Tuesday, August 28, 2007

Searching for Gold in Geometry - An Investigation

Those of you who found your way here via Carnival of Math Edition XVI, may also want to read some of the more recent posts:
A Preview of an Interview with Lynn Arthur Steen
Algebra 2 End of Course Exam - Latest Info
Singapore Math - Part III - Info from the 'Source'
Another Problem from Singapore Grade 6B Placement Test

If you're interested in more geometry investigations, look through the labels on the sidebar. There are many geometry activities and challenge problems for your students or for your pleasure...

[As promised, scroll down to the end of this post to see some links to the best references I have found for the golden triangle and related topics.]

The following detailed investigation is designed for geometry students. Questions 7 and 8 are designed for students who have had some basic trigonometry, which is sometimes included in a geometry course. You may want to save this for an extended activity or project for later in the year, but if you want to inspire your group earlier, show them what they will be able to solve by the time the course is over! Feel free to comment and make corrections and suggestions. There are many excellent online references for the triangle below. I will provide these links after you've had a chance to explore and to feel the 'Gold Rush'.

The isosceles triangle shown above doesn't seem very special but you may feel differently after the journey we're about to embark on.
Here are the given: ΔPQR is isosceles, PQ = PR. RS bisects ∠PRQ. ∠P = 36 degrees, sides PQ and PR have length x and base QR has length 1.

1. Draw a regular pentagon and all of its diagonals. How many triangles can you find that are similar to ΔPQR?
2. Determine the measures of all the angles in ΔPQR.
3. Explain why PS = SR = QR.
4. Here is the key step: Using similar triangles, show that x satisfies the equation:
x2 = x+1.
5. Solve to show that x = (1+√5)/2.
6. You've now shown that the ratio of the longer side of the isosceles triangle to its shorter side is (1+√5)/2. Note that this number is also the ratio of PS to SQ (why?). You've struck gold! This number is known as the Golden Ratio and is usually denoted by the Greek letter Φ (phi), pronounced 'fye'. Research this number and write five fascinating facts about it.
7. Think you're done? You've only scratched the surface. We will now relate all of this to a trigonometric value of 36°. You recall that cos 30° = √3/2, cos 45° = √2/2 and cos 60° = 1/2. Well, these are not the only angles whose cosine or sine can be expressed in exact form using whole numbers and square roots! Here's your challenge:
Show that cos 36° = Φ/2 using 2 methods:
(a) The Law of Cosines in ΔPSR
(b) Let M be the midpoint of segment PR. Now work with with ΔPSM. Show your results are equivalent.
8. Since we're having fun with trig, let's show that sin 18° = (Φ-1)/2 using 2 methods:
(a) Bisect ∠SRQ and use a triangle.
(b) No geometry - just use a half angle identity. Show that the results are equivalent!
9. If you feel there's much more to be mined here, you're right. Perhaps you could find the sin or cos of other angles that are multiples of 18°! Have fun!!

(a) cos 36
Click on 'One solution" to get more details. Prof. Wilson from UGA has an awesome problem-solving page that is accessible to high school students as well.
(b) Exact trig values
An exceptional site from UK that develops the theory of exact representations of certain trig values using radicals. Worth reading through it. I found this after the fact but I learned a great deal.
(c) Pentagram and the Golden Ratio
Another wonderful site with excellent diagrams and an historical account showing the development of the golden ratio. Once you start reading this, you will not want to stop - bookmark this one!


Eric Jablow said...

One more historical note: the original proof of the existence of irrational numbers was probably related to φ and the pentagram, not with √2.

Dave Marain said...

That's a fascinating fact since most of us assume that √2 was the precursor.
Also, investigating trig functions of halves, quarters, eighths of 18° leads to a sequence of nested radicals similar to an earlier post of mine. Wouldn't it be great if all middle and secondary students get to see the extraordinary interconnectedness and beauty of mathematics like this. I know for sure that some do. I'm hoping these investigations open this world up for many more. I didn't get to see any of this until my undergraduate courses and then only a small glimpse. There are many many links to the golden ratio, there are videos available that many districts order, and I believe NOVA had a show dedicated to the golden ratio and its connection to art and architecture. I wanted to give a slightly different view here. I hope everyone enjoys 'mining.'

mathmom said...

