Suppose you were an intelligent, small animal (such as an ant), and you lived on

Question

Suppose you were an intelligent, small animal (such as an ant), and you lived on the surface of a beach ball. Also suppose that you only were aware of two dimensions (forwards and backwards; left and right), such that the concept of up and down did not exist. You decide to test whether you live on a curved surface or whether your surface is flat, like the surface of a large billiards table. Rather than walking in a straight line for a very long time and seeing if you end up where you started, you go out and buy a protractor. Then you draw three lines, making a big “triangle.” With the protractor, you measure the angles of the triangle and add them together. If the angles add up to 180 degrees, you conclude your world is “flat,” and if not, you conclude your world is “curved.” (a) Since you live on a beach ball, will the sum of the three angles be greater than or less than 180 degrees? (b) Discuss why you might come up with the wrong conclusion about the curvature of your world if you only make small triangles

Explanation / Answer

Curved "up" and curved "down" are usually referred to as "positive" and "negative" curvature.

If we draw a triangle on a positively curved surface, then the sum of its angles is greater than 180 degrees.

If we draw a triangle on negatively curved surface, then the sum of its angles is less than 180 degrees.

(a) Since we live on a beach ball, the sum of the three angles will be less than 180 degrees.

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