Rectangles Inscribed in a DodecahedronDate: 06/19/99 at 10:07:59 From: Ed Condren Subject: I teach medieval lit at UCLA I'm currently writing a book on a single medieval manuscript. Research so far convinces me that it was organized according to mathematical principles, esp. the Golden Section. What I need now is an equation for determining the size of the rectangles inscribed within a regular dodecahedron. I know there are three of them; they are identical; they are at right angles to each other; their corners touch the midpoints of the pentagonal faces; and their sides are in a phi-relation. But I need a formula that includes the dimension of the dodecahedron. Perhaps if I knew the precise angle each pentagonal face makes with the other faces with which it shares an edge, I could figure it out. From the model I built, it looks like 117 degrees, but my model is far from precise. Any help anyone can give me will be much appreciated. Date: 06/22/99 at 14:25:19 From: Doctor Melissa Subject: Re: I teach medieval lit at UCLA Hi Ed, According to our Formulas FAQ, the angle you want is delta = arccos(-sqrt[5]/5) = 116 degrees and 34' For future reference, the URL for the FAQ is http://mathforum.org/dr.math/faq/formulas/ Since I know little else about three-dimensional geometry, I'm going to leave this question in the area where other doctors may see and respond to it. Good luck - this sounds fascinating! - Doctor Melissa, The Math Forum http://mathforum.org/dr.math/ Date: 06/23/99 at 03:51:46 From: Doctor Pete Subject: Re: I teach medieval lit at UCLA Hi, What you are looking for is the ratio of the side of the dodecahedron to the length of a short edge of one of its inscribed golden rectangles. But this ratio is the same as that between a side of the dodecahedron and a side of its inscribed dual, the icosahedron; this is because the twelve vertices (4x3) of the three golden rectangles are the twelve vertices of a circumscribing icosahedron. This icosahedron, having twenty equilateral triangular faces, is inscribed in the dodecahedron such that each vertex of the former is the center of a face of the latter. That said, some computation shows that this ratio is 2 : (1 + 3/Sqrt[5]), or Sqrt[5] : p^2, or 1 : 1.170820, where p is the golden ratio (1+Sqrt[5])/2. So the short edge of the inscribed golden rectangle is slightly larger than the edge of the dodecahedron. If you are really fascinated by all this, then let me tell you how I derived these numbers. I didn't use the dihedral angle at all; in fact, I used a fascinating property of the regular dodecahedron, which is that it is possible to choose 8 vertices of the dodecahedron which are also vertices of an inscribed cube. Furthermore, there are 5 such cubes. These cubes have edge length p in relation to the edge length of the dodecahedron, because each cube edge is a line joining two non-adjacent vertices on a pentagonal face. If we draw a side of one of these cubes, and construct a portion of the dodecahedron that sits atop one of these sides, it becomes clear that the distance we wish to find is the distance between the centers of two adjacent pentagons. But this line is parallel to the face of the cube, and so we may draw a pair of similar triangles which compares the desired length to the length of the cube's edge. The details of this are a bit too tricky to describe in words, as a diagram would do far better; but needless to say, it is only through some simple algebra that one arrives at the answer above. I invite you to draw some diagrams, in particular, to draw on your dodecahedron model 12 lines which describe a cube whose 8 vertices coincide with 8 of the dodecahedron's vertices, and whose 12 edges (hint! 12 cube edges, 12 faces of the dodecahedron) coincide with the 12 faces of the dodecahedron. Incidentally, if you want the dihedral angle of the dodecahedron, it is ArcCos[-1/Sqrt[5]], that is, the angle between 0 and 180 degrees whose cosine is -1/Sqrt[5], which is approximately 116.5650 degrees. You were quite close, I'd say. - Doctor Pete, The Math Forum http://mathforum.org/dr.math/ Date: 06/23/99 at 09:13:03 From: Edward Condren Subject: Re: I teach medieval lit at UCLA Hello Doctor Melissa, Thank you for responding. I've had this message out on the Net and in several chat rooms for about three weeks, and you are the first one to respond. I'll check the FAQ site you gave me, but the 116 and 34' may be all I need. My numeracy is low, but the first article I'm cheekily publishing on this intricate and fascinating subject will appear in the early fall, in Viator 30, under the pretentious title "Numerical Proportion as Aesthetic Strategy in the Pearl Manuscript." If you're having trouble getting to sleep some night, you might give it a try. Ed Condren Professor of English, UCLA Date: 06/24/99 at 01:00:51 From: Edward Condren Subject: Response from Dr. Pete Dear Doctor Pete, I want to thank you for your kindness in spending so much time responding to my shot in the dark about the mysteries of the dodecahedron for a book I'm writing on a medieval manuscript. What moved me in particular - apart from your expertise, of course - was your sense that my researches have provided me with a great deal of joy and fascination, despite my having only rudimentary skills at mathematics. I thought I'd risk overstaying my welcome by mentioning briefly what my project is about. A unique fourteenth century manuscript, British Library Cotton Nero A.x., has four poems, two of which are the brilliant Pearl and Sir Gawain and the Green Knight. It has always been thought to be a mere anthology, probably by the same author. My book argues that it is in fact a single artifact unified by the same mathematics that, from the Neo-Pythagoreans onward, have been held to demonstrate the unity of all creation. The line counts of the four poems are, respectively, 1212, 1812, 531, and 2531. This highly artful arrangement, minus the signature twelves in the first half and signature 31s (the tenth prime as the Middle Ages reckoned primes) in the second half, gives us two halves of 3000 lines each. More intriguing still, the two outer poems divided by the two medial poems give the Golden Section. The first poem, Pearl, has 20 sections of 5 stanzas each, with each stanza containing 12 rhyming lines. Moreover, the first line of each stanza replicates the last line of the preceding stanza, with the first line of the poem echoing the last line of the poem. It seems clear to me that this poem's 1212 lines create a verbal dodecahedron: 20 vertices; 12 faces; each face a five-sided pentagon. The pentagon, of course, is highly dependent on the phi ratio. The three remaining poems seem 2-D, rather than 3-D. In other words, the poet may have conceived them as the three inscribed planes. The first of these remaining three, Purity, is constructed almost entirely of phi-related sections which encode a five-pointed star - another phi-dependent figure - which happens to be the main symbol on Sir Gawain's shield. The next poem I haven't entirely cracked yet (hence my query about the size of the inscribed rectangles). It's a retelling of the story of Jonah. But its 5 sections have intriguing sizes: 60, 184, 60, 104, 123. When we remove the overages, which total 31 (4 + 4 + 23), we are left with two 60s, two 100s, and 180. This last is the radius of the circumscribed circle surrounding the five-pointed star laid out in Purity. Enough! You'll never again answer a humanist's math questions. But I trust you can see how much pleasure all this figuring has afforded me. While I've still been completing my manuscript I thought I'd stake out my claim on this complex thesis by publishing a small article on it in Viator, vol. 30, to be published in early September. If you would tell me where I could send an offprint, I'd be happy to send you one so you could be amused at how a literary critic thinks he's discovering the wheel, when you chaps have been playing with these phenomena for centuries. Thanks, again, for all your help. Ed Condren Professor of English, Univ. of California, Los Angeles (UCLA) Los Angeles, CA 90095 |
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