Date: May 11, 2001 6:21 PM
Author: Kirby Urner
Subject: [math-learn] Another Short Talk on Geometry
Another Short Talk on Geometry
by Kirby Urner
Oregon Curriculum Network
May 11, 2001
[Note: This talk is a continuation of an earlier one, archived here:
* * *
You'll remember from yesterday that we were discussing the importance
of phi, the divine proportion or golden ratio. I mentioned the
golden rectangle and golden cuboid as two geometric figures which
employ this ratio, but where else does it show up?
Phi gets into the picture when we get to Phive Phold Symmetry --
another silly spelling (recall Phirst Phractal). A regular triangle
may be spun on a spindle through its center in 120 degree increments
and end up looking the same -- it "fits into itself". Likewise a
square rotates at 90 degree intervals. 120 goes into 360 thrice,
and 90 goes in four times. We're dealing with 3 and 4-fold symmetry
here. Enter the pentagon: it fits into itself with rotations of
72 degrees, or 360/5, and is therefore 5-fold symmetric (rotationally
Let's look at the regular pentagon and analyze its dimensions. Make
all edges = 1, that's easiest. You can always do this, as whatever
they give you, just declare the edge to be your "unit" -- all other
edges or segments will now have a specific measure vis-a-vis your
unit, and you'll have a conversion constant for others to use if
they want to get the same answers you do.
Now lets draw a diagonal of the pentagon, meaning we skip a vertex
and go to the next one (there are only 5, so this is the only kind
of diagonal there is, as skipping one means skipping two on the
other side, and those are your only options for interior, vertex-
skipping segments, a.k.a. diagonals).
So now we have an isosceles triangle of edges A,A,blah, where A=1 and
we'd like to find blah, our diagonal. Consider that a regular
pentagon has spokes from its center to the vertices, making these
triangular wedges. The central angle, or apex of each wedge is
72 degrees, the increment of rotational symmetry, and these are
isosceles triangles, so 180-72 = 108, meaning we have angles of
108 degrees between perimeter segments, or 108/2 = 54 degree base
angles for our wedges.
Here's a diagram:
So now we've got this diagonal of unknown length across a wedge, and
an opposite angle of 108, meaning our base angles will be (180-108)/2
= 36 degrees. Draw a bisector to make two right triangles, with
hypotenuse = A = 1 = edge of pentagon, and unknown/2 (half of blah)
being a leg. If you know trig, you know this is where cosine comes
in: cosine(36)=blah/2, or blah = 2*cosine(36).
>>> radians = 36.0*(math.pi/180)
We need radians for our cosine function, or if you're using a
calculator, you should have a degree option...
Well, well, well. Looks like our old friend phi. So here's a
faster way to get phi on your calculator: 2*cos(36 degrees) --
fewer keystrokes than entering (1+sqrt(5))/2. Or use 2*sin(54)
if you prefer.
So the pentagon's diagonals are phi-ratioed vis-a-vis the perimeter
segments, or about 1.618 times the length of an edge.
The pentagon shows up in polyhedra -- most characteristically in
the last of the Platonic Five, the pentagonal dodecahedron (12
pentagonal faces). It's dual, another of the Platonic Five, is
the icosahedron (20 faces), which, upon further analysis, turns
out to contain 3 golden rectangles oriented in the XYZ (i.e.
mutually perpendicular) directions. So phi is coming out big time
vis-a-vis these two Platonic polyhedra, the pentagonal dodecahedron,
and the icosahedron, both of which have 5-fold rotational symmetry.
I mentioned "dual" a minute ago. This is an important operation in
polyhedra-ville. To form the dual of a polyhedron, you put a vertex
at the center of each of its faces, and then connect up these vertices
with edges running transverse to the ones you already have. Since
every edge runs between two faces, putting a dot in each of these
faces results in the same edge simply turning to connect two new
points -- your dual ends up with the same number of edges as it
You can think of duals in terms of Euler's Law for polyhedra,
which states that V + F = E + 2. Since the Vs and Fs are simply
exchanging places (every F becomes a V, every V becomes an F --
as new edges form a perimeter around what used to be a vertex)
so E must stay the same. We'll look at a movie in our web
browser now (projected screen image):
The Platonic Five may be organized in terms of dual pairs. The
pentagonal dodecahedron and icosahedron are duals. The cube and
octahedron are duals. That leaves the tetrahedron, the simplest
of all polyhedra (and therefore known as a "simplex" in some
contexts). The tetrahedron is its own dual -- its 4 faces turn
into 4 vertices and its 4 vertices turn into 4 faces, while the
6 edges remain 6 edges, but now at 90 degrees to what they were
Now a dual needn't be a specific size relative to the initial
polyhedron (the one we're taking the dual of), but if you like,
you can scale the dual so that the edges of the two (i.e. of the
original and its dual pair) intersect one another. This is
often at edge midpoints, but doesn't have to be. Some geometers
call these "structural duals" (when the edges are made to
Notice that a polyhedron + its structural dual defines yet a
new polyhedron. The tetrahedron + its structural dual (itself),
forms the cube. The cube plus its structural dual (octahedron),
forms the rhombic dodecahedron. The rhombic dodecahedron plus
its structural dual (cuboctahedron), forms what? This will be
a computer lab topic. We'll use Qhull, a freeware convex hull
finder that accepts XYZ coordinates, and returns the maximum
matching polyhedron or convex hull (it'll leave out interior
points if it has to, but it this case they'll all be used).
