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

A Short Talk on GeometryA key thing to understand about any discipline is how people have used it over time.  Geometry has everythingto do with navigation, getting from here to there safely,especially by sea.  But it also have everything to do witharchitecture.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 insize.  Do you know that if you have a triangle, squareor any shape, and double all of it's edges, what happensto its area?  How about if you have a pyramid, and blowit up (in the photography sense -- to make bigger, notto 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 volumeblah, and you double all its edges, it now has volume8*blah, where 8 is 2 times 2 times 2 or 2 to the thirdpower.  That's important to keep in mind, and will helpyou at many points in the future.Anyway, one of the key ratios that people have found fascinating is called the divine proportion or the goldenratio or phi.  You can tell just from the name that it'simportant.  So what is it?  First, it's important to keepin mind that it's a relationship.  It doesn't matter howlong segment A is, only how long it is in relationshipto segment B.  So A might be a mile, an inch, a millimeteror 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 thatis in divine proportion to it, in the same way as A relates toB.  And likewise (A+B) will relate to a yet longer segment by the same scale factor.  And what is that scale factor?  We canuse algebra:    A/B = B/(A+B)    cross-multiply:  A(A+B) = B^2Let's just have A=1 for simplicity, and find B (remember, it makesno difference what A is by itself -- it's the relationship to B that matters):   1(1+B) = B^2 or B^2-B-1=0What are the solutions?  You can use completing the square or thequadratic formula or maybe you don't remember these or haven't learnedthem.  We can attack the problem empirically at first:  We know fromthe sketch that B is bigger than A, but not _that_ much bigger.  Solet'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'splay 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 (whatlittle we teach) that 'poly' means many.  That's one of those rootsit 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 arecoefficents 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 - 1Now we pass in values for our x.  Behind the scenes, Python isevaluating the string 'x**2 - x - 1' and using whatever argumentwe pass for x.  ** means "to the power of" i.e. x**2 = x to the2nd 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)   -0.017899999999999805We'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*aA 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, withoutgoing 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 forthe 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 polynomialB^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 thatin a hand-out.  We can go over that later.  But we don't want to losethe 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 traditionalmath 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  1.6180339887498949Yep, same answer.  So our scale factor is (1+sqrt(5))/2.  That's thegolden ratio, the divine proportion, or phi.  Given how it goes up and down the size spectrum, ever smaller or larger, it's kind of likea fractal.  The self-similarity of A:B shows up at every level.  You might even say "Phi is the Phirst Phractal" (see the handout forhow I do the spelling -- haha, pretty silly, but memorable).The golden rectangle is a rectangle of dimensions A X B, i.e. B is1.6180339887498949 times longer than A, making for a taller or widerrectangle that you find all over the place in architecture.  Of courseour measurements, when working in inches or meters or whatever, don'tneed that kind of precision, so the architect will typically just use1.618 -- close enough for folk music.  This has been going on for centuries.  People even find phi in the proportions of the human bodyitself -- 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 intowalled-off-from-each-other compartments, the way we do today).Note that if we say the area of this rectangle is 1.618, and thenscale 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 andphi.  Or .618, 1 and 1.618.  1/phi has the pleasing property ofbeing 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 staythe same.  The *shape* is unchanged because *shape* relates to angles, and these are constant through a scaling operation.  Butthe 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 ofthe 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'redealing 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, becauseit 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 canbe 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.KirbyTo unsubscribe from this group, send an email to:math-learn-unsubscribe@yahoogroups.com Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
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