Alexandria Jones and the Strange Attractor

[Feature photo above: Clifford Attractor by Yami89 (public domain) via Wikimedia Commons.]

Alexandria Jones collapsed onto the couch with a dramatic sigh. Her father, the world-famous archaeologist Dr. Fibonacci Jones, glanced up from his newspaper and rolled his eyes.

“I don’t even want to hear about it,” he said.

Alex’s brother Leonhard was playing on the floor, making faces at the baby. He looked up at Alex and grinned.

“I’ll take the bait,” he said. “What happened?”

“Mom called my bedroom a Strange Attractor.”

“Oh? What does it attract?”

“I don’t know. Mostly books and model horses. But what’s so strange about that?”

The Mathematics of Chaos

Animation of a double compound pendulum showing chaotic behaviour.

Dr. Jones laughed and put down his paper. “Strange attractor is a technical term from the branch of mathematics called dynamical systems analysis — often called chaos theory.”

“So my bedroom is a math problem?”

“No. I think Mom meant your bedroom was chaos.”

“Oh.” Alex looked like she might pout, then she shrugged. “I guess she’s right, at that. So what is a strange attractor, really?”

“Well, when scientists first drew graphs of classical, non-chaotic systems — like a planet’s orbit or the flight of a football — it was surprising how often they got an ellipse or parabola or some similar curve,” Dr. Jones explained. “For some reason, nature seemed to be attracted to the shapes of classical geometry.”

Babymath: Story Problem Challenge III

<a href="http://www.flickr.com/photos/goetter/2352128932/"Photo by Raphael Goetter via Flickr

Alex and Leon enjoyed their baby sister, but they were amazed at how much work taking care of a baby could be. One particularly colicky night, everyone in the family took turns holding the baby, rocking the baby, patting her back, and walking her around before she finally succumbed to sleep.

Then Alex collapsed on the couch, and Leon sank into the recliner. They teased each other with these story problems.

Graph-It Game

[Photo by Scott Schram via Flickr.]

For Leon’s Christmas gift, Alex made the Graph-It game. She wrapped a pad of graph paper and wrote up the instructions:

To play Graph-It, one person designs a picture made by connecting points on a coordinate graph. He reads the points to the other player, who tries to reproduce the picture.

Renée’s Platonic Mobile

Alexandria Jones struggled to think of a Christmas gift that a one-month-old baby could enjoy, but finally she got an idea.

She cut empty cereal boxes to make regular polygons: 6 squares, 12 regular pentagons, and 32 equilateral triangles. Using small pieces of masking tape, she carefully formed the five Platonic solids. Then she mixed flour and water into a runny paste. She tore an old newspaper into small strips and soaked them in the paste. She covered each solid with a thin layer of paper.

A Football Puzzle

[Photo by rdesai.]

The MIT Mathmen got the ball on their own 20-yard line for the last drive of the game. They were down by 2 points, so they needed at least a field goal to win the game.

If quarterback Zeno and his offense advanced the ball halfway to the opposing team’s end zone on each play…

And the Baby Is . . .

[Photo by gabi menashe.] This story is continued from Alexandria Jones and the Eighty-Yard Drive

There was a time-out on the field, and the Jones family sat down for a brief rest. Sam asked, “How do babies decide when it’s time to be born?”

“Well, son, it has to do with numbers. You see,” Uncle Will explained, “the baby spends his first month thinking about the number one.”

“That’s not much to think about,” Sam said. “But I suppose he can’t handle much at that age.”

Alexandria Jones and the Eighty-Yard Drive

[Photo by West Point Public Affairs.]

Alexandria Jones pulled the last sheet of chocolate chip cookies from the oven and inhaled deeply. Mmm! Perfect. And just in time — Mom was calling her to the car. She slid the cookies into a plastic bowl but left the lid off so the steam could escape. The Jones family was going to meet Uncle Will and Alex’s cousin Sam for a tail-gate picnic before the big football game.

Alex Deals Out Equations

Image by incurable_hippie via Flickr

Looking around the room, Alex saw kids and parents moving from one table to another. Everyone seemed to be enjoying the Homeschool Math Carnival. She had six junior-high and high school students at her table, waiting while she shuffled her deck of cards.

“Okay,” she said. “These are Math Cards. I took out the face cards, so we just have numbers.”

Cousin Sam’s 15 Challenge

Uncle Will drove in from the tree farm to drop off Alex’s cousin, Sam, so he could go to the Homeschool Math Carnival.

“Hey, Sam,” Alex said. “What’s in the sack?”

Sam smiled. “A secret puzzle.”

“Aw, c’mon,” Leon whined. “We’ll be busy with our own games at the carnival. Can’t you show us now?”