Using Qhull's results, we'll be able to render the polyhedron
in question using Python + POV-Ray, the ray tracer.
Anyway, you're probably starting to see how the geometers of old
tended to relate the polyhedra to one another, in terms of size
and even orientation. The cube would be related to the tetrahedron
as the combination of a tetrahedron and its structural dual, an
8-pointed stellate our friend Kepler called the Stella Octangula.
Animated GIF: http://www.teleport.com/~pdx4d/hierarchy/cubeanim.gif
(tetrahedra are orange and black, cube a.k.a. regular hexahedron
The rhombic dodecahedron might be defined as the combination of
that very cube and *it's* structural dual, the octahedron.
Animated GIF: http://www.teleport.com/~pdx4d/hierarchy/dodecanim.gif
(Cube is green, octahedron Red, Rhombic Dodeca Blue -- sphere brown,
we'll get to it in a minute).
When you organize your polys in this way, the volumes come out
very nicely, as geometer Buckminster Fuller harped on incessantly:
tetrahedron:cube:octahedron:rh_dodecahedron :: 1:3:4:6
How easy! How perfect!
But how shall we connect up the pentagonal dodecahedron and
it's structural dual? Together they make the 30-faced rhombic
triacontahedron (yes, it has a dual too, and so on).
The standard solution, since ancient times, has been to nest
the cube in the pentagonal dodecahedron as face diagonals. The
pentagonal dodecahedron has 12 faces, and the cube has 12 edges:
each face contributes one phi-diagonal to the cube (assuming
edges of pentagonal dodecahedron are unit -- but what if the
cube's edges are unit? Answer: dodeca's = 1/phi).
Animated GIF: http://www.teleport.com/~pdx4d/hierarchy/pentaanim.gif
This way of relating the polyhedra has a lot going for it. The
rhombic dodecahedra turn out to be space-fillers, and if you put
a sphere in each one, tangent to the 12 faces, as these diamond-
faced dodecas fill space, the spheres will pack into the all-
important FCC arrangement, also known as the CCP -- the basis of
any number of chemical compounds (we'll talk about Kepler's
Conjecture another time).
Pentagonal dodecahedra don't fill space on their own, but when
you pack the inscribed cubes together, you get a pattern of
pentagonal dodecas and complementary stellates known as concave
pentagonal dodecahedra, which *do* fill space together. This
theme has been the focus of artist Jim Lehman, a class behind
the famous Ken Snelson at Oregon State University (they knew some
of the same teachers). You might want to take a break to study
some of Jim's work, starting with the explanations I've concocted:
Don't worry about understanding all the terminology. But be
sure to check out these nifty POV-Ray + Python -generated pictures
of the concave pentagonal dodecahedron:
You'll be doing some like this yourself (if you stick with me
Another interesting project is to pick up on the rhombic dodecahedron
and its structural dual, the cuboctahedron. In Fuller's concentric
hierarchy of nested polyhedra, we break with the pattern of using
a structural dual and let the cuboctahedron scale up to a relative
volume of 20 (remember: 1:3:4:6 -- and now :20), where it coincides
with the CCP sphere packing lattice, i.e. 12 spheres tightly packed
around a nuclear sphere form the corners of a cuboctahedron. But
there's a nice volume relationship between the rhombic dodeca and
it's structural dual as well. I worked with danu on that awhile back.
Danu operates in the sacred geometry tradition and I don't understand
his stuff at all -- very alien to my way of thinking. But you
might want to browse his site just to see what geometry leads to for
some -- and why a lot of teachers are still afraid to teach
the humanities side of the subject -- they don't want to get sucked
into some cultish vortex (and who can blame 'em?):
Note to teachers: I think student interest in geometry would
increase if we wired it up with the relevant humanities, as in the
old days, even if we don't go overboard in this direction. More
on that here:
On the spatial front, I encourage a kind of "mental geometry"
to develop alongside "mental arithmetic", and do so by exploiting
the ancient tradition of nesting polyhedra to form a "maze" or
"matrix". It's something the sacred geometers were into, down
through the theosophists in our own time.
Given this history, our graphics and visuals may tend to raise
some eyebrows. But that's OK. I think it's more interesting
to students as well, to connect to these arts and traditions,
while developing their right brain abilities to consider
geometry in the mind's eye. Opportunities to bring up historical
topics, which offer insight into how we came to be who we are,
should not be ignored.
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