Invariant
Named by 5 essays across 5 fields — each of them below, with the objects they name alongside it.
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.
Eight ways to leave a square alone
A square can be picked up and put back so that nothing looks different. There are exactly eight ways to do it, and the number is not asserted here — it is what a search through all twenty-four relabellings of the corners comes back with.
The rule that forgets where it came from
A walk between a few states, with the next step decided by the current one and nothing else. Run it long enough and the starting point stops mattering — but only when two conditions hold, and both of them have a picture in which they fail.
Area by counting dots
Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.
Nine points on one circle
Three midpoints, three feet of altitudes and three more midpoints. Nine points defined in three unrelated ways, on an arbitrary triangle, and all nine sit on one circle — checked here on two hundred and forty triangles as well as on the drawn one.
Named alongside it
The objects these essays reach for when they reach for this one.
CounterexampleCounting argumentSymmetryAltitudeAreaBoundaryCircleClosureConvergenceCrossing numberCyclic groupDihedral group