I like this little investigation. Very nice, and accessible on many levels. Next time I visit similar triangles, I'll try to work this in somehow. (Probably won't be this year -- lots of graduates last spring means a very young "middle school" group for me this spring -- mostly 5th and 6th grade ages.)

mathmom said...

btw, could you add a "similar triangles" tag to this post to make it easier for me to find it when I am ready for it?

Dave Marain said...

I updated that tag - thanks!
I'm glad you liked it. Actually these 'little' activities are extraordinarily labor-intensive. I'm developing these literally from the 'ground up'. I have ideas that pop into my head usually triggered by someone else's comment or something I see online. Then I work through the details painstakingly. Although there are many resources online that provide an expository piece about this triangle or that ratio or whatever, the actual development of an activity with methodical questions is arduous. I revise, edit, tear it up, then start over again until I'm satisfied with the end product. Even then, only my readers can truly edit it. Knowing how some of these actually work in a classroom is of great interest to me as these activities are my creation. I'm glad you appreciate them.

mathmom said...

Thanks for the tag! I hope you realize that I in no way intended to belittle your effort by referring to this as a "little" investigation. I have tried making up some activities like this on my own, but usually rely on things I can find that others have already developed. I do appreciate all you put into all of your activities, and have bookmarked many of them to use, when the time is right. As I said, I have a young group this year, so a lot of your stuff may be a little out of reach for now, but I'll have them ready for it in a year or two. :)

Jackie said...


Your works are appreciated. I've been printing these & giving them to one of our honors geometry teachers (technophobe). She loves them!

Dave Marain said...

thanks mathmom and jackie1
I definitely have to work on my 'sensitivity' and 'defensiveness' issues and should know better than to react to anything after 4 PM! Since I get up at 4:30 every morning, there's no question that my brain functions better and I can see things more clearly at that time. If I wait until later in the day, my comments are more obtuse or tend to have an edge to them. That's why I tell my wife to ask me to help with the household chores before the PM hours! After that, I'm a basket case - isn't aging a wonderful process...

Of course, I know, in my lucid moments, that my faithful readers appreciate my efforts, but, every once in a while, I guess I need that validation, being the insecure person that I am. Thanks for being so understanding. This particular post on the golden ratio was particularly challenging, since I could have gone off in so many different directions. Traditionally, mathematicians develop the properties of phi via the regular pentagon and the regular pentagram (formed from the diagonals of the pentagon). I chose to develop it via the triangles which reside in the pentagon, because there's something particularly appealing to me about this triangle and invites the trig connection more naturally. There's actually considerable 'meat' in this activity that allows for much further exploration. The underlying concept of which angles lead to 'exact' representations (using only whole numbers and square roots) in their trig functions is profound stuff and is related to Fermat primes and constructibility. Jonathan's comment on another post regarding the 'coincidence' of the pattern for the sin and cos of 30,45 and 60 actually prompted me to write this post. Thanks, Jonathan!

I will soon post links to some of the best references for the golden triangle, regular pentagons and the golden ratio that I have found which directly relate to this post. I always assume that any of you can google to find these treasures, but the truth is that it takes time and I do appreciate it when someone else has taken that time to identify the 'best of the web' for a particular topic.

Eric Jablow said...


You've spent time on primes in the past, and you're spending time on φ and polynomials now. Why don't you put the two together! Start giving some of the more curious students a glimpse of algebraic number theory and number rings.

1. Introduce the idea of extending the integers and the rationals by adjoining an algebraic irrational:

Z[√2] is the set of numbers you get by taking integers and √2 and adding, subtracting, and multiplying.

1a. Why do you get only numbers of the form a+b√2?

Answer: Just multiply two numbers of that form.

Q(√2) is the set of everything you can get by taking rationals and √2 and doing the basic algebraic operations.

1b. Why do you get only numbers of the form a/b + (c/d)√2?

Answer: use the standard trick of clearing the denominator: 1/(s + t√2) = (s-t√2) / (s^2 - 2t^2).

2. Define rings and fields. This is why we talk about the ring of integers Z and the field of rationals Q.

3. Now, Q(√2) is a field, just as Q is. Q contains Z, the ring of [rational] integers. Great terminology, huh? Is there an idea of a 'ring of integers' inside Q(√2)? Your students may guess it's Z[√2], and they'd be right.

