Consecutive Dates

Photo credit: USA Today

Well, a big consecutive date is now in the past. 12/13/14 has come and gone, and now we look ahead to the next major milestone of this kind. Most people note that because 13/14/15 isn't going to be possible next year, and indeed none of the integers that follow will ever be possible (as December is the 12th month), the next major such date will be 01/02/03, in 2103, a little over 88 years from now.

Photo credit:
KeepCalm-a-Matic

But never fear, Fibonacci is here! We actually have a Fibonacci date coming up in just under seven years! First let's look back at the last Fibonacci date. It would have been 05/08/13, or May 8th of last year. Our European friends would even argue that where they live, that date would have been August 5 of 2013, even more recent.



So when will the next Fibonacci date be? On August 13, 2021, the date will be 08/13/21. Sorry, Europeans. No such date for you. So make your plans early, and prepare for the last Fibonacci date celebration of the century!

Happy Fibonacci Day!

I want to wish each and every one of you a very Happy Fibonacci Day, today, November 23. Why, you ask? The Fibonacci Sequence is a pattern of numbers that begins 1, 1, 2, 3, ... The key to the sqeuence is that you simply add any two consecutive number to produce the next number. Today, November 23, is 11/23 and for that reason we celebrate this powerful pattern.

Take a look through this blog for more about the Fibonacci Sequence, and be sure to look for my new book Fibonacci Zoo, due on shelves in April 2015!

Lesson #6 - Make Your Own Fibonacci Sequence

Ok, we have come to the final lesson in this journey. Your final task is a little more open-ended, but should be a fun one. I would like you to come up with your own Fibonacci Sequence.

A Triangular Number Sequence
Imagecredit: Math is Fun.com

I know what your first reaction may be: How is that possible? Isn't there only one Fibonacci Sequence?

And indeed, technically you may be correct. But think for a second about what makes this sequence so simple. All you do is add two consecutive numbers in the sequence to find the next number. That's pretty simple, right?

The Fibonacci Sequence (the real one) begins with the numbers 1 and 1. From there, 1+ 1 =2, 1 + 2 = 3, and so on.

A Pentagonal Number Sequence
Image credit: astarmathsandphysics.uk

What I would like you to do is to pick two different numbers to start your sequence. They can be the same numbe
r twice, just like Fibonacci did. Something like 2, 2, perhaps. Or they can be different numbers. Maybe 2, 3. Or you could pick two totally outlandish numbers, like -5 and 7.

It doesn't matter which two you pick. What's most important is that you figure out how to produce the next values in the sequence and then go exploring to see what the sequence holds in store.

Some questions to ponder:

A Hexagonal Number Sequence, of sorts
Imagecredit: drking.org.ca

• What common ratio does your sequence eventually lead to? The Fibonacci Sequence leads to 'phi', or approximately 1.618, which is often called the Golden Ratio. Does your sequence approach a similarly small value? Is it close to 'phi'?
• Is there any way to create a sequence of numbers that eventually matches the original Fibonacci numbers, if not all, at least some? 
• Is it possible to produce a spiral like the one the Fibonacci Sequence produces that could possibly be mirrored in nature?

This is the time to use your creativity and see where it takes you. Who knows? Maybe you could tell a story like Fibonacci did about rabbits that leads to a powerful new number pattern. And then, just maybe, in another thousand years (or less) math students of the future could find themselves studying the (Your Name Here) Sequence.

Good luck - good math!

-Tom Robinson

Pascal's Triangle (Stay tuned for Pascal's Orchard!)
Image credit: MtDouglas.ca

Here is a link to a Math Is Fun site that explores other number sequences you may be interested in. They are not Fibonacci sequences, but might be fun to discover on your own.

Lucas Numbers

I came across this video by chance, though I love the Numberphile guys (and gals) and their work. It explains the concept of Lucas numbers, which are directly related to Fibonacci numbers, but are pretty intriguing in an unexpected way. I won't spoil the fun, but do take some time to watch - you will be surprised by what you learn, though I do take exception to his claim that Lucas Numbers are actually better than Fibonacci. Call me biased...