Alexandria Jones and the Mathematical Carnival

Maria Jones hung up the phone and collapsed at the kitchen table. She buried her head in her hands and groaned. Alex looked up from her game of Solitaire.

“Let me guess,” she said. “Can’t-Say-No Syndrome, again?”

Mrs. Jones nodded. “This time I volunteered to plan an activity for next month’s homeschool group meeting.”

Leon wandered in and pulled an apple from the fruit bowl. “Ha!” he said. “She means she volunteered us to plan an activity, right?”

Mrs. Jones smiled. “That’s my motto: When in doubt, delegate!”

Math History Tidbits: The Battling Bernoullis

July 27th is Alex’s birthday. She shares it with Johann Bernoulli, an irascible mathematician from the late 17th century. This coincidence intrigued her enough that she wrote a research paper on Johann and his mathematical brother, titled “Jeering Jacob and Jealous Johann.”

Of course, to make the alliteration work, she had to mispronounce Johann’s name — but she figured he kinda deserved that. Read the historical tidbits below to find out why one writer said the Bernoulli brothers were “the kind of people who give arrogance a bad name.”*

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. If you’re stuck, read the hints. Then go back and try again. Figure them out for yourself — and then check the answers just to prove that you got them right.

This post offers hints and answers to puzzles from these blog posts:

Probability and Baby Blues

[Photo by audi_insperation.]

[In The Birthday Surprise, Alex discovered her family was expecting a new member...]

What will the baby look like, Alex wondered. “Dad, is there any way to tell whether the baby will have blue eyes like I do, or brown like the rest of the family?”

Dr. Jones shuffled the papers on his desk and found a blank page. “Over 100 years ago, the Austrian monk Gregor Mendel studied genetics, or how various traits are passed down from one generation to another.” He began to draw a diagram as he talked.

Alexandria Jones and the Birthday Surprise

[Photo by D Sharon Pruitt.]

[July 27th is Alex’s birthday, which she shares with Johann Bernoulli, an irascible mathematician from the late 17th century.]

The guests had gone. Alex and her family sat around the table, sharing the last tidbits of birthday cake and ice cream. Alex smiled at her parents.

“Thanks, Mom and Dad,” she said. “It was a great party.”

Maria Jones, Alex’s mother, leaned back in her chair. “I do have one more surprise for you, Alex. But you will have to share this one with the whole family.”

Leon groaned. “I know what it is: Let’s all pitch in to clean up.”

“That wouldn’t be a surprise,” Alex said.

Math History Tidbits: Agnesi, Euler, and China

I’ve fallen behind on my project of transcribing my Alexandria Jones stories. Finally, here are a few more tidbits from math history, along with links to relevant Internet sites and a few math puzzles for your students to try.

I hope you find them interesting.

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

Euclid’s Geometric Algebra

Euclid’s Geometric Algebra

Picture from MacTutor Archives.

After the Pythagorean crisis with the square root of two, Greek mathematicians tried to avoid working with numbers. Instead, the Greeks used geometry to demonstrate mathematical concepts. A line can be drawn any length, so straight lines became a sort of non-algebraic variable.

You can see an example of this in The Pythagorean Proof, where Alexandria Jones represented the sides of her triangle by the letters a and b. These sides may be any length. The sizes of the squares will change with the triangle sides, but the relationship $a^2 + b^2 = c^2$ is always true for every right triangle.

The Pythagorean Proof

[In the last episode, Alexandria Jones received a letter from archaeologist Sofia Theano, asking for help with a Pythagorean puzzle.]

The Mosaic Tile Mystery

Dear Alexandria Jones,

We continue to excavate the ancient building complex, which I believe may have been Pythagoras’s school. Yesterday, one of our digging crews uncovered a mosaic tile floor in the courtyard. The pattern of the tiles alternates between two square designs. (See enclosed sketches.)

During your family’s recent visit, you expressed an interest in the mathematical ideas of Pythagoras. Could you or your father offer us any insight into what these tile designs may represent?

I look forward to your response.

Sincerely,
Sofia Theano, Ph.D.
Crotone, Italy

Alexandria’s Dog is Now a Teacher

Photo by alex-s.

A reader of Indian descent has been kind enough to write to me about a difference in our cultures. In the US, or at least in the parts with which I am familiar, it is common to name one’s pet after a famous person. In India, however, to name a dog after a human is a very deep insult.

I am sorry! It was not intended that way.

Therefore, I have re-named Alexandria Jones‘s dog after a Westerner. I would like to keep the tradition of naming the characters in my Alex stories after people in math history, so I am hoping this one will not offend anyone.

Game: Avoid Three, or Tic-Tac-No!

