Date: May 11, 2001 2:40 PM Author: Kirby Urner Subject: [math-learn] A short talk on geometry

A Short Talk on Geometry

A key thing to understand about any discipline is how

people have used it over time. Geometry has everything

to do with navigation, getting from here to there safely,

especially by sea. But it also have everything to do with

architecture.

Since ancient times, people have been fascinated by

various dimensional relationships or ratios. You all

know what it means for a line segment to be doubled in

size. Do you know that if you have a triangle, square

or any shape, and double all of it's edges, what happens

to its area? How about if you have a pyramid, and blow

it up (in the photography sense -- to make bigger, not

to explode). What happens to its volume? <pause>

Yes, area goes up as a 2nd power of the linear change,

and volume as a 3rd power. So if a pyramid has volume

blah, and you double all its edges, it now has volume

8*blah, where 8 is 2 times 2 times 2 or 2 to the third

power. That's important to keep in mind, and will help

you at many points in the future.

Anyway, one of the key ratios that people have found

fascinating is called the divine proportion or the golden

ratio or phi. You can tell just from the name that it's

important. So what is it? First, it's important to keep

in mind that it's a relationship. It doesn't matter how

long segment A is, only how long it is in relationship

to segment B. So A might be a mile, an inch, a millimeter

or a furlong. Doesn't matter. The important thing is

that A:B as B:(A+B).

In other words, start with a single segment and make a

mark dividing it into A and B. A is the shorter part,

and it relates to B in the same way that B relates to

the whole segment:

---------------|-------------------------

A B

-------------------------|----------------------------------------

B A+B

A:B = B:(A+B)

This progression goes on in a self-similar manner in both

directions: bigger or more macro, and smaller, or more micro.

In other words, relative to A, there's some shorter segment that

is in divine proportion to it, in the same way as A relates to

B. And likewise (A+B) will relate to a yet longer segment by

the same scale factor. And what is that scale factor? We can

use algebra:

A/B = B/(A+B)

cross-multiply: A(A+B) = B^2

Let's just have A=1 for simplicity, and find B (remember, it makes

no difference what A is by itself -- it's the relationship to B

that matters):

1(1+B) = B^2 or B^2-B-1=0

What are the solutions? You can use completing the square or the

quadratic formula or maybe you don't remember these or haven't learned

them. We can attack the problem empirically at first: We know from

the sketch that B is bigger than A, but not _that_ much bigger. So

let's assume B is double the length of A: 2^2-2-1 = 4-3=1. Too big

-- we need the result to be 0. We know B=1 is too small: 1-1-1=-1.

So the answer we're looking for is somewhere between 2 and 1. Let's

play around at the command line for a sec.

Booting Python here (command window projected on screen):

Python 2.1 (#15, Apr 16 2001, 18:25:49) [MSC 32 bit (Intel)]

on win32

Type "copyright", "credits" or "license" for more information.

IDLE 0.8 -- press F1 for help

>>> from mathobjects import *

>>> p = Poly([1,-1,-1])

We're building a polynomial. Remember from your ancient greek (what

little we teach) that 'poly' means many. That's one of those roots

it pays to remember. Shows up so often: polyhedron, polymorphic,

polygamous, polygon, polymath, polywannacracker (bad joke)...

Polynomials, you may remember, are made up of monomials, which are

coefficents multiplied by your variable (often x or t, by convention)

to some power. The highest power in play is the degree of your

polynomial. B^2 - B - 1 is an example (variable is B). Here we

switch to x -- doesn't matter.

>>> p

x**2 - x - 1

Now we pass in values for our x. Behind the scenes, Python is

evaluating the string 'x**2 - x - 1' and using whatever argument

we pass for x. ** means "to the power of" i.e. x**2 = x to the

2nd power or x*x.

>>> p(2)

1

>>> p(1)

-1

...the examples we used.

>>> p(1.5)

-0.25

>>> p(1.6)

-0.039999999999999591

>>> p(1.7)

0.18999999999999972

>>> p(1.65)

0.072499999999999787

>>> p(1.63)

0.026899999999999924

>>> p(1.61)

-0.017899999999999805

We're narrowing the search for B, the solution to the polynomial,

simply by going higher and lower, getting better and better

approximations. We now know that B is in the neighborhood of 1.61.

That's what you need to scale A by to get B, or scale B by to get

(A+B) -- approximately.

