geometry's ironic assumption

I have emphasized over and over again to my Geometry students that "eyeballing" is never an acceptable justification for any argument. Today, I tried to make my stand a bit more compelling by presenting a series of optical illusions, a move to point out how unreliable our perceptions may be. Two lines that look congruent may actually be different lengths, or vice versa. An angle that looks like a right angle may not measure 90 degrees at all. Is that quadrilateral really a square? "Well, it looks like it is!" they tell me, exasperated with my unwavering prompts of "But do you know for sure?"



The reality is that our eyes and the sense that our brains try to impose on the world around us are completely unreliable. How we interpret the world may not be an accurate representation of the world. Or rather, what we "see" is simply that -- a representation. The images presented here are just two of a series of optical illusions I showed to my students, though these two most closely address the types of incorrect assumptions they might make with their work.

Ironically enough, students are allowed to make the assumption that a line is straight. Yet the following illusion suggests that even with straight lines, what we see is not actually truth.


Are the two red lines in the middle straight or curved?

geometry around the house

What comes to mind when you think of geometry? How do you define it? What is it that people study when they study geometry? How does an understanding of geometry inform our everyday experiences?



For this assignment, simply look around your house or neighborhood. Search for things that rely in some way on geometry for their function, design, or aesthetic appeal. For example, you might point out a manhole cover in the ground; the fact that it is circular in shape is very important to its function. On the other hand, pointing out a stop sign as an octagon does little to help us understand why, if at all, it is interesting.



Give me a sense of your approach to geometry by answering the questions that I posed at the start of this post. Then, list three things that you found, and for each, write a short paragraph about the ways in which you think geometry plays a role in its function, design, and/or aesthetic appeal.

Don't forget to check out the "Geometry at Home" gallery!