Projection
One cone, four curves
The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.
A sphere is a plane plus one point
Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.
The dot product is a shadow
Multiply the matching coordinates and add them up. That rule explains nothing, and it hides the fact that the answer is a length — how far one arrow reaches along another, times how long that other one is.
Three moves, and what they cannot undo
A knot is a closed loop of string, and two knots are the same if one can be wiggled into the other. Reidemeister reduced all possible wiggling to three local pictures — which is what makes it possible to prove that a knot is knotted.
Named alongside it
The objects these essays reach for when they reach for this one.
AstrolabeBijectionCircle preservingConformal mapContinuityCosineCrossing numberDandelin spheresDegenerate conicDot productEllipseEquivalence