Lucas Numbers - from Numberphile's YouTube Channel 

featuring Matt Parker

Lesson #5 - Creating Integers From Fibonacci numbers

Objective: Students will use Fibonacci numbers and the four arithmetic operations to produce integer values from 1-20, 1-50, 1-100.

Photo credit:
DevCentral

Task: Using only Fibonacci Sequence numbers and the four arithmetic operations (addition, subtraction, multiplication and division) produce each integer value from 1-20. Older students may be challenged to produce numbers through 50, or even 100.

Example: The number 10 may be produced in the following ways using Fibonacci numbers:
5+5
8+2

Photo credit:
MathIsFun.com

13 - 3

The number 17 may be produced by doing the following:
13 + 5 - 1
8*3-13 + 3*2

• How many different ways can you find to make each target number?
• Which numbers can you find using only addition?
• Which numbers can you find using only addition and subtraction without repeating any numbers?
• Which numbers can you find using all four operations?

Lesson #4 - Find a Rectangle

Rectangles are all around us. There are posters and framed photo on our walls, highway and roadside signs, buildings, computer monitors and smartphones, and the list goes on and on.

But some of these rectangles seem to look more, well, RIGHT, than others. And sometimes it's hard to tell why they do. Consider the following two rectangles:

Photo credit:
American Pools Ipswitch

Photo credit:
Creative Beacon.com

       

Does one look 'better' than the other? Why do you find yourself drawn to that one? Chances are, it's because of its dimensions, which likely are very close to the Golden Ratio, found in the Fibonacci Sequence.

Lesson #4 - Find a Rectangle

Objective: Students will identify rectangles whose dimensions closely match 'phi', or the Golden Ratio, which is approximately, 1.618.

Task: Look around your house, your community, or larger town and identify five rectangles which seem to match the dimensions of the Golden Ratio. That is, their lengths are approximately 1.618 times their widths.

Take a photo of the rectangle for your collection.

If possible, use a measuring device to measure the actual length and width of each rectangle. If the rectangle is located on private property, be sure to ask permission before taking measurements. Compare the ratio of your rectangle's length to its width to the Golden Ratio. How close were your values?

Lesson #3 - Find a Flower

Sorry for the delay. I was working in Italy this past month, and was fortunate enough to spend time in Fibonacci's home town (Pisa). In fact, here is a photo of me at the top of the Leaning Tower with my friends, Chris, Bern, and Mike, the coaching staff of the University of Pennsylvania women's basketball team.

Standing on top of the Leaning Tower of Pisa, home of Leonardo Pisano,
better known as Fibonacci (that is, Pisa was his home, not the tower!)
That's me on the left! August 2014

Lesson #3 - Find a Flower

Objective: Students will locate and pick flowers with petals that represent one of the Fibonacci numbers.

Task: Look around your house, your school, or your local community for flowers with petals. Make sure you are allowed to pick the flowers. If you are not sure, just take a picture of it. If you are completing this task at at time of year in which there are no flowers blooming, look for plants or trees with leaves if possible.

Count the petals and try to find one example of each of the first few Fibonacci numbers:

1, 1, 2, 3, 5, 8, 13...

After you have selected your flowers, present them to your teacher. Finally, rate yourself using the scale below:

1 flower: Way to go - you found one!
2 flowers: Excellent! Keep your eyes open for more Fibonacci numbers around you.
3 flowers: Outstanding! Fibonacci would be proud of you!
4 or more flowers: Incredible! You might be the next Fibonacci!

 

Lesson #2 - The Fibonacci Breakfast

Lesson Objective: Students will use the Fibonacci Sequence to determine how many pieces of cereal are

Photo credit: Wikipedia

necessary to produce a satisfying bowl of breakfast cereal.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, ...

Task #1:
Use a cereal with individual pieces that are easily counted.
Good examples include choices such as Cheerios, Trix, Cocoa Puffs, Kix, Lucky Charms.
Bad examples include choices such as oatmeal, granola, Rice Krispies.

Your task is to determine, by adding pieces of cereal in increasing order of the Fibonacci Sequence, which number finally makes what you would consider to be a full bowl of cereal.

