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A Mathematical Review of the Rainbow Bridge: Rainbows in Art, Myth, and Science John A Old Dominion University ODU Digital Commons Mathematics & Statistics Faculty Publications Mathematics & Statistics 2002 Like a Bridge Over Colored Water: A Mathematical Review of The Rainbow Bridge: Rainbows in Art, Myth, and Science John A. Adam Old Dominion University, [email protected] Follow this and additional works at: https://digitalcommons.odu.edu/mathstat_fac_pubs Part of the Algebraic Geometry Commons, and the Fluid Dynamics Commons Repository Citation Adam, John A., "Like a Bridge Over Colored Water: A Mathematical Review of The Rainbow Bridge: Rainbows in Art, Myth, and Science" (2002). Mathematics & Statistics Faculty Publications. 157. https://digitalcommons.odu.edu/mathstat_fac_pubs/157 Original Publication Citation Adam, J. A. (2002). Like a bridge over colored water: A mathematical review of the rainbow bridge: Rainbows in art, myth, and science. Notices of the American Mathematical Society, 49(11), 1360-1371. This Article is brought to you for free and open access by the Mathematics & Statistics at ODU Digital Commons. It has been accepted for inclusion in Mathematics & Statistics Faculty Publications by an authorized administrator of ODU Digital Commons. For more information, please contact [email protected]. Like a Bridge over Colored Water: A Mathematical Review of The Rainbow Bridge: Rainbows in Art, Myth, and Science Reviewed by John A. Adam The Rainbow Bridge: Rainbows in Art, Myth, and other important and explicit themes: the connec- Science tions with mythology, art, and science. To a de- Raymond L. Lee Jr. and Alistair B. Fraser gree this is inevitable, if such a weighted metric is Pennsylvania State University Press , 2001 to be used. Lee and Fraser are intent on exploring 393 pages, $65.00 bridges from the rainbow to all the places listed ISBN 0-271-01977-8 above, and in the opinion of this reviewer they have succeeded admirably. The serious reader will This is a magnificent and scholarly book, exquis- glean much of value, and mathematicians in par- itely produced, and definitely not destined only ticular may benefit from the tantalizing hints of for the coffee table. It is multifaceted in character, mathematical structure hidden in the photographs addressing rainbow-relevant as- and graphics. Following an account of several pects of mythology, religion, themes present in the book that I found particularly IJ:iE RAI BOW BRIDGE the history of art, art criticism, appealing, some of the mathematical structure un- the history of optics, the theory derlying and supporting the bridge between the of color, the philosophy of sci- rainbow and its optical description will be empha- ence, and advertising! The qual- sized in this review. ity of the reproductions and I have included several direct quotations to photographs is superb. The au- illustrate both the writing style and the features of thors are experts in meteoro- the book that I found most intriguing. Frankly, logical optics, but their book I found some of the early chapters harder to draws on many other subdisci- appreciate than later ones. This is a reflection, no plines. It is a challenge, there- doubt, of my own educational deficiencies in the fore, to write a review about a book that contains no equa- liberal arts; but upon completing the book, I was tions or explicit mathematical moved to suggest to the chair of the art department themes for what is primarily a that it might be interesting to offer a team-taught graduate seminar in art, using this book as a text. RAYMOND L LEE. JR. AND ALISTAIR B. >IM.SCR mathematical audience. However, while the mathemat- He is now avidly reading my copy of the book. ical description of the rainbow may be hidden in this The ten chapters combined trace the rainbow book, it is nonetheless present. Clearly, such a re- bridge to “the gods” as a sign and symbol (“emblem view runs the risk of giving a distorted picture of and enigma”); to the “growing tension between what the book is about, both by “unfolding” the hid- scientific and artistic images of the rainbow”; den mathematics and suppressing, to some extent, through the inconsistencies of the Aristotelian description and beyond, to those of Descartes and John A. Adam is professor of mathematics at Old Do- Newton, and the latter’s theory of color; to claims minion University. His email address is [email protected]. of a new unity between the scientific and artistic 1360 NOTICES OF THE AMS VOLUME 49, NUMBER 11 Figure 1. A sector illustrating the rainbow violet configuration, including both a primary bow (on enterprises; the evolution of scientific models of the bottom) and a secondary bow (at the top). the rainbow to relatively recent times; and the The bright region is not shown to scale. red exploitation and commercialization of the rain- bow. All these bridges, the