-- "Perhaps [Madre] didn't know his cane was shortened?" Bob asked.
-- "That's right," Mr. Mason replied.
We're still in the middle of our mystery. Recall that Bob is our shy student who doesn't participate in traditional math lessons, yet is engaged by this logic story. The students continue to ask questions:
-- "Then, when he stood with it he was very upset, but why?" Bob wondered.
-- "Because it didn't fit him. He was too tall for it!" Carol shouted.
-- "So far, so good," the teacher said.
-- "But why would thinking he had grown taller upset him so?" Gary asked the other students.
-- "This time I really got it," Tom yelled. "Madre was a midget."
-- The teacher shouted, "YES! What else?"
At this point the student named Terri believes that she has the complete answer. She begins to tell her story in order for the teacher to confirm it -- but her story doesn't end until page 271.
And so in order to give you, the readers of this blog, one last day to figure out the entire mystery, I'll save all of Terri's explanation until tomorrow. Stay tuned for our thrilling conclusion....
TO BE CONTINUED
Today is Day 30, and so it marks the end of the first hexter and midpoint of the first trimester. This was relevant last year at my middle school, but this year I'll be subbing mostly at high schools, which don't observe hexters or trimesters.
Before we return to Ogilvy's book, there's something I want to mention here. Today Google is celebrating its 19th birthday by bring back some of its classic games. And guess what that means....
The Fischinger player is back!
When Google first posted the Fischinger player, it was during a time when I wasn't blogging, due to my disappointment when I learned I wasn't hired for the new school year. By the time I felt like blogging again, the Fischinger player had been taken down -- yet I continued to write about the player for weeks afterward, during the Pappas music pages.
Now that it's back, let's recall how it works. We can play four different notes -- C, D, E, and G -- in three different octaves. In theory, we should be able to play C, D, E, G in the lowest octave; c, d, e, g in the middle octave, and c', d', e' in the highest octave. But actually, the middle e and g actually play one octave higher as e' and g'. This means that we can play the bass notes C, D, E, G, c, d, and then the treble notes c', d', e', g'. There's a way to change the key from C to any other key, but this tends to mess up the octaves.
Here are some songs that I've been wanting to play on the Fischinger player:
The Fibonacci Song
Here is the Fibonacci sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
Now it's easy to play the first six numbers as notes in a C major scale:
C-C-D-E-G-c
Notice that these are all Fischinger playable notes. But what should we do with the 13? Others who have converted Fibonacci into music often continue the pattern, so 9 is D (in some octave), 10 is E, 11 is F, 12 is G, and 13 is A. But the note A isn't Fischinger playable in any octave.
Well, the number 13 consists of the digits 1 and 3. These are the notes C and E -- and hey, notice that both of these are Fischinger playable! In fact, we can convert the first eleven Fibonacci numbers into sixteen notes -- which is the number of notes that fit on the player:
C-C-D-E-G-c-C-E-D-C-E-F-G-G-c-d
Of these 16 notes, 15 of them are Fischinger playable. The lone exception is that F there. Earlier, I suggested that we can just play the note G there instead of F:
C-C-D-E-G-c-C-E-D-C-E-G-G-G-c-d
This works. But there are still a few changes we can make here. First of all, I've always liked to avoid 89, which takes us past the octave. Instead, we add rests (that is, we play no note at all) between the two G's of 55. This way, it's easier to tell when the player is about to repeat the song:
C-C-D-E-G-c-C-E-D-C-E-G-G-r-G-r
Second, earlier we changed 34 into 35 since F isn't playable. But after I hear the music, I think it actually sounds better if we change 34 into 23. Then the sequence C-E-F-G (34 and its neighbors), which contains four different notes, becomes C-D-E-G. Of course, if this were a real instrument rather than the Fischinger player, we'd keep it as C-E-F-G in accord with Fibonacci:
C-C-D-E-G-c-C-E-D-C-D-E-G-r-G-r
Remember, the #1 rule of music is that it should sound good. And to me, these notes above are the best-sounding version of the Fibonacci song.
Notice that these notes are to be played in the bass octave (which is on top). That is, the top line is C, the second line is D, all the way to c on the fifth line. Normally, we like to play the melody on the top octave and the harmony in the bass, but here the melody is in the bass.
