Magic Octagon Coin

The Magic Octagon involves two vectors (arrows) that share a fixed angle of 90°, and since each is imprinted on the two sides of a coin, only one arrow’s direction can be viewed at a time. The challenge is to determine the orientation of the second arrow when the first arrow’s orientation is changed, and this determination involves considering three separate rotations: two vertical flips, and one 45° rotation about a centered axis perpendicular to the face of the coin. It turns out that this is not something that comes easily to the brains of a majority of us, and while science confirms this deficit, the reason for it remains a current subject of research in cognitive psychology.

However, the fact that most are quite surprised by the outcome makes this “discrepant event” a valuable teaching tool in the geometry classroom-- virtually guaranteeing a raucous and robust classroom investigation as well as a memorable learning experience, and providing a confirmation that the angle between two vectors is indeed preserved under rotations.

Coin designed in collaboration with Raymond Hall (@physicsfun) and engraved by Shire Post Mint.