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:

http://www.mathforum.com/epigone/math-learn/bimferdkha ]

* * *

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

symmetric).

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:

http://inetarena.com/~pdx4d/ocn/graphics/pentagon.gif

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...

>>> radians

0.62831853071795862

>>> 2*math.cos(radians)

1.6180339887498949

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

started with.

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):

http://www.fluidiom.com/about.jsp?

droplet=droplets/cinema/dualpoly.html

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

before.

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

intersect).

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

is green).

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!

Slide: http://www.teleport.com/~pdx4d/hierarchy/vols.gif

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:

http://www.inetarena.com/~pdx4d/ocn/lehman.html

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:

http://www.inetarena.com/~pdx4d/ocn/concavedodeca.html

You'll be doing some like this yourself (if you stick with me

that is).

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?):

http://www.critpath.org/~strucky/danu/

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:

http://www.inetarena.com/~pdx4d/ocn/overview.html

Excerpt:

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.[3] 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.

Discussion?

Kirby

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