So we must use the high octave for the harmony. I've decided to play the harmony on every fourth note, as if this were 4/4 time and the high note is played at the start of each bar. And hey, why don't we use Fibonacci again and use the first four numbers, 1, 1, 2, 3, to give us the treble line?
c'-r-r-r-c'-r-r-r-d'-r-r-r-e'-r-r-r
Don't forget that the bottom line on the player is e', then above that are d' and c'. And voila -- that's our Fibonacci song:
Bass line (melody): C-C-D-E-G-c-C-E-D-C-D-E-G-r-G-r
Treble (harmony): c' -r -r -r -c' -r -r -r -d' -r -r -r -e' -r -r -r
Body Music: The DNA Song
For the DNA song, let's move (using Modify near the bottom of the screen) to the key of F. The playable notes are now F, G, A, C, and three of these correspond to DNA bases (guanine, adenine, cytosine). The note F corresponds to the last base, thymine, since there is no musical note called T.
To create the song, I decided to choose random numbers 1-5 for the notes of the F major scale. Now the F major scale begins F, G, A, Bb (B-flat), C, but of course Bb isn't playable. So instead, the number 4 corresponds to a rest. Here are the notes I chose:
c'-g'-c'-g'-r-g'-c'-c'-r-g'-g'-f'-g'-r-f'-a'
This time, we put the melody in the highest octave. Notice that Fischinger now plays the c' as the lowest note in the octave and a' as the highest note (as opposed to C major, where the highest note was g' despite the bottom line playing e'). This should at least make the melody lines easier to find, as now a' is the bottom line, and counting up from it gives g', f', and c'.
To form the harmony, we use the DNA rules for base pairs -- G pairs with c', C pairs with g', A pairs with f' (since there's no t'), and F pairs with a':
Bass line (harmony): G-C-G-C-r-C-G-G-r-C-C-A-C-r-A-F
Treble line (melody): c'-g'-c'-g'-r-g'-c'-c' -r-g'-g'-f'-g'-r-f'-a'
Again C plays as the lowest note in the octave, even though the top line is F. To make it easier, we start with G as the second line from the top, and then always keep five blank notes between the melody and harmony (so the harmony is the melody translated up six steps).
This will end up making the last harmony note sound as middle f instead of bass F. We can live with this, or just remember to go to the top line for the last bass note. Alternately, we can just leave the last note off. Then there will be two rests near the end of the song, just as with 55 in the Fibonacci song -- and moreover the last note played in the melody is the tonic F. But then the melody and harmony contain only three of the four DNA bases each. It seems so much better to have each part contain all four DNA bases.
The Unit Rates Song
I wish to play one of my songs from last year on the Fischinger player. Of course, most of the songs contain all seven notes of the scale and so aren't Fischinger playable. A few of the songs I wrote had only six notes -- for example, the Benchmark Tests song had only C, D, E, F, G, A. In practice, I ended up dropping the F when I sang it, to leave the pentatonic scale C, D, E, G, A. But neither G nor A appear in the first two lines, which are thus Fischinger playable (in the key of C):
c'-d'-c'-d'-c'-c'-c'-r-c'-d'-e'-d'-c'-d'-c'-r
For the bass line, let's just play the note C on every fourth beat (beginning with the first note).
But I decided that I want a full song that is Fishinger playable. Today is Day 30, and so I choose the song from Day 29 last year, the Unit Rates song. This isn't the same as the UCLA fight song parody, which contains the entire major scale and so is Fischinger unplayable. (Also, I didn't have a song on Day 30 last year, as Day 30 both last year and this year are Wednesdays, when I had no music break.)
My original version of the song was written in the key of G major. I often used G major since this key was convenient for the guitar, where the three main chords G, C, D are easy to play. So we will use Modify to move to the key of G. Like the Benchmark Tests song, the Unit Rates song was originally written with six notes (here G, A, B, C, D, E). But the last two lines contain only Fischinger playable notes, which I reproduce here:
b'-b'-b'-b'-d'-d'-d'-r-g'-g'-a'-a'-g'-g'-g'-r
Again the note d' is played lower than the b' here. My original version actually had the D note as the highest note. But I like keeping the lower d' here, since the descending major sixth here makes the song sound different from the ordinary.
