parallel lines

PARALLEL LINES CAN MEET?

Alien comic strip:

Gork: Zork, Earthlings are taught that parallel lines never meet!

Zork: They'll be surprised when we remind them they live on a sphere, Gork.

Parallel lines run side-by-side, in the same direction.

How could they ever meet?

Think of train tracks.

Do they ever meet?

A riddle

Share this riddle with family and friends.

  • Zork is camping.
  • She steps out of her tent.
  • She walks south 1 km.
  • She walks west 1 km.
  • She sees a bear and gets scared.
  • She runs north 1 km.
  • … and arrives back at her tent!
  • How is that possible?
  • And, what colour was the bear?

Here’s a hint: look at this globe of the Earth.

If parallel lines never meet, then the Earth is flat, like a wall map.

Euclid

Parallel lines have a really interesting history!

The ancient Greek mathematician Euclid stated a parallel lines theorem that went something like this:

  • Draw a straight line, and a point near it.
  • There exists exactly 1 line that crosses the point and does not cross the first line.
  • The two lines are parallel.

Click on the image below to see a 3-D animation of our interview with Euclid (who claimed that parallel lines never meet)

What geometry do you live in?

What do you think of Euclid’s theorem?

How many “straight” lines can you draw through the point that don’t cross the line:

  1. no lines?
  2. one line?
  3. an infinite number of lines?

Surprise! The answer depends on the geometry you have in mind.

Below are images of 3 different geometries: Euclidean, Spherical and Hyperbolic. Can you match each of these with the three answers above?

The shortest path

One way to define a “straight” line is to say that it is the “shortest” path between two points.

Which lines on the globe create the shortest paths?

  • the lines of latitude?
  • the lines of longitude?
  • some other lines?

Imagine a bug walking “straight ahead” on the globe.

  • What path will it travel?
  • Could it be a line of latitude?
  • Could it be a line of longitude?
  • Could it be a different line?

Let’s consider an airplane flying from Tokyo to New York City. Would it fly along a line of latitude? Or would it take a different route?

Triangles on a flat surface


What is the sum of the angles in a triangle?

If you draw a triangle, cut off its 3 angles, and then put them together to form a combined angle, that combined angle would be 180 degrees.

Triangles on a sphere

We can draw a triangle on a sphere using 3 great circles (the equivalent of straight lines), such as a two lines of longitude and the equator, as shown below.


What is the sum of the angles in the triangle shown above?

What did you learn?

What did you learn about parallel lines?

  • What surprised you?
  • What math insights did you experience?

Share your learning with others, so they can experience math surprise and insight as well?