Revolution (Enneper 2)

Another special feature of Enneper’s surface is that it is intrinsically rotationally symmetric. This means that if you had a marble version of it, and a paper copy (made of curved paper, that is) sitting on top of it, you could rotate the paper copy smoothly by 360 degrees just by bending the paper, but without tearing or stretching. Amusingly, there is no truly rotational symmetric in Euclidean space that is isometric to Enneper’s surface.

Enneper5
Enneper’s surface shares this surprising feature with a few other minimal surfaces, like the one with five ear lobes instead of just two above. By the way, that the lobes touch is an artistic choice. The surface extends indefinitely, intersecting itself, which has led to its partial demise. There are also intrinsically rotationally symmetric minimal surfaces with two ends, like the plane and catenoid, or the more amusing one below with a planar end and an Enneper style end at the center.

PlanarEnneper5

This rotational symmetry gets lost when you stack two equal Enneper surface on top of each other, like so:

Doubleenneper7

In mathematics, when you give up something, you typically can gain something else. In this case, you gain flexibility. You can change the distance between the two wiggly Enneper ends and bring them so close together that cleaning in between becomes impossible. The version below would make an interesting wheel. Use at your own risk.

Wheel

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