authors claim, are united great faith in observations simplified and clarified by the human appreciation of the rainbow’s com- by the power of mathematics.” Newton, on the Alexander’s dark band pelling natural beauty. And who can disagree? An other hand, eschews Descartes’s cumbersome appendix is provided (“a field guide to the rainbow”) ray-tracing technique and “silently invokes his comprising nineteen basic questions about the rain- mathematical invention of the 1660s, differential calculus, to specify the minimum deviation rays of bow, with nontechnical (but scientifically red accurate) answers for the interested observer. the primary and secondary rainbows.” Later in the This is followed by a set of chapter notes and a book Lee and Fraser remark that Aristotle and later violet super- bibliography, both of which are very comprehensive. scientists in antiquity “constructed theories that pri- numerary bows The more technical scientific aspects of the distri- marily describe natural phenomena in mathemati- bution of light within a rainbow are scattered lib- cal or geometric terms, with little or no concern for erally throughout the latter half of the book; indeed, physical mechanisms that might explain them.” This the reader more interested in the scientific aspects contrast goes to the heart of the difference between bright region of the rainbow might wish to read the last five Aristotelian and mathematical modeling. chapters in parallel with the first five (as did I). The I was pleasantly surprised to learn that the Eng- technical aspects referred to are explained with lish painter John Constable was quite an avid am- great clarity. ateur scientist: he was concerned that his paintings But what is a rainbow? Towards the end of this of clouds and rainbows should accurately reflect review, several mutually inexclusive but comple- the science of the day, and he took great trouble mentary levels of explanation will be noted, to acquaint himself with Newton’s theory of the reflecting the fact that there is a great deal of rainbow (many other details can be found in physics and mathematics behind one of nature’s pp. 80–7 of the book). In a similar vein, the writer most awesome spectacles. At a more basic obser- John Ruskin was a detailed observer of nature, his vational level, surely everyone can describe the goal being “that of transforming close observation colored arc of light we call a rainbow, we might into faithful depiction of a purposeful, divinely suggest, certainly as far as the primary bow is con- shaped nature.” The poet John Keats implies in his cerned. In principle, whenever there is a primary Lamia that Newton’s natural philosophy destroys rainbow, there is a larger and fainter secondary the beauty of the rainbow (“There was an awful rain- bow. The primary (formed by light being refracted bow once in heaven: / We know her woof, her tex- twice and reflected once in raindrops) lies beneath ture; she is given / In the dull catalog of common a fainter secondary bow (formed by an additional things. / Philosophy will clip an Angel’s wings”). reflection, and therefore fainter because of light loss) Keats’s words reflected a continuing debate: did which is not always easily seen. There are several scientific knowledge facilitate or constrain poetic other things to look for: faint pastel fringes just be- descriptions of nature? His contemporary William low the top of the primary bow (supernumerary Wordsworth apparently held a bows, of which more anon), the reversal of colors different opinion, for he wrote, sin i sin r = in the secondary bow, the dark region between the “The beauty in form of a plant n two bows, and the bright region below the primary or an animal is not made less nred =13318. bow. This observational description, however, is but more apparent as a whole i probably not one that is universally known; in by more accurate insight into r its constituent properties and some cultures it is considered unwise even to look bi at a rainbow. Indeed, in many parts of the world, powers.” it appears, merely pointing at a rainbow is consid- In a subsection of Chapter 4 l i ered to be a foolhardy act. Lee and Fraser state entitled “The Inescapable (and D that “getting jaundice, losing an eye, being struck Unapproachable) Bow”, the au- by lightning, or simply disappearing are among thors address, amongst other the unsavory aftermaths of rainbow pointing.” things, some common misper- The historical descriptions are in places quite ceptions about rainbows. Just D = π(p−)+21 i− 2pr (p= 2here) breathtaking; we are invited to look over the shoul- as occurs when we contemplate ders, as it were, of Descartes and Newton as they our visage in a mirror, what we work through their respective accounts of the see is not an object, but an rainbow’s position and colors. While the color image; near the end of the chap- Figure 2.
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