Teaching the Golden Ratio
How many students are there today? Yes, an even number. If nobody else shows up late, this should work without my having to go to the board.
Today weíre going to have a beauty contest. A mathematical one. The most beautiful person will be determined by a ratio: the ratio of the height of their belly button to their total height. To have the contest, you need to group yourselves into pairs. Each pair goes to the board with a meter stick. Each of you measures your partner. Mark on the board the height of your partnerís belly button, and then the top of your partnerís head. Then measure both heights and write those measurements by the marks. Finally, calculate the ratio of the height of the belly button to the total height, and write that ratio, along with your partnerís initials, on the board.
The usual confusion. Help them out as needed. Repeat the instructions when asked. Most of us wonít remember that many instructions, especially if we donít understand the purpose very well.
Do we measure in inches or centimeters?
Aha. Thereís my favorite question! And it comes from Scott, who doesnít say much. Someone usually asks the question if the meter sticks are also marked in inches. I must keep up the investigative approach to encourage curiosity.
Hmm. Good question, Scott. Why donít you two try it both ways and see what happens?
More confusion. More repetition of instructions. Thatís OK. Another question of which units to use. Same answer. Even though the questions sound the same, they arenít the same because the second person wasnít ready to hear the first personís question or my answer. Even if they could have heard it over the noise. Only a few stragglers left, including Scott and Idile. But theyíre almost finished. There they go.
How should we judge these? Which is the best? Which is most beautiful?
Glad they know me well enough by now to know that Iím kidding about there being a best in some absolute way. But theyíll go along with the game.
I think Tiffany is the best.
Well, Ryan would. He isnít afraid to show he likes her. Iíll take it seriously, though.
Well, sheís just the best.
Does everyone agree? Yeah, you donít want to insult her. David?
I donít know.
Good. If you knew, thereíd be nothing to learn. Becky?
You keep saying we should make a mathematical model. But I donít see how to do that here.
You mean sheís been paying attention? Ah, Andrew wants to say something.
Maybe just represent the total body with a straight line, and the belly button as a point. Whatís the best way to divide a straight line by a point?
Should I criticize Andrewís language?
You mean a line segment.
I donít need to insist on careful terminology as long as Jeff is around! Wish heíd say more, though.
Nice point, Jeff. Do you have an idea about the most beautiful way to divide a line segment into two pieces?
How about in half? Then Sara would be the best, because her ratio is closest to .5.
Not very close, though.
Glad theyíre talking to each other without waiting for me to mediate. Weíve come a long way since the first day we met! I need to make sure Matt contributes, though.
Does that make sense to you, Matt?
Sort of. I like halves.
Why do you think we all like halves so much?
Fairness through equality.
Laughter. Theyíre kidding me, but theyíve been listening.
But the numbers here look more like 2/3. And if you divide the segment into 2/3 and 1/3, then the shorter piece is half the longer one, so you still have fairness through equality.
That was Jason. Nice point.
Thanks, Jason. Other ideas? Jennifer?
I like halves.
I like the 2/3 idea.
And I like controversy! Great!
They still laugh at that. Controversy does indeed get them engaged. And theyíve heard before from me that much mathematics grows out of controversy, even if itís conflict within one personís intuition about whether or not a conjecture is true.
So letís vote. How many prefer 1/2? How many prefer 2/3?
Who hasnít contributed? Maybe Katie.
Which was it you preferred, Katie? 1/2? Why?
Fairness through equality.
I like 2/3 because itís a little more elegant. I mean, you get at the 1/2 in a more subtle way.
Again using my words from earlier.
Thanks. Did you vote, Gina?
No. I donít think the ratios are very close to 2/3 either.
You mean you think both models are flawed? We need a better definition of beauty?
What if instead of the 1/3 being half of the 2/3, it was 2/3 of it?
Is Julie getting at what I think she is? How can I get her to use a variable to describe her idea?
Are you asking what if 1/3 were 2/3 of 2/3, Julie?
But itís not.
I agree, Tiffany. But letís see if Julie can elaborate.
Well, of course 2/3 of 2/3 is 4/9, not 1/3. But what if it were 1/3?
Liz seems to be listening carefully, and sheís usually pretty articulate.
