Mathematics · the experiment
Find the only round point
Points where a surface curves the same way in every direction, lines that turn around them, two conjectures that fall together. Four questions, from beginner to expert, on the surface with a single umbilic.
1 · Count the umbilics
Beginner
How many umbilics does the closed, smooth, strictly convex surface built by the authors have?
2 · Work out the number of turns
In the authors' local construction, the directions of strongest bending turn 1 + k/2 times around the centre, for every positive whole number k.
Intermediate
What is the index of the umbilic for k = 6?
3 · Tell the two conjectures apart
Picture a closed convex surface with exactly two isolated umbilics, one of which has an index of 3/2.
Expert
Which of the two conjectures does it contradict?
4 · Explain the umbilic at the north pole
The surface is a sphere whose support function receives a small multiple of the damped seed. Everywhere except the north pole, the seed rules out an umbilic.
Expert
Why is the north pole itself an umbilic?
Round point found
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