# Flatland Exploration

From EscherMath

**Objective:**
Get a better understanding of the geometric concepts one encounters in Flatland.

Note: Read Flatland outside of class and come to class prepared to discuss these topics.

- When was Flatland written? Who was E. A. Abbott?
- Why do Flatland houses look like pentagons, not triangles or squares?
- How do Flatlanders recognize each other?
- Describe the class structure of Flatland society.
- Are there any irregular shaped Flatlanders? How are they treated, and what do they correspond to in human society?
- What do you think of the portrayal of women in Flatland? Is it satire or sexism?
- Why can’t the King Of Lineland ever touch his wives?
- Mr. A. Square is visited by the Sphere from spaceland. What if he was visited by a Cube? What would A. Square see? Explain.
- Explain how A. Square, with the help of the Sphere, could escape from his jail cell.
- Flatland is inhabited by polygons. “People” recognize each other by the angles at their vertices.
The interior angle of a regular n-gon is given by $ \frac{n-2}{n} 180 $

a. Complete the following table:Angles of Regular Polygons (Regular) Polygon interior angle 4-gon (square) 5-gon (pentagon) 6-gon (hexagon) 7-gon 8-gon 9-gon 10-gon 100-gon 500-gon 600-gon 700-gon

b. As the number of sides goes up, it becomes harder and harder to distinguish the polygons. Explain why, using the table above. c. As the number of sides of the polygon increases, they look more and more like circles. If our square broke the rules and “felt” a circle, what angle do you think he would detect? Explain. - When the Sphere removed A.Square from Flatland, what did A.Square see?
- Suppose a 4-dimensional being removed you from this space. What would you see if you looked "down" on our 3-dimensional world?

**Handin:**

After class discussion, hand in carefully written explanations to these questions.