GeoGebra Lab #4: Non-Euclidean Geometry

Author:Dr. Abigail Brackins, soundmanbrad

This GeoGebra Lab is due at midnight on May 11 (Wednesday). I will make comments and give feedback in the evening of May 8 (Sunday), so you may want to have your lab done by this date.

Non-Euclidean Geometry

We've spent this semester studying the geometry of flat planes and straight lines, starting with Euclid's undefined terms (like a point: "that which has no part") and postulates (for example, "Between any two points we can draw a straight line segment"). This geometry is very useful for solving problems about many of the basic physical structures in our world-- but it's not the only game in town! Below you'll find some examples of different types of geometry.

Spherical Geometry

This type of geometry is very necessary for thinking about distances and areas on the globe. For small distances and areas, we can assume that the land is flat, and use Euclidean geometry to measure, but for very large distances, the curvature of the earth is an important consideration. Take a look at the triangle on the globe below, with a vertex on the north pole, another in the Indian Ocean, and a third in Indonesia:

In Euclidean Geometry, the sum of the measures of angles in a triangle is ALWAYS 180 degrees. In spherical geometry, this is not the case: The triangle pictured has angle sum 230 degrees. (On the other hand, the zoomed-in triangle in Japan has an angle measure sum of 180 degrees because, zoomed in that far, the curvature of the earth makes very little difference to the appearance of triangles.)

Question #1

Make a conjecture (an educated guess) about the sum of triangle measures in a triangle on a sphere.

Poincaré Disks

This is an example of a made-up geometry system. While this system doesn't have a whole lot of practical applications, it is an important source of new mathematical theory. In a Poincaré disk, lines look like arcs of circles, and must always meet the boundary of the disk at a 90 degree angle. The applet below shows the line between two points A and B in a Poincaré disk. The points C and D are the intersection of the line AB with the boundary of the disk. Move A and B around to see how that changes the position of the line. You can also make a new line with the special "hyperbolic" tools under the wrench to the right of the tool menu. Try the "Hyperbolic Perpendicular Bisector" tool under the second wrench.

The famous mathematician/artist M.C. Escher used the Poincaré disk as the basis for many of his tessellations. Here are a few examples of this type of artwork, which Escher called "Circle Limits." Can you see the types of lines from above in these works?

Question #2

Is the Parallel Postulate true in the Poincaré disk? That is, given the hyperbolic (curved) line AB and a point C in the disk not on AB, is it true that there is EXACTLY ONE hyperbolic (curved line) through C that does not intersect with AB? Experiment with the applet before you answer.

Fractal Geometry

Fractals are pictures that contain copies of themselves. From Wikipedia: "A fractal is any pattern, that when seen as an image, produces a picture, which when zoomed into will still make the same picture." To understand what this definition means, let's look at some examples.

Koch snowflake

This fractal is made from an equilateral triangle. In each step, each edge gets a new triangle-shaped bump on it. This process is repeated infinitely many times.

The final result winds up looking something like this:

(This is several Koch snowflakes pictured together.)

Dragon curve

This fractal is made by repeatedly rotating an image by 90 degrees. Each stage of the fractal's growth is twice as big as the stage before it.

Fractals in the movies

This is my favorite example of a fractal, from the Mel Brooks movie Spaceballs.

Fractals are super important in the natural world. To learn more, watch this AWESOME project from a 4th grade student.

Fractals in Nature

Question #3

What's your favorite example of fractals in nature?

Sierpinski triangle

Your final activity will focus on a very well known fractal called the Sierpinski triangle. In this fractal, we start with a filled in triangle and punch a smaller triangle out of the middle. Then, we repeat that process for each of the three remaining shaded-in triangles. The fractal is the image that we would get from repeating this process forever.

Question #4

At each stage, how many shaded-in equilateral triangles are there? (Your answer should include four numbers, one for each of Stage 0 - Stage 3.) How many triangles will there be at Stage n? (That is, find a formula in terms of the variable n that represents the number of shaded triangles at each Stage.)

Question #5

Suppose that the area of the original (Stage 0) triangle is 1 square unit. What's the area of each smaller component triangle in each subsequent stage? (For example, the area of one shaded triangle in Stage 1 is 1/4.) What will the area of one small shaded triangle be at Stage n? (That is, find a formula for the area of one small triangle, in terms of n.)

Question #6

What is the total shaded area at each stage in the Sierpinski triangle? What will the total shaded area be at Stage n? (That is, find a formula for the total shaded area in terms of n.)