WHERE PARALLEL LINES MEET
story by George Gadanidis & Janette M. Hughes
1. MENELAUS OF ALEXANDRIA
Molly and Alexander have travelled back in time in search of their grandfather. Grandpa dropped his magic pocket watch on his last trip and is now stranded somewhere in the past. Molly and Alexander used the magic pocket watch to visit the mathematician and philosopher Rene Descartes (1596-1650). However, they have not yet found Grandpa in their travels.
Molly and Alexander are now searching Grandpa’s office for his journal, to give them clues of his whereabouts.
“It’s not in his desk,” says Alexander. “I’ve looked through all the drawers twice.”
“I’m almost done looking through the bookshelves,” Molly responds.
“There’s a piece of paper sticking out of the last book on the top shelf,” says Alexander, pointing.
Molly reaches for the book. “It’s The Time Machine, by H.G. Wells.” Pulling out the piece of paper stored in the book’s pages, she adds excitedly, “It’s a list. A list of names!”
Alexander closes a drawer and rushes to her side. “Rene Descartes has been crossed out. So have a few other names.”
“This is Grandpa’s list of people to visit,” claims Molly.
“There isn’t a pattern to which names have been crossed out,” adds Alexander, examining the list.
“Let’s visit one of the people that Grandpa has not crossed out. How about Menelaus of Alexandria?” suggests Molly. “It says ‘Greek mathematician/astronomer’ beside his name.”
“I like his name,” says Alexander as he reaches for a math history book on Grandpa’s shelves. He looks through the index while Molly examines Grandpa’s list more closely.
Alexander opens the book to page 215 and reads.
Menelaus of Alexandria was a Greek mathematician and astronomer. One of the craters on the moon has been named “Menelaus” in his honour. He was born around the year 70, and spent his youth in Alexandria, Egypt. Then he lived in Rome. Menelaus was the first mathematician to recognize the great circles on a sphere as being similar to straight lines on a flat surface. His only surviving book, Sphaerica, deals with the geometry of triangles formed by arcs of great circles.
“What are ‘great circles’?” asks Alexander.
“They are the biggest circles you can draw on a sphere,” responds Molly. “Like the equator or a line of longitude.”
“So they are the smallest circles through which you can fit the Earth?” asks Alexander.
“That’s a neat way of looking at them, Alexander,” suggests Molly.
“Why are the great circles like straight lines?” asks Alexander as he walks over to Grandpa’s globe. “They don’t look straight to me,” he says pointing to a line of longitude.
“Let’s see,” says Molly as she sits at Grandpa’s desk and Googles ‘great circles’ using his laptop.
A few moments later, Molly calls Alexander over. “Look at this video of Megumi Harada. She’s a math professor at McMaster University. She says that ‘straight’ can be defined as the shortest path between two points. She uses string to show that the shortest path between two points on a sphere is always along a great circle.”
Molly takes another look at Grandpa’s list. “Let’s visit Menelaus of Alexandria,” she suggests.
“OK! But let’s dress up as Romans so we fit in.”
“How do we do that?” asks Alexander.
“We have tunics and sandals from last year’s school play,” says Molly. “I know where they are. Come with me.”
When Alexander and Molly return to Grandpa’s office, they look like Roman children.
Alexander sets Grandpa’s pocket watch to the correct time and places it on the page about Menelaus of Alexandria.
A thick fog quickly descends around them. Then an opening appears, like a gate.
Through the gate, Molly and Alexander see an evening country scene. On the top of a hill, a man is standing, pointing something towards the sky. There is a young girl standing beside him.
Molly grabs the
booklet and jumps through the gate. Alexander follows, clutching Grandpa’s
pocket watch tightly in his hand.
2. THE STARGAZERS
Molly and Alexander land on the side of the hill, a few metres behind the man and the girl, on a patch of parched grass. The air is hot and dry. There is not a cloud in the sky.
“Look at all of the stars!” says Molly excitedly.
“I’ve never seen the stars so bright,” responds Alexander.
The man on the hill notices their presence. “Are these friends of yours?” he asks the girl standing beside him. He gestures for Molly and Alexander to join them.
Molly and Alexander walk to the top of the hill. “My name is Alexander, and this is my sister Molly,” Alexander says.
“I am Menelaus. This is my daughter Iphigenia,” says the man.
“Hello,” says Iphigenia. “The occultation is about the start,” she adds, pointing to the sky.
“What is an occultation?” asks Molly.
“It’s when one heavenly body hides another. The moon is about to pass in front of one of the stars in the constellation of Scorpius.”