Procedure:
1. Place ONE  piece of cereal into your bowl.

• Is this enough cereal to make a full bowl?

2. Next, place ONE more piece of cereal into your bowl (making a total of two pieces).

• Is THIS enough cereal to make a full bowl?

3. This time, place TWO more pieces of cereal into your bowl (making a total of four pieces).

• Is THIS enough cereal to make a full bowl?

4. Continue this process, each time adding the next Fibonacci number in the sequence until your bowl is filled to an acceptable level. How many numbers did it take? Was it more or less than you predicted before you began?

Task #2:
In this task your job will be to predict which exact Fibonacci number will represent the correct number of pieces of cereal necessary to make a full bowl.

For example, are 55 pieces enough? What about 89? Something more than that? This task differs from Task #1 because your answer to Task #1 will not likely be an actual Fibonacci number. For this task, you are predicting, using only the numbers in the sequence, how many pieces are necessary to fill a bowl.

Make a prediction, then count that many pieces into your bowl. If you happen to be incorrect, just change your prediction and keep trying!

Extension activity: Try changing cereal. How does changing the cereal type and/or size affect your answer to this task?

Cover Art Sneak Preview!

I am so very pleased to announce that the cover art for Fibonacci Zoo is here! Check it out and look for my book on the shelf of your favorite bookstore (or in cyberspace) in February 2015.

Lesson #1 - Solutions

Lesson #1 - Solutions

Part I:
a) Pattern: add 2; Next two terms: 11, 13
b) Pattern: add 3; Next two terms: 17, 20
c) Pattern: add 5; Next two terms: 34, 39
d) Pattern: add 10; Next two terms: 60, 70
Bonus: Pattern: add 2.5; Next two terms 15.5, 18
Wrap-up: Each sequence involves ADDING the same value each time.

Part II:
a) Pattern: subtract 5; Next two terms: 5, 0
b) Pattern: subtract 4; Next two terms: 5, 1
c) Pattern: subtract -6; Next two terms: -10, -16
d) Pattern: subtract 2; Next two terms: -7, -9
Bonus: Pattern: subtract 1.5; Next two terms: 1.5, 0
Wrap-up: Each sequence involves SUBTRACTING the same value each time.

Part III:
a) Pattern: multiply by 2; Next two terms: 64, 128
b) Pattern: multiply by 3: Next two terms: 324, 972
c) Pattern: multiply by 1/2 (also, divide by 2, but for consistency, we'll stay with multiplication); Next two terms: 2.5, 1.25
d) Pattern: multiply by 5; Next two terms: 625, 3125
Bonus: Pattern: multiply by 1.5; Next two terms: 50.625, 75.9375
Wrap-up: Each sequence involves MULTIPLYING each term by the same value each time. (Note that on Question C, dividing by 2 is the equivalent of multiplying by 1/2.)

Part IV: At the conclusion of these exercises, it may be a good time to introduce children to the Fibonacci Sequence, which is given at the end of the solutions. See if they can identify the sequence as similar to those given in this section.
a) Pattern: add up any two consecutive numbers to find the next number; Next two terms: 23, 37
b) Pattern: add up any two consecutive numbers to find the next number; Next two terms: 45, 73
c) Pattern: add up any two consecutive numbers to find the next number; Next two terms: 142, 230
d) Pattern: add up any two consecutive numbers to find the next number; Next two terms: 16, 26
Bonus: Pattern: add up any three consecutive numbers to find the next number; Next two terms: 68, 125
Wrap-up: Each sequence is defined by adding up previous numbers to find the next number.

Fibonacci Sequence: 1, 1, 2, 3, 5, 8, 13, 21, ....

Lesson #1 - Number Patterns

Learning Objective: Students will identify number patterns and use conjectures

 to predict the next terms in the pattern

Part 1

Describe (in words) the pattern in each sequence of numbers and use that pattern to fill in the next two numbers in the sequence.

a. 1, 3, 5, 7, 9, __, __

Pattern ______________________

Next two numbers ____, _____

b. 2, 5, 8, 11, 14, __, __

Pattern ______________________

Next two numbers ____, _____

c. 4, 9, 14, 19, 24, 29, __, __

Pattern ______________________

Next two numbers ____, _____

d. 10, 20, 30, 40, 50, __, __

Pattern ______________________

Next two numbers ____, ____

Bonus: 3, 5.5, 8, 10.5, 13, __, __

Pattern ______________________

Next two numbers ____, _____

Wrap up: What do all of these number sequences have in common?

