Its existence shows that, unlike the one-dimensional curve-shortening flow (for which every embedded closed curve converges to a circle as it shrinks to a point), the two-dimensional mean-curvature flow has embedded surfaces that form more complex singularities as they collapse.
[1] However, as early as 1990, Gerhard Huisken wrote that Matthew Grayson had told him of "numerical evidence" of its existence.
[1] By considering the behavior of geodesics that pass perpendicularly through this axis of reflectional symmetry, at different distances from the origin, and applying the intermediate value theorem, Angenent finds a geodesic that passes through the axis perpendicularly at a second point.
Other geodesics lead to other surfaces of revolution that remain self-similar under the mean-curvature flow, including spheres, cylinders, planes, and (according to numerical evidence but not rigorous proof) immersed topological spheres with multiple self-crossings.
[1] Kleene & Møller (2014) prove that the only complete smooth embedded surfaces of rotation that stay self-similar under the mean curvature flow are planes, cylinders, spheres, and topological tori.