Molly looks to where Iphigenia is pointing. “Is the Scorpion’s tail on the left?” she asks.
“Yes, Molly,” responds Iphigenia. “The moon will pass in front of the bright star on the Scorpion’s left claw. It’s almost there. See?”
All four watch as the shady part of the moon slowly extinguishes the light of the star.
“Wow!” says Alexander. “I’ve never seen this before. How often does it happen?”
“Several times each year,” responds Menelaus.
“Father,” says Iphigenia, “the bright sliver of the moon is light from the sun. But I can also faintly see the rest of the moon. Where does that faint light come from?”
“I think it comes from the Earth, Iphigenia,” responds Alexander.
“How can that be?” wonders Iphigenia. “The Earth is not a sun. It has no light.”
“I think it’s the sun’s light reflecting off of the Earth,” adds Molly. “Just like the sun’s light reflecting off of the moon reaches the Earth and helps us see at night.”
“That’s a very clever observation, Molly and Alexander,” says Menelaus. “You will make excellent astronomers.”
“What about me, Father?” asks Iphigenia.
“You are already an astronomer, Iphigenia,” he responds as he puts his arm around his daughter.
“Look!” says Iphigenia pointing. “The star is becoming visible again.”
The four stargazers watch the night sky as the hidden star reappears behind the moon.
Molly feels Alexander tapping on her shoulder. He is trying to get her to look behind them. Walking up the hill and looking at the night sky they had not noticed the silhouette of the city behind them.
“That’s Rome!” says Molly. “There’s the Coliseum.”
“We’re standing on one of the hills of Rome,” adds Alexander.
“Where are you from?” asks Iphigenia, puzzled.
“This is our first time in Rome,” explains Molly.
“We’re looking for our grandfather,” adds Alexander. “His name is Timothy Harley. Have you seen him?”
“No,” replies Iphigenia.
“It’s late at night and you should not be roaming the countryside on your own,” says Menelaus. “Come to our home,” he points to a villa further down the hill.
Alexander looks at Molly. She nods in agreement. “Thank you. We’d love to come,” responds Alexander.
As they walk down the hill, Iphigenia squeezes between Molly and Alexander and takes hold of their hands. “Let’s run ahead,” she says.
Two torches light the entrance to the villa. As they enter, Iphigenia says, “Come to the atrium and see my fish.”
The atrium is a large rectangle inside the villa, with no ceiling. In the centre of the atrium there is a rectangular pond, with plants and flowers all around.
“This is beautiful,” says Molly.
“It’s my favourite part of the house,” responds Iphigenia. “Except when it rains,” she adds.
“Look at all the fish!” exclaims Molly.
As Molly and Iphigenia disappear down a hallway, Alexander walks over to a marble sphere held up by three female statues. As he places his hand on the sphere, Menelaus walks up beside him.
“Is your grandfather an astronomer?” asks Menelaus.
“He is a mathematician, like you, and a poet,” responds Alexander.
“I don’t remember telling you that I am a mathematician,” notes Menelaus.
Alexander wonders whether he should tell Menelaus that he and Molly are from another time. He decides not to share that information yet. “You are Menelaus of Alexandria. I have heard about you from my grandfather. You wrote the book, Sphaerica,” he replies.
“Really?” asks Menelaus, surprised.
“Yes. You are the first mathematician to see the great circles on a sphere like straight lines on a flat surface,” adds Alexander.
“I have been thinking about writing Sphaerica,” says Menelaus, looking intently at Alexander, “but I have not written it yet.”
“Oh,” says Alexander, blushing.
“Come and sit by the pond and tell me about yourself and about your grandfather,” says Menelaus.
Alexander sits beside Menelaus and recounts the story of his Grandpa, his magic pocket watch, and their attempts to find him.
Menelaus listens with interest. “You and your sister are very brave,” he comments. “I’m afraid that I can’t be of help. I have not met your grandfather.”
“I’m sorry I tried to deceive you,” apologizes Alexander.
“You have done nothing wrong, Alexander.”
“Thank you, Menelaus.”
“Tell more about Sphaerica,” winks Menelaus.
“It’s about the geometry of triangles drawn on a sphere,” says Alexander pointing to the marble sphere.
“Triangles drawn using arcs of great circles?”
“Yes,” replies Alexander.
“One thing I’ve noticed,” adds Menelaus, “is that the sum of the angles in triangle, when drawn on a sphere, is greater than 180 degrees. Also, this sum can change depending on how the triangle is drawn.”
“It can?” asks Alexander puzzled.
Menelaus walks back to the marble sphere. “Look,” he says to Alexander. “If we draw a great circle like this, and then two other great circles at right angles to it, you have 180 degrees just in the two angles at the base.”