________________________________________________________

Enrichment: Make up your own number sequence that follows the same kind of rule you identified in this section.

________________________________________________________

Part II

Describe (in words) the pattern in each sequence of numbers and use that pattern to fill in the next two numbers in the sequence.

a.  25, 20, 15, 10, ___, ____

Pattern ______________________

Next two numbers ____, _____

b.  25, 21, 17, 13, 9, __, __

Pattern ______________________

Next two numbers ____, _____

c.   14, 8, 2, -4, __, __

Pattern ______________________

Next two numbers ____, _____

d.  1, -1, -3, -5, __, __

Pattern ______________________

Next two numbers ____, _____

Bonus:   9, 7.5, 6, 4.5, 3, __, ___

Pattern ______________________

Next two numbers ____, _____

Wrap up: What do all of these number sequences have in common?

________________________________________________________

Enrichment: Make up your own number sequence that follows the same kind of rule you identified in this section.

________________________________________________________

Part III

Describe (in words) the pattern in each sequence of numbers and use that pattern to fill in the next two numbers in the sequence.

a.   2, 4, 8, 16, 32, __, __

Pattern ______________________

Next two numbers ____, _____

b. 4, 12, 36, 108, __, __

Pattern ______________________

Next two numbers ____, _____

c. 40, 20, 10, 5, __, __

Pattern ______________________

Next two numbers ____, _____

d.  1, 5, 25, 125, __, __

Pattern ______________________

Next two numbers ____, _____

Bonus: 10, 15, 22.5, 33.75, __, __

Pattern ______________________

Next two numbers ____, _____

Wrap up: What do all of these number sequences have in common?

_______________________________________________________

Enrichment: Make up your own number sequence that follows the same kind of rule you identified in this section.

________________________________________________________

Part IV

Describe (in words) the pattern in each sequence of numbers and use that pattern to fill in the next two numbers in the sequence.

a.  1, 4, 5, 9, 14, __, __

b.  5, 6, 11, 17, 28, __, __

c. 14, 20, 34, 54, 88, __, __

d. 0, 2, 2, 4, 6, 10, __, __

Bonus:  1, 2, 3, 6, 11, 20, 37, __, __

Wrap up: What do all of these number sequences have in common?

_______________________________________________________

Enrichment: Make up your own number sequence that follows the same kind of rule you identified in this section.

________________________________________________________

Publishers Weekly Sneak Preview for 2015!

Exciting news from Publishers Weekly!

ARBORDALE

Arbordale pricks up its ears for Sounds of the Savanna by Terry Catasús Jennings, showcasing a habitat alive with noise; Achoo!: Why Pollen Counts by Shennen Bersani, featuring an allergic baby bear; Fibonacci Zoo by Tom Robinson, illus. by Christina Wald, in which a boy discovers a pattern at an unusual zoo; Wandering Wooly by Andrea Gabriel, focusing on how Little Wooly uses her senses to find her way back to her family; and Tornado Tamer by Terri Fields, illus. by Laura Jacques, an adaptation of The Emperor’s New Clothes in which a weasel promises to build a special cover to protect the town from tornadoes.

Here's the full link to the preview:  Publisher's Weekly 2015 Sneak Preview

Common Core State Standards

I'm not suggesting that kindergartners and their parents will pick up this book with the sole intent of using it to meet the new national math standards known as the Common Core State Standards (CCSS). However, Arbordale Publishing and I both agree on an important philosophy - that learning math and science is not only possible through reading, but is actually enhanced through reading. And so, during those critical primary school years (grades K-3 or so) while children are making such great strides in both their reading fluency and comprehension, we believe this is a unique opportunity to encourage them to develop their math literacy (sometimes called 'numeracy').

