Math Chat: Mirrors and Roadways

Old challenge (Fred Wedemeier)

Why do mirrors reverse left and right and not up and down?

Answer

This is one of the most confusing questions I know, and there were some very ingenious wrong answers: "The mirror does reflect from top to bottom, but your brain recognizes that the shape should be right-side-up, and flips it for you."

"The image in a mirror is rotated about the vertical axis because a person, when turning to regard another person, rotates himself about the vertical axis." (But if we stood on our heads to look behind us, would mirrors suddenly start working differently?) "This is just the way your eyes see it. If your eyes were one above the other, the mirror would reflect up and down." (But if you close one eye, do mirrors work differently?)

Bill Cutler rightly explains that mirrors "don't reverse left and right; they reverse front and back. Stand facing north with a mirror in front of you. Wave your left (west) hand. The image in the mirror waves its west hand too, so there is no left-right reversal.

But, your nose is pointing north while the nose of the image is pointing south. "This is what makes you interpret your reflection's west hand as a right hand. See the excellent discussion in Martin Gardner's "The New Ambidextrous Universe" (Chapts. 1 and 3). (Other winners: David Dorman, William Hansen, John Kruschke, and Peter Yff.)

Tic-Tac-Toe

Tic-Tac-Toe was the subject of a Nov. 24 seminar at the Institute for Advanced Study in Princeton, N.J. You may think the subject would not interest mathematicians, since the standard tic-tac-toe game always ends in a tie. But what about playing in higher dimensions? If you play for 4 in a row in a 4x4x4 cube (with 64 little boxes to put Xs in), the first player can always win.

Prof. Jozsef Beck of Rutgers University talked about going for 5 in a row in a 5x5x5x5 4-dimensional hypercube (with 625 little boxes) or more generally going for n in a row in a huge d-dimensional hypercube. He proved that even for d much bigger than n, the second player can sometimes force a tie. Understanding game theory can also help us understand international political and economic competition for example.

New challenge

What is the shortest road network connecting the four corners of a square? What about the six vertices of a regular hexagon?

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