WittSem 100L: Patterns in Nature
Assignment 3, due Tuesday Sept 12 at the beginning of class
 

0. Not to be handed in, but reading for Tuesday:

A Fearful Symmetry, Stewart & Golubitsky, pp. 1-17 (stop where marked)

You should be prepared to discuss:

            The difference between pattern and symmetry

            What mathematicians mean by a “transformation”

            What the point of the Plato quote is in this context

            What we mean by “symmetry breaking”

            What Curie's Principle is

            Why Curie's Principle is wrong (or at least needs modification)

            What this all has to do with splashes

 

 

The following are to be handed in on Tuesday.
You may work together with anyone in the class, and you may get help from me or the tutors in the Math Workshop, but you should write up your answers in your own words and with your own understanding (don’t simply quote from the reading or copy someone else’s answers). Your answers must be written up neatly and readably on separate sheet(s) of paper, and should be correct, complete, and understandable. Word processing is fine, but not required (unless your handwriting is unreadable). 
 

1. As we've seen, the Fibonacci sequence starts with 0 and 1 and then develops the next value by adding the two previous values. This gives

    0 1 1 2 3 5 8 13 ...

Using Excel, we saw that if we take the ratio of each number to the one preceding it (Fn+1/Fn), the ratios approach the Golden Ratio (Phi, or Φ, 1.618025751073....) as a limit:

Fibonacci # Fn

Ratio Fn+1/Fn

1

 

1

1

2

2.000000000000

3

1.500000000000

5

1.666666666667

8

1.600000000000

13

1.625000000000

21

1.615384615385

34

1.619047619048

55

1.617647058824

89

1.618181818182

 
And so on....
Now, what happens if instead we take the ratio of each number with the one following it (Fn/Fn+1)? That is, 1/1, ½, 2/3, ….
Do these ratios approach a value? What (to at least 4 decimal places) is the limiting value of this series? You may do this in Excel or with a calculator. Write out enough results (as above) to show how the limit is approached.
Bonus: do you see anything interesting about this number?
 

2. The 19th century French mathematician, Edouard Lucas, while investigating Fibonacci number patterns, found (or invented) a similar series called the Lucas series: 1, 3, 4, 7, 11, ...

a) What are the next two numbers in the Lucas series?

b) Researchers have found that Fibonacci spirals aren't the only patterns that can emerge in plants. Some plants produce leaves/needles/seeds such that the numbers of spirals in each direction are not Fibonacci numbers, but the closely related Lucas numbers. Let’s investigate how this finding relates to what we found about the “best” way to arrange leaves and seeds so that there isn't very much overlap or wasted space. Find the ratio of each number in the Lucas sequence with the one before it, either using Excel or with a calculator (3/1, 4/3, …)—you can stop when the pattern is clear. What value do these ratios approach as you go farther down the series? Show your work and reasoning.

 

3. Livio writes that Phi can be expressed as a continued fraction composed entirely of 1's—that is, as

 

 

a) To evaluate this fraction, we can calculate it using successively better and better approximations:

1 = 1

 

 

 

 

Continue for two more approximations. What's the pattern here (hint: think Fibonacci!)?

b) Why are we justified in saying that if the fraction were continued to infinity, it would approach Phi = 1.618025751073.... ? (hint: think about your answer to the first question)

 

4. Choose one of the following two questions to answer (do either 4A or 4B). (You can do both and receive a bonus.)

A) We've seen that the ratio of two successive Fibonacci numbers gets closer and closer to Phi as we go farther down the sequence. Here's how we can prove that Phi is actually the number that is the limit of these ratios.

We represent the nth value of the Fibonacci sequence by Fn. Then the next two values are Fn+1 and Fn+2. Assume that we're far enough along in the sequence that Fn+1/Fn and Fn+2/Fn+1 are very close to each other (in other words, the ratio is changing very little). So, we can say

Fn+1/Fn=x

and

Fn+2/Fn+1=x.

We also know that Fn+2=Fn+Fn+1 (by definition of the Fibonacci sequence).

Combine these equations to come up with the famous quadratic equation that has Phi as a root:

x2-x-1=0. (Show your work.)

 

OR

 

B) As we've seen, the Fibonacci sequence starts with 0 and 1 and then develops the next value by adding the two previous values. This gives

    0 1 1 2 3 5 8 13 ...

We can continue the Fibonacci sequence backwards by finding the number that, added to the first number in the sequence, gives the second number.

That is, x + 0 = 1 gives the number x=1, so the number before 0 in the sequence is 1.

So the sequence, extended backward, is

  1 0 1 1 2 3 5 8 13 ...

Continuing one place backwards, the "next" number satisfies x + 1 = 0, so x = -1.

Then the sequence is

  -1 1 0 1 1 2 3 5 8 13 ...

The next step is x - 1 = 1, so x = 2. The sequence becomes

  2 -1 1 0 1 1 2 3 5 8 13 ...

Continue this to find the "next" three numbers going backwards. You may do this by hand, with a calculator, or using Excel. Regardless of how you do this, explain how you found the values (like I did above for the first two numbers). (Doing something like continuing a sequence backwards is a common thing to do in math, where we might be interested to see whether patterns persist if we push a rule into new territory.)

Then, take the ratio of pairs of numbers in this “going backwards” series. The first ratio is

1/(-1)=-1, then the next one is -1/2 = -0.5. Find the next three ratios. What number is this ratio approaching? (Hint: when we solved the quadratic equation x2-x-1=0 for phi, we found another root--what was that root?)