This is where Fibonacci Zoo comes in. This book takes its readers on an adventure through a most unusual zoo. Children will meet, in order, an alligator, a bison, camels, dolphins, elephants, flamingos, gorillas, hippos, and iguanas. Many will notice that these animals arrive in alphabetical order so even the youngest children can begin to practice their ABCs, even while they grow their reading skills. But Fibonacci Zoo continues as children meet the Fibonacci sequence, and learn to count, explore number patterns, make predictions, and then see their predictions tested. These 'inductive reasoning' skills will become powerful tools for these children as they grow into pre-adolescents and adolescents and begin their formal mathematics careers in earnest.

And so, in the spirit of celebrating Fibonacci Zoo as "more than just a picture book," I have identified some of the CCSS math standards for grades K-4 that this book may address. Please understand that by no means am I suggesting that the book on its own is sufficient to meet all of these standards completely. In fact, my hope is that parents, teachers, math coaches, administrators, librarians, and indeed children themselves use it as a tool for enrichment, for exploration, and for extension of the basic skills they are already learning. Over the next several posts, I will be sharing lesson ideas (around ten when we get done) for teachers and parents to use at their leisure to develop the skills necessary for success in mathematics for their children. In time, I envision this book becoming a critical component of elementary school mathematics instruction from Alaska to Florida, from Maine to Hawaii.

Thanks for coming on this journey with me. It promises to be quite a ride!

Common Core State Standards - Math - Grades K-4

CCSS - Kindergarten

CCSS.MATH.CONTENT.K.CC.A.2

Count forward beginning from a given number within the known sequence (instead of having to begin at 1).

CCSS.MATH.CONTENT.K.CC.B.4.A
When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object.

CCSS.MATH.CONTENT.K.CC.B.5
Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.

CCSS - First Grade

CCSS.MATH.CONTENT.1.NBT.C.4

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

CCSS.MATH.CONTENT.1.NBT.B.2.A
10 can be thought of as a bundle of ten ones — called a "ten."

CCSS.MATH.CONTENT.1.NBT.B.2.B
The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

CCSS.MATH.CONTENT.1.NBT.B.2.C
The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

CCSS.MATH.CONTENT.1.OA.A.1
Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.¹

CCSS.MATH.CONTENT.1.OA.C.6
Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

CCSS.MATH.CONTENT.1.OA.D.7
Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

CCSS - Second Grade

CCSS.MATH.CONTENT.2.NBT.B.5
Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

CCSS.MATH.CONTENT.2.NBT.B.9
Explain why addition and subtraction strategies work, using place value and the properties of operations.¹

CCSS.MATH.CONTENT.2.OA.A.1
Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.¹

CCSS.MATH.CONTENT.2.OA.B.2
Fluently add and subtract within 20 using mental strategies.² By end of Grade 2, know from memory all sums of two one-digit numbers.

CCSS - Third Grade

 CCSS.MATH.CONTENT.3.OA.D.9

Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.

CCSS - FourthGrade

CCSS.MATH.CONTENT.4.OA.C.5

Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule "Add 3" and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way.

CCSS.MATH.CONTENT.4.NF.C.6
Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.

CCSS.MATH.CONTENT.4.NF.C.7
Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

The Parthenon

Though I've never been to Greece, I definitely would like to visit sometime. And for all the white sandy beaches, the exquisite food offerings, and the culture that oozes out of every pore in that country, one of the top sites on my list of things to see is the Parthenon, pictured here.

 

The Parthenon - photo credit: Wikimedia Commons

To most eyes (mine included, if truth be told) this building represents little more than an ancient ruin, an artifact from an era long gone, and generally, a really cool building to see, even more remarkable that it's still standing.

But from a mathematician's standpoint, some say the Parthenon is one of the great wonders of the world as it represents one of the earliest examples of builders and architects understanding dimension, perspective, and ratio. Many believe the Parthenon was built according to the Golden Ratio, or phi.

The Parthenon is considered to have been a most significant temple for the ancient Greeks, build on the hill called the Acropolis. It was a symbol of military victory, and was built to honor the city's (Athens) patron deity, Athena. It was an architectural wonder, mixing Dorian and Ionic columns, and it was decorated by sculptures, the likes of which had not been seen before.

