Win Corduan

PAGE 2: PHI IN GEOMETRY

Euclid's Golden Triangle

Perhaps you feel that I’m spending too much time and bandwidth on more or less relevant peripheral matters. But, you see, I’m trying to provide a background, as limited as it will be at that, which allows us to do more than simply gush at the remarkable presence of phi in nature. I guess one could view my effort here as trying to counter the attitude of "I don't know anything about math, but I know that I like phi." I teased you at the outset of this series with a clone of the famous Fibonacci series, and I will return to it. However, right now I’m trying to disconnect phi from the Fibonacci numbers as much as possible. The Fibonacci series is a remarkable function, and it is true that it converges closer and closer to phi the further you compute it. However, phi is not directly derived from the Fibonacci function—never has been and never will be. But, as I said, don’t worry; we’ll come back to it.

To the best of my knowledge, the first written record concerning the ratio that we now express with the number phi stems from Euclid of Alexandria (ca. 300 BC) in his best-seller The Elements, a book that stood for several millennia as the final authority in geometry. (Please also note that this book predated Fibonacci's Liber Abaci by about 1,500 years.) I will attempt to describe (please note—not prove!) Euclid’s discovery of phi, though taking a slightly different sequence of steps. (I’m following pretty closely on the heels of Mario Livio, The Golden Ratio, pp. 78-82.) But, as alluded to above, please make sure you realize that Euclid was not writing about a number, but about the proportions of lines in certain geometric figures expressed in words.

1. The Golden Triangle.

Let us examine some features of a triangle, which is sometimes called the “Golden Triangle.” Eventually I will tell you how we can derive such a triangle from another geometric form, but for the moment we can just assume that it was not invented strictly for its own sake. It is a somewhat pointy triangle (“acute”) and its two longer sides are of equal length (“isosceles"). We’ll label the corners A, B, and C and call the three sides a, b, and c respectively. Let's also label the three angles α, β, and γ.

My readers undoubtedly know that one feature of any (Euclidean) triangle is that its three angles add up to 180 degrees. The Golden Triangle’s base angles measure 72 degrees each, which leaves 36 degrees for the top angle. You see a picture of it on the right copied from Wolfram Alpha and modified--or wantonly bedazzled--by me. (By the way, the basic version of the program is free to use on their website. Chances are that, once you've used the free version, you'll want to get the "Pro" edition, and I must honestly say, that I find it indispensable when I start to toy around with math.)

We could print out this picture, measure the sides of the triangle and, perhaps, come to some interesting conclusions. However, proofs in math or geometry based on physical measurements don’t count for much. Standards of measurement are invented by people, and you’re going to get different numbers if you use, say, centimeters rather than inches. Ideally, the relationships, such as the ratio of one measured line to another will be the same regardless of the calibration of our physical rulers. For example, 1 inch divided by 2 inches and 2 ½ cm divided by 5 centimeters both come out to ½, but the measurements on which this ratio is based will still depend on the accuracy of our instruments, which is always limited, and so we couldn't really be certain that the ratio we obtained is correct. By contrast, if ever in the course of carrying out some little household carpentry project, I wound up within 1/100 of an inch precision for all of my cuts, it would be a miracle. But in math the difference between 2.49, 2.50, and 2.51 can be crucial. There is no "level of tolerance" when it comes to pure calculations independent of any measurements.

However, we can resort to a mental ruler, whose fundamental unit is completely exact for our purposes, and on which we can all agree, regardless of what the usual units of length in our daily lives may be. We can posit that 1 mental unit equals exactly the length of b, the base of this triangle without taking account of its size when we drew it or even whether we drew it correctly. Obviously then, we should not be surprised by the fact that the length of b is 1 since we made it that way. Therefore, by assumption,

b = 1.

Having done that much, we still do not know the lengths of the two other sides, which are, of course, equal and, for the moment at least, we can only represent them with a variable, calling on the ever-prepared “x” to do its usual duty.⁴ Thus,

a = x and c = x.

Thus, so far this is what we know about our triangle.

2. The First Proportion: a to b

Let us now figure out the proportion of one of the sides (we’ll pick a) to the base, b. Given the convenience of our mental ruler, the proportion ^a⁄_b is equal to ^x⁄₁.

^a⁄_{b =} ^x⁄₁

Normally we would remove the useless 1 from this expression, since there is no difference between ^x⁄₁and x, but let’s allow it to stand in the redundant form, ^x⁄₁ for the moment because it will illustrate the ensuing point a little more clearly.

3. Opening Up the Triangle to Create Line g.

Now we will open up our triangle.

We shall disconnect line a at point C and, using point A as a hinge, swing it downward until it has become a continuation of the horizontal base, b.