Math concepts: slope, logical strategy
Number of players: 2 or more
Equipment: 4×4 or larger grid, pebbles or other tokens to mark squares

Set Up

Alexandria Jones and her brother Leon played Avoid Three with pebbles on a grid scratched in the sand, but you can also use pencils or markers on graph paper. You need a rectangular playing area at least 4×4 squares large. The bigger your grid, the longer your game.

Answers to Leon’s Figurate Number Puzzles

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

Puzzle: Figuring Out Figurate Numbers

Photo by frumbert.

Alexandria Jones‘s parents decided that the family needed to relax after the excitement of tracking Simon Skulk, so they spent the next day at a beach on the Mediterranean coast. Leon collected pebbles and tried to build up figurate numbers — numbers that make a figure, or shape — the way Dr. Theano had shown them.

The Puzzling Pythagorean Pebbles

Feature photo above by meichimite (CC BY 2.0).

Alexandria Jones and her family flew to Italy for spring break. Her father, the famous archaeologist, had to visit an excavation.

It was late when their plane landed in Crotone, a small coastal city near the instep of Italy’s boot. Dr. Jones had used the Internet to find a hotel that allowed pets, so Alex was able to snuggle down with her favorite pillow — her trusty dog, Ramus.

The Original School of Mathematics

The next day, Dr. Jones introduced his family to Sonya Theano, a former student of his and the director of this dig. “Come, let me show you around,” Dr. Theano said. “We’ve uncovered several buildings of a small compound, set apart from the city of Crotona, as it was called then. From the pottery and trade goods, we estimate these buildings were in use around 550-500 BCE.”

Answers to Alex’s and Leon’s Puzzles

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original posts. Figure them out for yourself — and then check the answers just to prove that you got them right.

Leonhard’s block puzzles

Alex’s & Leon’s homeschool puzzle

Alex’s & Leon’s Homeschool Puzzle

Photo by gotplaid?.

While checking out the book table after a homeschool group meeting, Maria Jones glanced up to see her children laughing with some kids she did not recognize. Driving home, she asked about the new family, but Alex and Leon had been too busy exchanging silly stories to even ask the strangers’ names.

“Well,” Leon said, “the boy told me he has twice as many sisters as brothers.”

No way!” said Alex. “The girl told me that she has the same number of brothers and sisters.”

How can that be?

Leonhard’s Block Puzzles

Leonhard Jones is Alexandria Jones’s younger brother. He enjoys woodworking, and he cut a wooden cube into 8 smaller blocks to make himself a puzzle.

Puzzle #1

Leon painted the 8 blocks with his two favorite colors: red and forest green. When he was finished, Leon could put the blocks together into a red cube, or he could switch them around to make a green cube.

How did Leon paint his blocks?

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer (relatively) soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

The Secret of Egyptian Fractions

Alex made a poster of Egyptian-style fractions, from 1/2 to 9/10. Many of the fractions were easy. She knew that…

$\frac{5}{10} = \frac{4}{8} = \frac{3}{6} = \frac{2}{4} = \frac{1}{2}$

Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents.

She had the most trouble with the 7ths and 9ths. She tried converting these to other fractions that were easier to work with. For example, 28 has more factors than 7, making 28ths easier to break up into other fractions with one in the numerator.

The Secret of Egyptian Fractions

Photo from Library of Congress via pingnews.

Archaeology professor Dr. Fibonacci Jones came home from a long day of lecturing and office work. Stepping inside the front door, he held up a shiny silver disk.

“Ta-da!” he said.

“All right!” said his daughter Alexandria. “The photos are here.”

They had to chase Alex’s brother Leon off the computer so they could view the images on the CD, but that wasn’t hard. He wanted to see the artifacts, too. Alex recognized several of the items they had dug up from the Egyptian scribe’s burial plot: the wooden palette, some clay pots, and of course the embalmed body.

Then came several close-up pictures of writing on papyrus.

Photo from MathsNet.net.

How to Write Egyptian Fractions

“I remember how to read the Egyptian numbers,” Alex said, “but what are these marks above them?”

Dr. Jones nodded. “I thought you would catch that. Those are fractions. The scribe places an open mouth, which is the hieroglyph ‘r’, over a number to make its reciprocal.”

“I know that word,” Leon said. “It means one over the number. Like, the reciprocal of 12 is 1/12, right?”

“That is right. 1/12 would be written as…”

The Rest of the Story

As I transcribed this article from my old math newsletter, I realized that it would require more graphics than I was willing to construct. LaTex does not handle Egyptian hieroglyphs — or at least, I don’t know how to make it do so. Instead, I decided to scan the newsletter pages and give them to you as a pdf file:

The file includes a student worksheet for Egyptian fractions from 1/2 to 9/10.

To Be Continued…

Read all the posts from the January/February 1999 issue of my Mathematical Adventures of Alexandria Jones newsletter.

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