4. But what is an algebraic integer? Change √2 to √5 for a moment. What are the 'integers' in Q(√5)? They will guess Z[√5], but they'd be wrong. It's Z[φ]. So, what is an algebraic integer anyway?

5. What do we know about rationals like 1/2 in Q? Take Z, append 1/2, and see what you can get just by multiplication. You get among other things 1, 1/2, 1/4, 1/8, 1/16, ... 1/1024, ..., 1/65336, ....

You'll never be able to get a result like that of 1a for this set.

6. Now, take Z and append φ. What happens if you take powers of φ? You get 1, φ, φ^2 = 1 + φ, φ^3 = φ + φ^2 = 1 + 2φ, and so on. Every positive integer power of φ will just be an expression like a + bφ with a and b in Z. For the cognoscenti, we say that Z[φ] is a finite-dimensional vector space over Z, but that will be beyond your students.

The important point is that at some point (here, n=2), φ^n is a sum of earlier powers of φ (with integer coefficients). In other words, φ is the root of a polynomial of the form x^n + lower-order terms with integer coefficients, and that leads to the definition

7. An algebraic integer is a number that is a root of a polynomial of the form x^n + ax^(n-1) + ... + k,
with the coefficients integers and the lead coefficient 1.

We call that a monic polynomial, by the way.

8. And now we have an amazing fact. If u and v are algebraic integers, u+v, u-v, and u·v are. The best way to prove this is to take powers of each and use the pigeonhole principle.

9. So, we can define the 'ring of algebraic integers inside Q(θ), where θ is an algebraic irrational.

10. Where we have 'integers', we should have 'primes' and 'composites'. But what is a prime?

11. Is 1 a prime number? We say not. In this circumstance we define units as algebraic integers that divide 1. For example, 1 and -1 are the units in Z. In Z[√2], 1+√2 is a unit, because (1+√2) (-1 + √2) = 1, and both factors are in Z[√2]. Because of this, powers of (1+√2) are also units.

12. What is a prime then? There are actually two related ideas. For Z, they mean the same thing, but for most algebraic integer rings, they don't.

13. In a ring of algebraic integers, a number u is irreducible if

u=v·w implies that v or w is a unit.

14. Similarly, a number u is prime if

u divides v·w implies that u divides v or u divides w.

15. So, find a prime or two in Z[√2]. In Z[√3]. In Z[φ]. It helps if they know about norms.

16. Where do things go wrong? The earliest example comes from Z[√-5]. Consider 6 in that ring. Your students may object to this one, actually; √-5 will confuse them. Tell them that you need to be a bit playful with ideas in math, and not to worry.

6 = 2·3 = (1+√-5) (1-√-5).

The four factors 2, 3, (1+√-5), and (1-√-5) are all irreducible, but not prime. In fact, the fundamental theorem of algebra fails here.

The earliest 'proofs' of portions of the Fermat conjecture depended on the types of primes. The n=3 case depended on Z[ω] with ω a complex cube root of 1. Since Z[ω] has unique factorization, the proof worked. But for large n, these fields lose unique factorization, and the 'proofs' failed.

Dave Marain said...

thanks, Eric!
You concisely summarized 75% of my Abstract Algebra course from 4 decades ago! You are awesome.
Actually, advanced hs students could definitely be able to handle the concepts of algebraic numbers, rings and fields. I've been thinking about writing a series of posts about this, but it would be more expository than the investigations I usually create. However, you're better at this than I am, so maybe we should collaborate! At this point, I cannot show these directly to students at this point as I am now officially out of the classroom. That's what makes writing these so much fun and so challenging.
Eric, please continue to offer your broader and more advanced views of my elementary problems. You also incorporate the history of mathematics, which I feel is inspiring to students. Thanks again!!

jonathan said...

Dave, thanks for this post. I finally got to the comments, so I need to thank Eric as well for the history and the extension. I am teaching the wrong courses this year for using this stuff, but I will hold it for the future. Eric, I've come within inches of beginning your extension... I see now that it would hold together so nicely.

Finally, where can I read more of Eric's history piece?