We’ll call its new point of origin on the left D. We’ll also disconnect line b at point B so that we are now only concerned with a single line instead of a triangle.

This line consists of two segments, line a (extending from point D to A) plus line b (between points A and B). We can also think of the new line as an entirety running from points D to B, and, rather than erasing the segment divider in the union of a and b, I will just draw this new line afresh and call it g.

4. The Second Proportion: g to a

Now, we’ve already brought up the proportion of a to b, which we expressed as ^x⁄₁.

Since line g is the sum of lines a (value: x) and b (value: 1), the length of g is x + 1. Thus:

g = x + 1

Since the variable we assigned to line a is x, the proportion of line g to line a is

Putting it all together, then, we can say that

5. The Golden Ratio

Now, it turns out that in the golden triangle these two proportions (a to b and g to a) come out as equal, and this is the "golden ratio." Euclid called it "the proportion of the mean to the extreme." Expressed in words, it says that

The proportion of the larger segment (a) to the smaller one (b)

is equal to

the proportion of the entire line (g) to the larger segment (a).

Or:

As soon as we are stating it in its algebraic form, we are doing something that Euclid probably never dreamed of.

6. Working Towards Finding a Value for ϕ

Obviously, it's not possible to do math and geometry without engaging in math and geometry. However, when we come to longish sections in which we deal with large equations, you may leap over them. Click here to skip the equations and go to the results.

Substituting the one variable (x) and the one value (1) we have for a, b, and g into the above equation we get:

Now that it has served its purpose of displaying its position in setting up the ratio, we can drop that silly 1 on the left side, and we get

Let’s eliminate the fraction on the right side by multiplying both sides by x. Then we have:

x² = x + 1.

We can rearrange this equation and get:

x²–x –1 = 0.

This kind of equation is called a "quadratic equation," and there is a standard formula for solving it. We will receive two solutions or "roots," one positive and one negative. (Keep in mind that, even with something as simple as the square root of 4, we get two solutions, even though our immediate intuition may just be to grab for the positive one. The number 4 has two square roots, namely 2 and –2.)

Quadratic equations have the general form:

ax²+bx+c=0

a is the coefficient for x²; if there is none, a 1 is implied;
b is the coefficient for x; again, if there is none, a 1 is implied.
c is called the modular constant; if there is none, then the value for c is 0.

In our case the equation is:

x²+(–x)+ (–1)= 0.

or, more simply

x²–x –1= 0.

and the coefficients are:

a=1, b=–1, and c=–1.

The two solutions (or “roots”) can be found with the formula below. You can solve quadratic equations in other ways, depending on the numbers you are given; this formula gives the "general" solution, regardless of what numbers are involved. It definitely looks intimidating, but that's what sophisticated calculators or programs are for. Note the “±" in the numerator. It is this dual operator that helps us find the two solutions, one positive and one negative.

Plugging in the values, we get

and

Once you go through the two stages of computation for each root (or let Wolfram Alpha do them for you), you arrive at the values for phi, as we shall list them below.

I've already given away that the result of computing the golden ratio is going to result in an irrational number, viz. one with a never-ending string of digits after the decimal points without lapsing into a predictable pattern. If you use the Wolfram Alpha program, it clarifies for you that its printed result represents only a "decimal approximation." It carries it to 52 places, but offers to find a longer string should you happen to need it. Livio gives us the first 2,000 digits. His list starts in the middle of page 81 and ends in the middle of page 82. Still, no matter how overwhelmed we may feel looking at this string, an approximation is what it is. This is ϕ(phi), our friend for some time to come.

Here is Wolfram Alpha's "approximation" for the positive root of the equation.

1.6180339887498948482045868343656381177203091798057628…

I stated a moment ago, that a quadratic equation comes with both a positive and negative root. Now when we look at the negative root, things start to get weirdly fascinating. The number starts with a 0 rather than a 1 before the decimal point, and it is a negative decimal fraction, but the decimal digits are the same as the ones for the positive solution.

–0.6180339887498948482045868343656381177203091798057628…

If you calculate the difference between the positive and negative roots of the equation, viz. find the absolute value between them, it comes out to

2.23606..., which just happens to be the square root of 5 (√5).

While we’re at it, let me also mention that there is a number called the "golden ratio conjugate," which does not really present us with any surprises. It is defined as ϕ–1. The result is anything but startling. After all, any number from which we subtract 1 will be 1 less in magnitude. E.g.1½–1 = ½. Not worth a letter to the editor or a tweet to the world. So,

1.61803... –1 =
0.6180339887498948482045868343656381177203091798057628…

Please don't start to yawn just because I mentioned one thing that happened be clear and obvious. Yes, there's no cause for astonishment in the fact that a number minus 1 will be one less than it was before. But don't miss out on the fact that this number is also the negative root, except that the sign is switched.

