Rectangles Inscribed in a Dodecahedron

Date: 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