“The angle at the top can change by moving the two great circles closer or further away from one another,” offers Alexander.
“Exactly,” agrees Menelaus.
Just then Iphigenia and Molly walk into the Atrium. “Father,” says Iphigenia, “Our evening meal has been prepared. Please join us in the dining hall.”
4. WHEN IN ROME
Alexander’s mouth gapes in surprise as they walk into the dining hall. A set of tables, about the height of coffee tables, but much longer, are arranged in the shape of a C. They are covered with various foods. Against the tables are cushioned couches for people to recline on.
“Come,” says Menelaus to Alexander. “When in Rome, do as the Romans do.”
Menelaus sits on a couch as a servant approaches with a basin of water. The servant proceeds to wash Menelaus’ feet. Then, using a smaller basin offered by the servant, Menelaus washes his hands.
Alexander’s cheeks flush red with embarrassment as the servant washes his feet. Alexander washes his hands and dries them on a towel hanging on the servant’s forearm.
“This is my wife, Zoe,” says Menelaus as a beautiful woman enters the room.
“I’m so happy to have you join us for our evening meal,” says Zoe as she strokes Alexander’s cheek. She then reclines on the same couch as Menelaus.
“I’m happy to be here,” replies Alexander. “There is so much food,” he adds.
“I hope you are hungry,” smiles Zoe.
“I am,” says Alexander.
Alexander sits awkwardly on a couch beside Molly and Iphigenia. Servants walk in and out of the dining room, bringing new platters of food.
“The Romans don’t use forks,” whispers Molly. “Use a knife or a spoon to pick up food. Or just use your hands.”
“Are you sure?” asks Alexander.
“I’m sure,” replies Molly. “But wait for Menelaus and Zoe to start eating before you eat. It’s not polite to start eating before the hosts.”
The meal has several courses. Menelaus and Zoe discuss the events of the day. Iphigenia and Molly talk and giggle as if they have been best friends for a long time. Alexander eats. And eats. And eats some more.
5. WHERE PARALLEL LINES MEET
“Alexander,” says Zoe, “Menelaus tells me that you’ve been talking about lines on a sphere.”
“We have,” says Alexander.
“Perhaps you can help me with a friendly disagreement that Menelaus and I are having.”
Alexander looks at Zoe, then at Menelaus, who nods his head at him and winks.
“You see,” continues Zoe, “Menelaus is famous for his study of triangles on a sphere. But what about parallel lines on a sphere?”
“There are no parallel lines on a sphere,” interrupts Menelaus.
“Then there are no triangles, either,” retorts Zoe. Menelaus frowns.
Zoe picks up an apple. “Look,” she says, holding a knife in her other hand. “Here is a great circle,” she adds as she etches a horizontal circle on the apple. Then etching a vertical circle, she asks Alexander, “What is the angle formed by these two great circles?”
Alexander considers this for a moment. “Is it 90 degrees?” he asks.
“Exactly,” concurs Zoe. “Now look,” she says as she draws another vertical great circle. “Here’s another line that crosses at 90 degrees.”
“Oh, I see,” says Alexander. “I’ve learned in school that if two lines cross another line at the same angle, then the two lines must be parallel.”
“That is true, Alexander,” says Menelaus. “Euclid discovered this many years ago. But it only applies to flat surfaces.”
“You are so enthralled by Euclid that you are missing my point, my love,” responds Zoe with a smile.
“So, on a sphere,” adds Alexander, “parallel lines meet.”
Zoe gets up and walks over to Alexander. She holds his face in her hands and plants a loud kiss on the forehead. “I love you, Alexander!” she smiles. “This is what I’ve been telling Menelaus, but he cannot hear me.”
Alexander smiles shyly as Menelaus reaches for the apple. “I’m starting to see your point,” he says. “But what about Euclid’s Parallel Postulate?”
“Euclid was a great mathematician,” replies Zoe. “But keep in mind, Menelaus, that he spent his life studying the geometry of flat surfaces while living on a sphere.”
“So Euclid is wrong?” asks Menelaus.
“I think he’s wrong that the geometry of flat surfaces is all that there is,” says Zoe. “And you have already proven this, even though you haven’t yet admitted it, with your triangles on a sphere.”
“You see, Alexander,” says Menelaus, “history might
remember me for my mathematics, but it’s really Zoe’s ideas that create the
After dinner, Alexander tells Menelaus that it’s time to go home. “I’ll walk Molly and Alexander back home,” says Menelaus to Zoe and Iphigenia.