For our bass line, this time I decided to play G on every second beat, except we change the sixth downbeat to D (when the melody is playing a', as these two notes sound more harmonious).
Bass line (Harmony): G-r-G-r -G-r-G-r -G-r-D-r -G-r-G-r
Treble line (Melody): b'-b'-b'-b'-d'-d'-d'-r-g'-g'-a'-a'-g'-g'-g'-r
Here are the lyrics to the Unit Rate song from last year, in case you want to sing along. Just sing the whole song to the two lines that keep repeating:
UNIT RATES
If you want to find unit rates,
There's one thing you must know.
To find a unit rate,
All you do is divide!
To see if it's proportional,
All you do is divide!
Write it as a fraction,
Reduce it then you're fine.
Graph it at (0, 0),
Then just draw a line.
If you want to find square roots,
There's one thing you must know.
To find an estimate,
4 and below, round down!
To find an estimate,
5 and above, round up!
1 place for tenths, 2 for hundredths,
3 for thousandths, you're fine.
Graph it between two values,
Right on the number line.
Recall that the second verse was for the eighth graders, who were learning about square roots rather than unit rates at the time.
If I had never left my old classroom and were writing a new song this week, the subject of the song might not be about unit rates, but about the playground installed at the new campus. Playworks (which I mentioned last year) helped support the new playground, and there was even a visit from our local pro soccer team, the LA Galaxy. (Is there an LA Galaxy fight song? Of course if there is, it's probably not Fischinger playable anyway.)
There are other musical games available on Google's birthday besides the Fischinger player. One of them allows users to recreate Beethoven's symphonies. Another one is a "theremin," an electronic instrument played by Clara Rockmore. The theremin is set up to play a full G major scale, with no notes left out. (So the original version of the Unit Rates song, as well as many of my other songs, would be theremin playable, if not Fischinger playable.)
Chapter 10 of Stanley Ogilvy's Excursions in Number Theory is called "Continued Fractions." But I've decided to skip to Chapter 11, the final chapter, "Fibonacci Numbers," in honor of the return of the Fischinger player and my Fibonacci song.
By the way, continued fractions are related to two of my favorite topics. One of them is music theory, but only indirectly. Indeed, continued fractions are about rational approximations of irrational numbers, while music theory uses irrational approximations (such as 2^(7/12)) of rational numbers (such as 3/2).
The other is calendar reform. In fact, today is Google's 19th birthday -- or first Metonic birthday -- and continued fractions explain why a Metonic lunisolar cycle has 19 years. In fact, there exists a pure lunar calendar -- the Yerm Calendar -- that was constructed wholly from continued fractions. I have decided, therefore, to postpone our reading Chapter 10 until the end of the year, when I have my usual Calendar Reform posts.
Ogilvy actually opens Chapter 11 with a little more on continued fractions, and so let's just skip that part and go directly to the sequence itself:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, ...
named for the 13th century Italian mathematician Leonardo (da Pisa) Fibonacci. We can see that every number in the sequence equals the sum of the two previous numbers:
F_n = F_(n - 1) + F_(n - 2)
Notice that Ogilvy actually uses the same notation for Fermat numbers and Fibonacci numbers. I've decided to write a roman F for Fibonacci and same the italicized F for Fermat.
Ogilvy begins by finding the sum of the first n Fibonacci numbers. He writes:
F_1 = F_3 - F_2
F_2 = F_4 - F_3
...
F_(n - 1) = F_(n + 1) - F_n
F_n = F_(n + 2) - F_(n + 1)
We add up these equations, and many terms are telescoping, or cancelling out. We are left with the sum as F_(n + 2) - F_2. But F_2 = 1, so the final sum is F_(n + 2) - 1.
The next property to prove is:
(F_(n + 1))^2 = F_n F_(n + 2) + (-1)^n
For example:
n = 6: (F_7)^2 = F_6 F_8 + 1; 13^2 = 8 * 21 + 1
n = 7: (F_8)^2 = F_7 F_9 - 1; 21^2 = 13 * 34 - 1
Ogilvy proves this by induction. The initial case is trivial, for n = 1, we have 1^2 = 1 * 2 - 1. So now we assume it for n = k:
(F_(k + 1))^2 = F_k F_(k + 2) + (-1)^k
Add F_(k + 1)F_(k + 2) to both sides:
(F_(k + 1))^2 + F_(k + 1)F_(k + 2) = F_k F_(k + 2) + F_(k + 1)F_(k + 2) + (-1)^k
Factoring:
F_(k + 1)(F_(k + 1) + F(k + 2)) = F_(k + 2)(F_k + F_(k + 1)) + (-1)^k.