To us, however, something even bigger lie in this building. It was built between 447 B.C. and 432 B.C. So it took fifteen years to build - not that unusual for the time. However, you may note that this period of time is around 1600-1700 years BEFORE Leonardo Pisano wrote his book about rabbits and the Fibonacci Sequence.

You see, the original dimensions (lost to us through the centuries) appear to follow a very similar pattern, as you can see in the image below. By isolating square and rectangular regions of the facade of the Parthenon, you can begin to see a Fibonacci spiral grow.

The Parthenon displays the Fibonacci spiral.
Photo credit: Goldennumber.net

Scholars have debated this, and whether it was built using the Golden Ratio or not may never truly be known. But one look at the spiral and you can see that there was something special about this building.

Taking it one step further, let's look at some relative measurements of the building. Here you can see that familiar 1.618 to 1 ratio value yet again. Remember, part of the building is no longer standing, so some of this may simply be speculation. But still, the Parthenon stands as a) one of the great architectural designs of the ancient times, and b) quite possibly the oldest-standing example of the Fibonacci sequence and Golden Ratio in the world.

Photo Credit: rgu.ac.uk

Now, my question to you is this: can you find other structures, either from the ancient world, or still standing today, that reflect similar dimensions? We explored this with rectangles, but are there buildings today that just seem to 'look right'? If you find one, drop me a note and let me know about it.

Note: This will likely be the final installment looking at applications and consequences of the Fibonacci sequence. From now on, be looking for quick, simple lesson plans you can use either in your classroom or even at home with your own children to explore Fibonacci. In keeping with the spirit of my upcoming book, these lesson ideas will probably differ from much of what you find online, as those lessons tend to be geared toward middle or high school students. I would like these lessons to be applicable to elementary-aged students, and if possible even the primary grades. Obviously, such lessons will rely heavily on parent or teacher support and guidance, but could easily be used in conjunction with the book itself, allowing students to read it multiple times, each time discovering a new activity that connects their reading to their learning.

Mona Phi-sa

By now you know that this number 'phi', which emerges as you look further down the list of Fibonacci numbers, explains some really unexpected relationships, from geometry to architecture, to the animal kingdom (Fibonacci Zoo, notwithstanding) and in the human body. One particular human, and technically, we aren't actually talking about a human, but a painting of a human, stands out as perhaps the most perfect of them all - Mona Lisa.

The Mona Lisa

This painting is arguably the most famous one ever made, certainly the most famous one Leonardo da Vinci (not OUR Leonardo!) painted. We aren't talking scupltures, so David is not in the conversation, by the way. The Mona Lisa hangs in the world-famous Louvre museum, and has captivated art buffs and casual fans for centuries. You can learn more about the painting at this Khan academy site.

So what makes this painting so famous? I've not had the chance to see it, but I understand it is actually quite small, and viewing it is a little stressful, with crowds gathering around it, cameras snapping right and left, and I am left to wonder, what's the big deal?

Well, I'm not going to make any claims about my art history credentials (which are none). Instead, I want to look at her face. Lisa, it is believed, was the wife of a man who asked da Vinci to paint her. That's it. But what makes her so interesting from a mathematical standpoint is that the dimensions of her face are nearly perfect. I guess maybe it's from one Italian Leonardo (Pisano) to another (da Vinci), but a close look at Mona Lisa reveals that her facial dimensions very closely match the Fibonacci spiral we saw in a previous post.

The Fibonacci Spiral in the painting of Mona Lisa

It turns out that some faces just look nicer than others, much like those rectangles we saw whose 'perfection' was hard to describe, but we knew it when we saw it. Facial recognition experts have long found the value of phi in faces that are widely accepted as being beautiful. Sample faces include Angelina Jolie, Jennifer Aniston, Beyonce, Johnny Depp, and Jessica Simpson, to name a few. At the Math of Beauty web site, you can see a little more into this phenomenon. And wouldn't you know it? Even Oprah got into the act on an episode that dealt with something called "The Beauty Equation.

So is it real? Do men and women with 'perfect' facial dimensions really look more attractive? I'll leave that to you to decide for yourself. As for me, my wife thinks so, and that's all that matters. Try searching for 'golden ratio' and 'beauty' and see what you find.