A bit of nomenclature. Any number used as the denominator of a fraction in which the numerator is 1 is called the "reciprocal" of the number. For example, the reciprocal of 7 is ¹/₇ , which can also be written as 7^-1. We can calculate reciprocals, of course, and for ¹/₇ the outcome in decimal notation is 0.14285714..., with that exact sequence repeating indefinitely (and thus making it a rational number).

Now let's look at the reciprocal of phi, 1/ϕ, and calculate its value. Here it is, once again availing ourselves of Wolfram's "decimal approximation."

0.6180339887498948482045868343656381177203091798057628…,

Yes, this is the same number as phi's conjugate. Phi minus 1 yields the same number as 1 over phi. No other number behaves that way. Thus:

ϕ–1 = 1/ϕ

Let's do one more thing, namely to square phi: The value of ϕ² is:

2.6180339887498948482045868343656381177203091798057628….

As you can readily see, it's precisely the same as phi, only larger by exactly 1.

So, these things are true of phi and no other number:

The positive root of the equation that computes it equals 1.618034... The negative root of the equation has the identical decimal points as phi. (-–0.618034...). The conjugate of phi is exactly 1 less than phi.(0.618034...). The reciprocal of phi is exactly 1 less than phi, and thus equal to its conjugate (0.618034...). The square of phi is exactly 1 more than phi (2.618034...).

Other numbers have their own peculiarities, which may be just as startling, but the traits that you see here belong to phi alone.

Are you beginning to see now why I’m talking about beauty within the numbers themselves? Why numbers sometimes appear to have personalities, at least for those of us who have a little bit of inclination towards fantasy? That a number can be distinct from all others in more than a trivial sense, and that some are more eccentric (or gifted?) than others? That you don’t have to count generations of rabbits or petals of a rose blossom to find fascinating and startling facts about phi (though it may be fun, and we'll get to that)? The beauty is already there in the number. The rest, which I don't want to minimize even though I'm ignoring it right now, is an additional bonus over and above the properties of the number alone.

Sorry, we're still not ready to talk about Fibonacci and his series of numbers; we need to stay with its geometrical home for a little longer. In this entry we have stipulated the Golden Triangle, knowing in advance how the ratios and numbers would fall out. The question we need to address is whether this geometric figure is anything other than something concocted by an ancient mathematician for the entertainment of his guests on long November evenings. Can we find the Golden Triangle somewhere where it is right in place, playing a significant role in geometry?^4a

Let's go to the Pentagon to find out!

Notes (4) Since we know that both a and c are of a different length than b, we could also use another approach and let x represent the difference between b and either a or c. Then we could say that a and c are of length 1+x, where x could be a negative number or zero, just in case that the "Golden Triangle should turn out to be either obtuse or equilateral. This is not a serious consideration, however, since we already know from the values for the angles that the triangle must be isosceles and acute. So, there's no need to add that complication. Back to the text.

(4a) Let me clarify my question so that I don't create the appearance of asking questions just for their entertainment value. It is not unknown for mathematicians to find numbers (some folks might say "invent" them) in order to make a specific point. For example, an important question in the nineteenth century was whether some numbers were "transcendental," a term refers to a number that cannot be produced by an "algebraic" equation. Algebraic equations are by definition limited to integer coefficients in polynomials and make use only of basic operators such as the four areas of arithmetic (addition, subtraction, multiplication, and division) or finding roots and powers (e.g., square roots or squaring a number, a2). Non-algebraic numbers are called "transcendental" numbers. Joseph Liouville (1809-1882) intentionally created a function (non-algebraic, of course) that would yield a transcendental number. Once he had shown that such numbers could exist, it was just a short step to the recognition of previously known numbers as transcendental, e.g pi and e. Phi is not in that category. Here it is for anyone who may be interested. This particular function is a "continuing fraction." On the next page we will encounter one that directly involves phi, but that's not quite as complicated as this one. Back to the text.

 

The Pentagon and the Golden Triangle

Let me issue an insincere apology to anyone who may have been misled by the term “Pentagon.” I was, of course, not referring to the building that houses the U.S. Department of Defense, and so I sup­­pose I should not have capitalized “pentagon.” Could it be that I was deliberately creating an ambiguity to get people to click on my blog?

Not this Pentagon!

As long as I’m pretending to issue apologies, let me hand out another one together with a rain check for the Fibonacci series, which figures so prominently in discussions on this topic. It has its own formula, Fn = Fn-1 + Fn-2, which is just a compact definition of the Fibonacci numbers and will tell us nothing new once I’ve explained the series. There also is a formula that we can use to determine directly the value of a specific Fibonacci number at any given point in the series, but to do so, we already need to know the value of phi and its negative reciprocal. As I continue to insist, as amazing as the Fibonacci series it, it yields phi only by convergence, not directly. Nothing wrong with convergence, but it’s not what I’m after at this time.