“Can I come too?” asks Iphigenia.
“It’s very late, Iphigenia,” says her mother.
“Will you come back soon?” pleads Iphigenia.
“I would like that very much,” says Molly as she hugs her.
They walk with Menelaus back to the top of the hill, where they first met.
“I hope you find your grandfather soon,” he says.
Alexander is not sure of the time, so he slowly turns the hands of Grandpa’s pocket watch. As the time reads 11:30, a thick fog descends around them again. Then the gate appears.
“It was a pleasure to meet you,” says Alexander.
“Thank you for inviting us to your home,” adds Molly as they jump through the gate.
As Molly and Alexander land back in Grandpa’s office, they hear their mother calling them. “Dinner is ready. It’s your favourite. Italian!” calls Mom.
“Oh, no!” exclaims Alexander, holding his tummy. “I can’t eat another thing.”
7. A PARALLEL LINES RIDDLE
An animated video with a riddle about parallel lines.
8. PARALLEL LINES RIDDLE in GRADE 2
Grade 2 students used globes of the Earth to investigate the parallel lines riddle, shown below.
Their comments were used to make lyrics to a song, which is in the animated video below.
9. MATH IS BIG, KIDS’ MINDS ARE BIGGER
We learn in school that parallel lines never meet.
A couple of millennia ago, Euclid tried to prove that parallel lines never meet. He was not able to. Neither were mathematicians after him.
The problem was that “parallel lines never meet” is an assumption, and not a theorem.
If we assume that we are talking about lines on a flat surface, then parallel lines do not meet.
However, if we make a different assumption, like lines on a spherical surface, parallel lines can meet. Like the lines of longitude.
The animated video below dramatizes 2 different student experiences with parallel lines. Which experience do you prefer?
10. MATH EXTENSIONS
Try the following activities along with Molly and Alexander. Have fun!
10.1 The Shortest path
“Here’s a puzzle, Alexander,” says Molly, unrolling a flat map of the Earth. “What is the shortest path between Vancouver, Canada and London, England?”
“They both are at approximately the same latitude,” comments Alexander.
“So, what do you think?”
“The simple answer seems to be to travel along the line of latitude,” replies Alexander.
“Is that your answer?”
“I think you are trying to trick me, Molly,” says Alexander.
“What do you mean?”
“You are asking the question while showing me a flat map of the Earth. On this map, if I take a string and stretch it taut between Vancouver and London, then the shortest path would be along the line of latitude.”
“The shortest path between two points is a straight line, Alexander. And the line of latitude looks like a straight line to me,” replies Molly smiling.
“But only on a flat map,” responds Alexander, as he walks over to Grandpa’s globe.
“So what’s your answer, Alexander?”
“I think if we join the two cities on the globe with a string, pulling the string tight, to make it taut, then we’ll find the shortest path,” responds Alexander.
“I like your idea, Alexander,” says Molly. Let’s try it,” she adds.
10.2 Great Circles
Molly and Alexander are sitting at the kitchen table. “Here’s a puzzle for you, Molly,” says Alexander.
“I’m ready,” says Molly.
“One way to describe a great circle,” continues Alexander, “is the circle we get when we cut a sphere in half.”
“Like this orange,” says Molly reaching for the fruit platter. Molly takes an orange and uses a knife to slice it in half.
“Exactly,” responds Alexander.
“What’s the puzzle?” asks Molly.
“How many other ways can you define a great circle?”
“Hmm. That’s a good one, Alexander.”
Molly ponders Alexander’s puzzle for a minute. “A great circle is the biggest circle you can draw on a sphere.”
“Like a line of longitude or the equator,” adds Alexander.
“It’s also the smallest circle through which the sphere can fit through,” adds Molly. “We talked about this before going to Rome.”
“I remember,” says Alexander.
“Is there another way?”
“I found two other ways. Here’s one hint: Where is the great circle’s centre?” Alexander pauses, then adds, “Here’s another hint: What path would a bug walk on a sphere, if it walked straight?”
10.3 Parallel Lines on a Sphere
“My turn to give you a puzzle,” says Molly. “Which are more parallel, lines of latitude or lines of longitude?”
“I like these fuzzy questions,” responds Alexander.
Alexander ponders the puzzle as he examines Grandpa’s globe. “The lines of latitude never meet, and they stay the same distance apart,” notes Alexander.
“Is that your answer?”
“But,” continues Alexander, “they are not straight lines. Straight lines on a sphere are the great circles.”
“What about the lines of longitude?”
“Well, they are great circles,” responds Alexander. “And, they all cross the equator at 90 degrees. I think this means that they are parallel to one another.”