The Golden Rectangle

We're fast approaching some really fascinating applications of this number 'phi' and the Fibonacci sequence. But before we get to looking at those applications, famous faces, buildings, etc., I'd like to introduce you to something new. We've talked briefly about the Golden Ratio, or the Golden Mean (the number 'phi', which is approximately 1.618...). But there is a very simple geometric shape that defines this ratio and which serves as a starting point for just about all of the geometric instances of Fibonacci.

Photo credit: jwilson.coe.uga.edu

The shape is a rectangle and this very special rectangle is one of those shown to the left. Can you figure out which one it is? Which rectangle looks 'just right'?

You see, some rectangles are long and skinny, while others are so even in their dimensions they almost look like squared (the green rectangle toward the bottom middle, for example).



But there are certain rectangles that just seem to look 'right'. I don't know what it is, but they are pleasant to look at and just catch your eye. One that immediately jumps out at me is near the top middle of this collection. It's orange, and it lies just below a long skinny orange rectangle.

Now why does this one rectangle stand out to me? What is it about its dimensions that make it seem so 'perfect'? Ah, this is where Fibonacci comes in.

Remember the Golden Ratio, the ratio of consecutive numbers in the sequence? For example, we looked at the ratios 8/5, 13/8, 21/13, 34/21, etc. Those values are not the same, and yet as we work our way through the sequence, the ratios all approach this special value we call 'phi', which is approximately 1.618. It's known as the Golden Ratio (among other names).

Here is a rectangle whose dimensions reflect the Golden Ratio, along with several that do not. That one in the middle is pretty sexy, isn't it?

Photo Credit: http://www.cet.ac.il/

Let's take a look quickly at how this 'golden' rectangle works. If you were to take a Golden Rectangle and cut from it a square whose length is the shorter side of the rectangle, it would look like this:

Photo Credit: Wikipedia

We now have a square, which measures b by b, and a rectangle. The thing is, we could then cut the new rectangle (A) into a square, again with side length equal to the shorter side of the rectangle, and that would leave another rectangle.

What's so interesting is that this pattern can continue forever. Each time you cut off a square, you end up with a new (smaller) rectangle. And that rectangle always has a length and width whose respective length ratio is phi.

Photo Credit: flickr.com

Now this is where it gets really cool, and we'll probably stop here for today. You can start way down inside the repeating pattern of rectangles and squares and think of that innermost rectangle(the bright pink one) as TWO squares that each measure 1x1. Next, the brown square right below it is then 2x2, and the blue square that follows is 3x3. See a pattern here?

The green square measures 5x5, the red one is 8x8 and now it shouldn't surprise you to learn that the orange square is 13x13, followed by the yellow square which of course now measures 21x21. In fact, you could continue this pattern forever, and eventually, the series of squares will all form into a spiral...a Fibonacci Spiral. More on that in my next post. For now, I will leave you with a drawing of the numbered squares that form the Fibonacci Spiral.

Photo Credit: wakeupworld.com

A Fibonacci Finger?

This kind of surprised me. If you look around online, you will find a number of claims that say that the ratio of the length of your forearm (elbow to wrist) to the length of your hand (same wrist to the tip of your middle finger), is about phi, or 1.618. I tried it and didn't come close. Maybe I have long fingers, I don't know. My ratio was 9:8, or about 1.125.

Photo credit: http://www.handanalysis.co.uk

But then I started measuring other parts of my hand and found this:

Looking at just my forefinger, I see that it is broken (no pun intended) into three segments. The tip, the middle section, and the longer main section, back to where my large knuckle is. From there, my hand continues down to the base of my thumb (not my wrist, but where my thumb kind of melds into my hand).

I brought out a ruler and here is what I found. Note that I am using rough estimates, but something easily enough seen on a standard ruler.

My fingertip (the last segment of my forefinger) is one (1) inch long. The second segment is also one (1) inch long. The third (base) segment of my forefinger (again, measured on top of my hand, back to the knuckle, is two (2) inches long. Finally, that last segment (not really my finger, but the rest of my hand) is three (3) inches long.