Do you understand what Julie is saying, Liz?
I think so. Like, if both pieces are 1/2, their ratio is 1. The ratio 1 is bigger than 1/2. If one piece is 1/3 and the other is 2/3, their ratio is 1/2. The ratio 1/2 is less than 2/3. Can you adjust it so that the ratio is the same as the longer piece?
Pretty good. Becky seems to understand, and she hasnít said anything for awhile.
Does that make sense to you, Becky?
Yeah. You want the ratio of the smaller to the larger to be the same as the larger.
Is that what you had in mind, Julie?
I think so.
Who was it who mentioned elegance earlier? Eric.
What do you think of that, Eric?
Itís pretty subtle. I like it.
How about you, Jennifer
I still like halves. Theyíre simpler.
Yeah, but the data arenít very close to 1/2.
Aha! Ryan is still with us. And using the word ďdataĒ in the plural! Wow!
So do you like this new idea, David?
What number between 0 and 1 seems to satisfy the condition?
Well, 1/2 is too small, and 2/3 is too big. Somewhere in between.
How could you find it?
Lots of blank looks.
Try out lots of possibilities, maybe.
Yes, Tiffany, that approach would help you close in on it and would probably give everyone a better feeling for whatís going on.
Other ideas? Scott?
After doing that kind of thing lots of times already, and then finding a shortcut with variables, why is nobody yet thinking of using a variable this time?
That sounds good to me.
Well, the idea of ratio is one of the most difficult in mathematics. So I suppose it wouldnít hurt to take a little more time on this than Iíd planned. The students should work in groups. How big? Thereís not much to do, so Iíll use pairs again.
OK. Why donít you work with your neighbor to home in on a number. When you get something youíre happy with, please put it on the board.
The usual hubbub. Several are going right to it. Others are having to explain what to look for to each other. Some are lost completely. Helping them out reminds me of how subtle the problem is, and how difficult ratios are, even for students at this level. Most of the pairs have put something on the board now. And most of their results are close to .38. Only two of them are closer to the ratios of heights.
Thank you all. What do you see?
Most of them are way off, but a couple are pretty close to the data we got.
Nice. Hadnít heard Matt use the word ďdataĒ before.
Who got the .618?
Nice to see Tiffany taking the lead here
How did you get it?
Itís just 1 minus what we got, isnít it?
I like it when they reduce my role to initiating the discussion and making sure everyone participates. It establishes a habit of discussing mathematics.
Yeah. I bet itís the length of the longer piece, and we got the length of the shorter one.
Well, yeah. I suppose so.
But itís also the ratio of the lengths of the shorter to the longer. Thatís what we were looking for, isnít it?
So by that number, Jeff is the most beautiful.
Of course, theyíre very interested in who won. As frequently happens, the winner is a tall, lanky guy. Fortunately, Jeff has a good sense of humor. Heís standing and bowing. Theyíre making the usual jokes.
Congratulations, Jeff. Or does it make sense to congratulate someone on success they didnít really do anything to earn? Like winning a door prize?
Anyway, as Iíve said once or twice before, a mathematical investigation isnít over when we come up with an answer. What are some questions we could ask?
What if not?
Is there a better way?
Under what conditions....?
Three of the most popular that weíve worked with in this course.
Good. Let me pick up on the second one first. A number of times this term weíve used successive approximations like this to come up with an approximate answer. But several of those times weíve found a better way and gotten an exact answer. Can we do that here?
You mean by using a variable to represent the unknown?
Yes, Idile is thinking.
Like, let x be the length of the shorter piece?
Donít we really want the length of the longer piece? Itís the same as the ratio we want.
Sure. Let x represent the length of the longer piece. Then what? Whatís the length of the shorter one?
1 - x.
How do you know the total length is 1?
That discussion was going so fast I donít even know who was saying what. But theyíve finally begun examining the assumption theyíve slid into. Now itís pretty quiet. Should I say something? I think Iíll count 40 seconds first. Almost always someone speaks up before I get there, if only to fill the silence.
When weíve been talking about 1/2 and 2/3 and 1/3 and all that, we were assuming that the length was 1.