“So, your answer is the lines of longitude?”
Alexander thinks about this for a moment. “The problem is that the lines of longitude meet at the poles. So, maybe they are not parallel.”
“What’s your answer, then?”
“I’m not sure,” responds Alexander. “What do you think?”
“Hey! This is my puzzle. You are the one who has to solve it,” replies Molly playfully.
“OK. Let me think about it some more,” smiles Alexander.
10.4 Triangles on a Sphere
“Menelaus talked to me about triangles on a sphere,” says Alexander.
“Was this before dinner, when I was with Iphigenia?” asks Molly.
“Yes. Did you know that, on a sphere, the sum of the angles of a triangle is more than 180 degrees?”
“Really? The sum of the angles of a flat triangle is always 180 degrees. What is the sum on a sphere?”
“I’m not sure. Let’s take a look at the globe,” responds Alexander.
Molly and Alexander examine the globe. They use their fingers to outline various triangles.
“If we use the equator as the base,” says Alexander, the two angles at the equator are each 90 degrees.”
“So, if we add the third angle, the sum has to be more than 180 degrees,” offers Molly.
“But how much more?” asks Alexander. “What is the biggest sum that we can get for the angles of a triangle drawn on a sphere?”
“Let’s imagine the two sides of the triangle moving further apart or coming closer together,” says Molly.
“So the sum of the angles in a triangle drawn on a sphere is not a single number,” comments Alexander.
“Exactly. The sum of the angles changes depending on the size of the angle at the vertex,” offers Molly.
11. SWEET PARALLEL LINES
Musical performance for kids and parents at the Brooklin Public Library. Lyrics by Veronica Smith.
12. MORE ABOUT PARALLEL LINES
The story Where Parallel Lines Meet and the activities that follow it are not grade specific. They can be explored by students at a variety of grade levels, and at different levels of complexity.
12.1 Life on a sphere
We live our lives on a sphere (or an approximate sphere). However, we often forget this in our everyday living and we tend to behave as if our world is flat. In fact, most of the geometry we study in school is Euclidean geometry, or the geometry of flat surfaces.
From a mathematical point of view, the geometry of a flat surface is quite different from the geometry of a sphere. Noticing and exploring the differences, as well as the similarities, helps children develop a richer understanding of geometry and a better appreciation of the beautiful and surprising nature of mathematics.
To explore the geometry of a sphere, it is helpful to have some physical models, like a globe or a ball, or a balloon. Balloons are quite useful in that you can use a pen to write on them. So you can use a balloon to draw and explore lines and triangles.
12.2 Straight lines on a sphere
You might be thinking that there are no straight lines on a sphere. However, mathematics is most interesting when it’s fuzzy, rather than black and white.
Looking at a sphere from the outside, from an extrinsic point of view, there are no straight lines on a sphere. However, from a life-on-a-sphere point of view, there are lines that feel straight and can in fact be considered as being straight.
Great circles, like lines of longitude and the equator, are lines that are ‘straight’ on a sphere. Here are some ways of defining a great circle:
- If a bug on a sphere walks straight ahead, it will walk the path of a great circle.
- The shortest path between two points on a sphere.
- The biggest circle that can be drawn on a sphere.
- The smallest circle through which a sphere will fit.
- A circle on a sphere that has the same centre as the sphere.
- The circle you get when you cut a sphere in half.
12.3 Parallel lines on a sphere
Are there parallel lines on a sphere?
One definition of parallel lines is “two straight lines that never meet.”
Lines of latitude appear to be parallel because they remain the same distance apart as they travel around the globe. However, lines of latitude are not straight lines on a sphere—they are not the shortest paths between points on a sphere—so they do not qualify to be parallel.
Lines of longitude, on the other hand, are straight lines on a sphere. Also, we might notice that lines of longitude cross the equator at 90 degrees. We know that on a flat surface, if two lines both cross another line at the same angle, then they are parallel. However, lines of longitude meet at the poles.
12.4 Triangles on a sphere
On a flat surface, the sum of the angles of a triangle is 180o. What about triangles drawn on spheres?
Imagine the triangle shown on the right. It is formed by the equator as the base and two lines of longitude for the other sides.
The smallest sum we can get is just over 180o, by making the angle at the vertex very small.
What is the largest sum? It turns out to be just under 540o. How is this possible? Here’s how we might do this: (1) Move the lines of longitude apart, until they are almost on opposite sides of the globe. This will make the angle at the vertex almost 180o. (2) Then, rotate the equator around the globe until it almost touches the South Pole. Then, each of the angles in the triangle will be almost 180o.