 Knowledge of the 3-sphere and at least part of its role in human history is a prerequisite for interest and appreciation of the “paint on canvas” God’s Image ?
dots are points in our visual 3-dimensional space, we may “walk around the assemblage of 25 dots” looking for “flat areas” or “missing parts.” The idea is illustrated in Figure 5.12. Fig. 5.12Circular, missing part, and flat-side views of 2-sphere dots. Keep in mind that each of the three “assemblages of 25 dots” pictured above is an image of a 2-sphere. The 25 dots contain subtle information. Since we know that the object in question is a 2-sphere, we know that the slant planes induced by the
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
of its subsets satisfy certain conditions.4 The point is simply that there are all kinds of examples of spaces that are not metric spaces, i.e., spaces where measurement of distances between points cannot be used to determine convergence. One classical example is the extended long line.5 So one naturally wonders whether at some future date, say when and if humans encounter a need to confront such physical non-metric spaces, will physics extend its history as a science of physical laws within
hyperbolic geometry. Against this backdrop, one could rightly say that this book concerns “fractal art” because the construction of the art depends on the 2003 mathematical discovery of how to move the 4-web fractal from 4-space into 3-space with its structure preserved. Is the meaning of “artist” being extended? One might wonder about the meaning of “artist” in the context of one whose brush strokes are guided by a fractal-based mathematical algorithm. Even more disconcerting is the fact that