Actually, the 1/2 and 2/3 were ratios of the whole body, the whole line segment.
So we were taking ratios of ratios? Weird!
You might as well assume that the whole is 1. It probably makes calculations easier.
Another lull in the conversation. I donít see any puzzled looks. They seem to think theyíve solved the problem. They probably have as deep an understanding as I can expect at this point. Some of them will be confused again about it later. Weíre running low on time, so Iíd better redirect their attention to the problem, maybe start writing on the board.
So if you assume that the length of the line segment is 1, how can you use a variable to represent an unknown? Did you say this: Let x represent the length of the longer piece. Then 1 - x is the length of the shorter one. How can we represent the ratio bit?
x over 1 - x equals x.
Should I correct her? Investigators donít have someone telling them whether theyíre right or wrong, but they need to learn ways of checking.
Thank you, Krissy. What do you think, Biana?
Can you say it in different words, Biana?
The ratio of the, uh, longer to the shorter equals the longer.
But thatís not right. Itís the other way around. 1-x over x equals x. Itís the ratio of the shorter to the longer equals the longer.
Do you see what Becky is saying, Biana?
So which do you want?
I think thatís right. The shorter over the longer is the longer.
Sheís puzzled. Watch the wheels turn. Will it click? Yes!
Yeah. I get it.
Then letís go with (1 - x)/x = x. Can we solve for x?
Well, 1 - x = x^2.
OK. So can we solve for x?
Itís a quadratic equation. x^2 + x - 1 = 0. So x = (-1 Ī sqrt(5))/2.
We donít always get to this so quickly, even in college calculus classes. Probably some of these students wouldnít have come up with it themselves. But theyíre following, and weíre short of time, so I wonít draw it out.
Thanks, Scott. Which value do we want?
The one with positive sqrt(5), since otherwise weíd have a negative result.
And that checks on a calculator.
So can we summarize our result, David?
The best way to divide a line segment is into two pieces so that the longer one has length
(-1 + sqrt(5))/2.
Thatís if the line segment has length 1.
So actually (-1 + sqrt(5))/2 is the ratio between the longer piece and the whole segment? And also the ratio between the shorter piece and the longer one?
They seem to be with me. Lots of nods. Time for some history.
Actually, the problem goes back a long way in history, at least to the artist Leonardo da Vinci. That ratio is sometimes called the Golden Ratio. More often, though, its reciprocal is called the Golden Ratio: the ratio of the longer to the shorter. Strangely enough, the reciprocal of .618 is 1.618. That is, the reciprocal of the smaller number is one more than the smaller number. The Golden Ratio has other interesting properties as well. Moreover, the Golden Ratio shows up in nature and in lots of art and architecture and music, sometimes though not always intentionally. And itís associated with Fibonacci numbers, which you have seen elsewhere.
Not only does the Golden Ratio connect mathematics to beauty in the arts. The thinking process you just went through, so typical of mathematical research, is an experience thatís similar in many ways to the creation of a work of art.
Anyway, why do you think dividing the line segment according to the Golden Ratio might be considered more beautiful than dividing it in other ways, such as in half? Jeff?
Because itís much more subtle. Even more than fairness through equality.
Do you agree, Kate?
It also gives a better mathematical model for beauty in humans. If Jeff can be considered beautiful, which I doubt.
We canít stop with answers, though.
What other questions can we ask?
What if not? What if not one dimension? Like, can you divide up a square or cube in some way?
Or, what if not two pieces? What if weíre dividing a line segment into three pieces?
Under what conditions is the Golden Rule useful? I suppose youíve started to answer that one.
You know, thereís one thing I donít understand. When we were measuring belly buttons to start with, you suggested that we measure in inches as well as centimeters. We got the same ratio both ways. Why?
Yes, ďWhy?Ē is a great question opener for investigations. Any ideas?
Thatís my favorite question because none of the answers I know satisfies me at this time. So I donít expect many ideas from them yet. We may come back to some of these questions if we have time later in the course. Or they may pursue them on their own.
Talk over these questions with your friends at lunch, and let us know next class of any ideas you have.
Not my friends!
And if your friends donít want to talk about strange ideas like the Golden Ratio, well, get some new friends!