1, 1, 2, 3....Weird, huh? Of course, if you REALLY wanted to get into it, we have 2 hands, each with 5 digits, and technically just 8 fingers (2 thumbs). All Fibonacci numbers.

While you ponder this remarkable set of findings, watch these kittens meet a guinea pig, thanks to Animal Planet. After, I will give you some links to sites that explain some of the simple human anatomy relationships involved with Fibonacci.

Video courtesy of Animal Planet

Here are some links you might enjoy:

Human Hand & Foot

Golden Section

Cool video about the Golden Mean / Golden Ratio / Phi

Just a quick post here. This video is about seven minutes long and it does a really nice job of connecting the numbers in Fibonacci to the Golden Ratio, which we will soon explore in great detail. Thanks to Angie Zigkiri for posting it to YouTube. I think you'll like it!

Intrigued? It's pretty freaky, isn't it? Stick around for more! Next up will be the Golden Ratio and the human body.

Golden Ratios in Fibonacci

The magical and mysterious number: Phi
Photo credit: education-portal.com

By now we've seen some interesting and fairly unexpected places where values in the Fibonacci sequence can be found. I confess that I could spend hours dissecting even more intricate applications and examples of these numbers (don't worry, I probably still will). But for now, let's get back to the sequence itself. Here are the first 12 values in the sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Eventually, even the most ardent of mathematicians get a little bored looking at number sequences and they start playing around with patterns. Because the sequence always grows (after the first two numbers), we could try and find the ratio of any value to the one just before it. 

You may be familiar with the idea of a GEOMETRIC SEQUENCE, or one in which the ratio of consecutive elements is always the same. We call it a 'common ratio'. Below is an example of a geometric sequence in which the common ratio is 2:

1, 2, 4, 8, 16, 32, 64, 128, ...

So what happens when we look at ratios in Fibonacci? Well, at first, nothing really exciting. Here are the first few:

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.67

See? Nothing really exciting ...yet. But let's look at little more closely at these values. The first ratio (1) is fairly small but the second one gets larger (2). Then the third (1.5) is smaller, followed by the fourth one (1.67) which is larger. It's as if the pattern of ratios is bouncing around, up, then down, then up a little less, then down a little less....

Let's see what happens when we keep going from the previous ratio, which was 1.67

8/5 = 1.6 (smaller)

13/8 = 1.625 (larger)

(and here I'm going to start using approximations because the ratios will not come out to be nice, clean integers, fractions, or decimals).

21/13 = 1.615 (smaller)

34/21 = 1.619 (larger)

See? The pattern continues - up, then down, then up, then down..but something else is happening at the same time. Can you see it? Not only are the values bouncing up and down, but they seem to be getting closer together. That is, each time they jump up, they don't jump up as much as they did the previous time. And each time they jump down, that jump isn't as big as the previous jump down. Here are a few more:

55/34 = 1.6176 (smaller)

89/55 = 1.61818 (larger)

144/89 = 1.617977 (smaller)

The pattern of ups and down continues, but by now, we are really closing in on what seems to be a magic number. But what is that number?

Ratios of consecutive Fibonacci numbers
Photo credit: knoji.com

The ancient Greeks figured out what it was. It's 'phi'! If you keep finding ratios of consecutive Fibonacci numbers (and by all means, feel free to do so...), you will find that the ratios continue both bouncing around (slightly above and slightly below) and getting closer to this number, phi.

The value of the number is approximately 1.618. I say approximately, because phi is a number like pi, in that neither decimal ever ends or repeats. One interesting difference between phi and pi (besides the fact that they are different letters in the Greek alphabet) is that pi (approximately, 3.14159) is TRANSCENDENTAL. That means it is not the solution to any polynomial equation you could ever come up with. But strangely, phi is such a solution. So phi is not considered transcendental. That one might win you Final Jeopardy - might want to tuck it away for future reference. You're welcome.

What does one do with phi? Oh, we're just getting started answering that question. Stay tuned. For now, just know that it's out there, waiting for you. You can't ever actually find it, but you can get ever so close to it - kind of like true love, right?