So, instead, let’s look at a pentagon in geometry. It’s easy to draw one with modern software programs; let’s remember, though, that in geometry anything that we draw will always be approximate. On the one hand, that means that you can’t just solve a geometric problem by measuring the lines. On the other hand, we also do not need not go beyond normal human abilities in drawing our figures. If­­­­ they turn out a little lop-sided, that’s okay, they’re only attempts to illustrate something that actually cannot be reproduced by a sketch with a paper & pencil or by a computer program.

So, let us think of a regular pentagon, viz. a two-dimensional figure with five equal sides.

Now let us bisect two side-by-side angles. For convenience, we’ll go with the bottom two in the drawing, those that are located at points C and D. They converge at point A.

Surprise! We have found the triangle ADC, and it turns out to be a golden triangle. (I’m skipping how that fact is derived from calculating the angles involved. See Livio, pp. 78-79 if you’d like to read more on that topic.)⁵

which comes out to phi on both sides of the “equals” sign. But that’s not all. Let’s take one of the triangle’s base angles, bisect it and run a straight line to the edge of the new triangle. We get a new triangle, DGC, and it, too, has the coveted “golden” proportions, as indicated by the number phi.

 

No reason to stop there. Go ahead and bisect another base angle of the latest triangle, and welcome another golden triangle (GHC) into the family.

We could go on and continue the process, but let’s go no further and celebrate our new discoveries. 1) We have found that a golden triangle is not just something created ad hoc for the sake of accommodating phi, but it is a direct property of a regular pentagon. 2) A golden triangle gives rise to further, but smaller, golden triangles each time one draws a straight line bisecting one of the base angles.

It’s not just the golden triangle that has the property of reproducing similar smaller versions of itself. Next time, we’ll look at the golden rectangle and take our first outing into the physical world by considering the chambered nautilus.

Notes

(5) If we were to continue bisecting all of the angles of the pentagon, the result would be a pentagram, which had a respectable existence for a long time prior to its appropriation by followers of the occult. After all, there are fifty of them on the American flag. Perhaps more interesting is the fact that the inside of this pentagon is constituted by another pentagon, from which we could derive another pentagram, and so on (theoretically).

Back to the text.

 

The Golden Rectangle

I left off last night by showing that phi manifests itself when you bisect two adjacent angles of a pentagon and, thereby, create a “golden triangle.” Then we were able to give birth to more and more golden triangles of diminishing size by bisecting one of their base angles each time. There is also a process that gives rise to new generations of golden rectangles, as we shall see below.

By now I’m sure you have figured out why this rectangle should be golden: the ratio of the longer side AB to the shorter side AD is the same as the combined sides AB+AD to the larger side AB.

Now, we can lop off a square of length AD from the one of the ends rectangle, and we have a smaller rectangle left. I have placed the square on the left side of the rectangle. There is no rule governing that placement, nor can there be, since one can always flip the figure without doing it any damage. I’m placing my squares so that I can use the ongoing generation of golden rectangles to make a specific point in a short while.  

The remaining rectangle (EBCF) now has the golden proportions. Let’s continue the process and remove another square designated by EBHG, and we have produced yet another golden rectangle, answering to the name of GHFC.

Are we done now? Only if you want to be. We can remove another square and enjoy the sight of golden rectangle GIJF.

And let’s do one more and call it GILK.

And so forth … This is another unique treat that phi brings to us: We can go on and on bringing out golden rectangles by removing squares from one of its side.

Let us now reverse this process by starting out with the smallest golden rectangle and adding squares to it so as to create a newer, larger one, which will yield another golden rectangle by means of the same procedure.

It is at this point that the placement of the square takes on significance. If I were to continue the enlargement procedure indefinitely according to my pattern, my arrangement will give us a spiral. In order to turn the tiniest of our rectangles into the next largest size, we’ll put a square underneath it. To reach the next size, we can place our square to its right. Moving on the next larger one, we can place the square on top of the one we have. Finally, to reach the largest size with which we began, we can expand it by means of a square on the left. Again, there is no point at which we have to stop, except for intrusions into our mathematical world, such as lack of available bandwidth, old age, or boredom. What you see is the beginning of a spiral. If we were to continue the process, the sequence can continue with the pattern of adding squares: down, right, up, left. Each time we get a new rectangle, it’s a golden one, and each one stands in proportion to all of the other by multiples of phi.

Here is a golden spiral that I downloaded from Wolfram Alpha.  I chopped it up and turned it into an animation. I'm saving a discussion of further mathematical properties of the golden spiral for later on in this series.

Continue to Page 3
Return to Page 1