Do you know the quadratic formula? We can solve any polynomial of

the 2nd degree simply by substituting as follows:

Given: a*x**2 + b*x + c = 0

x1 = (a**2 + (b**2 - 4*a*c)**0.5)/2*a

x2 = (a**2 - (b**2 - 4*a*c)**0.5)/2*a

A 2nd degree polynomial has 2 solutions, although they might not

be real numbers, and certainly they don't have to be positive

numbers. But we're looking for a positive, real number, i.e.

something that makes sense as "the length of B" (remember A is 1,

so A*scalefactor = B, and A=1, so B=scalefactor, numerically

speaking).

Let's write this as a function and pass the coefficients as

arguments:

>>> def quadequa(a,b,c):

x1 = (a**2 + (b**2 - 4*a*c)**0.5)/2*a

x2 = (a**2 - (b**2 - 4*a*c)**0.5)/2*a

return [x1,x2]

I'm entering this function directly at the command line, without

going into edit mode. Guido's IDLE lets me do this. quadequa is

now a part of my working namespace...

>>> quadequa(1,-1,-1)

[1.6180339887498949, -0.6180339887498949]

I've entered my coefficients, and here we have a positive value for

the variable. By the way, don't confuse b the coefficient of the

2nd monomial with B, what we used as a variable in reference to

segment B. B is our x, our unknown, our variable, in the polynomial

B^2 - B - 1 = 0. b was our coefficient in the 2nd term, i.e. -1.

We derive the quadratic equation using algebra and I've included that

in a hand-out. We can go over that later. But we don't want to lose

the thread here: the importance of certain ratios in architecture,

in design science, and the fact that geometry has everything to do

with these fields, as a discipline with a long history.

We also want to simplify the solution for the scalefactor

algebraically. (a**2 + (b**2 - 4*a*c)**0.5)/2*a in more traditional

math notation would be (1 + SQRT(1 + 4))/2 once you do the

substitution. And that simplifies further to (1 + SQRT(5))/2.

We can check that:

>>> import math

>>> (1+math.sqrt(5))/2

1.6180339887498949

Yep, same answer. So our scale factor is (1+sqrt(5))/2. That's the

golden ratio, the divine proportion, or phi. Given how it goes up

and down the size spectrum, ever smaller or larger, it's kind of like

a fractal. The self-similarity of A:B shows up at every level.

You might even say "Phi is the Phirst Phractal" (see the handout for

how I do the spelling -- haha, pretty silly, but memorable).

The golden rectangle is a rectangle of dimensions A X B, i.e. B is

1.6180339887498949 times longer than A, making for a taller or wider

rectangle that you find all over the place in architecture. Of course

our measurements, when working in inches or meters or whatever, don't

need that kind of precision, so the architect will typically just use

1.618 -- close enough for folk music. This has been going on for

centuries. People even find phi in the proportions of the human body

itself -- evidence, to some, of a divine creator (the so-called

sacred geometers imbued their math with a lot of philosophy and

metaphysics -- didn't separate all these aspects of thought into

walled-off-from-each-other compartments, the way we do today).

Note that if we say the area of this rectangle is 1.618, and then

scale all the edges by 3, the area goes up by a factor of 9, i.e.

(3**2)*1.618 = 14.562.

The golden cuboid is like a brick, and has edges of 1/phi, 1 and

phi. Or .618, 1 and 1.618. 1/phi has the pleasing property of

being phi-1. Indeed (phi-1)=1/phi is restatement of the basic

algebra, i.e. may be transformed into our polynomial: B^2-B-1=0.

B is phi here, since A was 1.

If you halve the edges of the golden cuboid, all the angles stay

the same. The *shape* is unchanged because *shape* relates to

angles, and these are constant through a scaling operation. But

the volume is affected. If it started at 1*phi*(1/phi)=1 cube

(same as a cube with edges 1), it will now be (1/2)**3 or 1/8 of

the same unit cube -- because volume changes as a 3rd power of

the change in linear dimensions (this is part of the 7th grade

standard in California in 2001, so you better know this if you

want to pass out of the 7th grade in California).

Note also that sqrt(5) is irrational, i.e. the algorithm behind it

will keep giving you decimal digits until you stop it. So we're

dealing with an infinite loop, a loop with no defined break point.

That's what an irrational number is -- an algorithm with no terminus.

So yes, phi is one of those. It's not transcendental though, because

it shows up as a solution to a polynomial with rational coefficients,

as we've just seen -- transcendentals don't do that.

In our talk tomorrow, we'll go into how polyhedra, not just line

segments, may be ratioed. This has a lot to do with how they can

be nested, or fit inside one another, in a pleasing, memorable way.

This kind or ratio-ing, or relationship formation, has also been very

important to geometers through the ages, and is a good place to pick

up as we continue to explore this discipline.

Kirby

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