Using the recursive definition of Fibonacci as F_k + F_(k + 1) = F_(k + 2):
F_(k + 1)F_(k + 3) = (F_(k + 2))^2 + (-1)^k
(F_(k + 2))^2 = F_(k + 1)F_(k + 3) - (-1)^k
(F_(k + 2))^2 = F_(k + 1)F_(k + 3) + (-1)^(k + 1)
which proves the n = k + 1 case. QED
By the way, this demonstrates why Fibonacci bases like 21 and 34 tend to have divisibility rules for many different primes. In base F_n, we can show that any prime divisor of F_(n - 2), F_(n - 1), F_n, F_(n + 1), and F_(n + 2) has a divisibility rule of type divisor, alpha, omega, or SPD (based on the square alpha or omega). Of course, the SPD rules in this range are usually impractical.
After telling us that the sum of rising diagonals (22.5 degrees) in Pascal's triangle add up to the Fibonacci numbers, Ogilvy now connects Fibonacci to Geometry. If we have a rectangle of sides consecutive Fibonacci numbers, then we can cut out a square and be left with another rectangle with Fibonacci sides. For example, an 8 * 13 rectangle gives us:
1^2 + 1^2 + 2^2 + 3^2 + 5^2 + 8^2 = 8 * 13.
Generalizing:
(F_1)^2 + (F_2)^2 + (F_3)^2 + ... + (F_n)^2 = F_n F_(n + 1)
Ogilvy continues with Geometry. We wish to find the point x on a segment of unit length such that the longer part is the geometric mean of the unit length and the shorter part:
1/x = x/(1 - x)
x^2 + x - 1 = 0
x = (-1 + sqrt(5))/2 (discarding the negative solution)
Now the ratio of the unit segment to the longer part is:
1/x = 2/(-1 + sqrt(5))
= 2/(-1 + sqrt(5)) * (1 + sqrt(5))/(1 + sqrt(5))
= 2(1 + sqrt(5))/(-1 + 5)
= (1 + sqrt(5))/2
which is Phi, the golden section. Earlier, Ogilvy uses continued fractions to show us that the ratio of consecutive Fibonacci numbers converges to Phi. (as in 1/1, 2/1, 3/2, 5/3, 8/5, and so on).
Indeed, if we have a Golden rectangle of dimensions Phi * 1 and take away the unit square, the remaining rectangle has sides in ratio 1/(Phi - 1). But then we start with the known equation for Phi:
Phi^2 - Phi - 1 = 0
Phi^2 - Phi = 1
Phi(Phi - 1) = 1
Phi = 1/(Phi - 1)
and so the remaining rectangle is also a Golden rectangle.
Ogilvy's final trick is to do the same with a 36-72-72 triangle ABC (vertex angle at A). We bisect angle C to create a new 36-72-72 triangle CBD, and then we see that:
AB/BC = BC/BD (corresponding sides of similar triangles)
AB/AD = AD/BD (as AD = DC = BC)
and so D divides AB in a golden section. The author tells us that this "Golden triangle," as it's often called, appears as part of a regular pentagon. Indeed, the ratio of the diagonal to the side of a regular pentagon is Phi.
Ogilvy ends the book as follows:
"The path is endless, but many rewards are offered along the way. One could do worse than follow the gleam of numbers."
Before I end this post, let me link to Fawn Nguyen's most recent post:
http://fawnnguyen.com/global-math-week-starts-on-october-10-2017/
Apparently, there's a "Global Math Week" coming up. Notice that the date is October 10th, which is also Metric Week (as in 10/10). Nguyen writes about something called "Exploding Dots," which somehow connects whole number operations to polynomial operations. Just as Ogilvy tells us that integer arithmetic is just polynomial arithmetic with x = 10, Nguyen informs us that we can even have fractional bases if we let x equal a fraction (but she doesn't specify which fraction she tried out).
Here is the Chapter 2 Test:
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