Transcription
[Music] an aversion turns a mathematical sphere inside out without ripping or pinching its surface. Our sphere Avion is different from earlier ones. It is computed automatically by minimizing the elastic bending energy.
To turn a sphere inside out, we bend it up, increasing energy until the immersed halfway stage, and then let it relax back down to the round [Music] sphere. Let's watch this aversion again without transparency. The outside of the sphere is painted yellow to red, while the inside is blue to purple. The computer uses polyhedral approximations to the smooth surfaces. The triangulation is updated as needed during the aversion. If we remove the center of each triangle, we can see the interior structure. Watch the double locus, the white curves along which the immersed surface intersects [Music] itself.
The computation actually starts at this halfway model, which is an immersed sphere with a four-fold symmetry that interchanges inside and out. We use second-order information about the energy to push away from this saddle point. Then the evolution proceeds downhill by gradient descent. The aversion constitutes an optimal route over the lowest pass from one side of the energy mountain range to the other.
Watch the double locus more closely. As the North Pole pushes past the South Pole, the gastr related sphere ceases to be an embedding, and the first double curve is born. As two loes near the neck penetrate each other, a second double curve is created. The aversion has two-fold symmetry throughout. Ribbon neighborhoods show how one curve must double back on itself so that at two symmetric places, three arcs converge and spawn a pair of triple points. The four triple points become the vertices of a tetrahedron that shrinks right through itself, forming a quadruple point. Halfway through the Minimax inversion, the whole process is inverted in time with a 90° twist. The triple points disappear in pairs, and then the two double curves shrink and vanish.
The critical events we have been describing, once called catastrophes, are moments when the regular homotopy passes through immersions which are not generic. These isolated events are modeled by a horizontal plane rising through a minimum, a maximum, or a saddle point of a second surface patch. When the patches are tangent, we see the creation, annihilation, or reconnection of a double curve. At the halfway stage, a reconnection happens on the symmetry axis opposite the quadruple point. This was called the ismos point in the first computer animation of a sphere version by Nelson Max. Four additional reconnections occur at the so-called ears of the halfway model. Two ears open just as the other two ears are closing.
When three surface patches meet along a common tangent line, a pair of triple points is created or annihilated. A quadruple point happens when four surfaces meet. The halfway model is an elaborately immersed sphere with a pleasing four-fold symmetry that reverses the orientation, interchanging the blue and orange sides. We use the surface of least bending energy with such symmetry. However, Wilmore's bending energy is invariant under conformal transformations of space. This one-parameter family of conformally equivalent surfaces always has the desired symmetry. Any one of these could have been our halfway model.
As the quadruple point is sent to infinity, we arrive at a cus minimal surface, which has an explicit vust representation. The surface has four flat ends which cut the sphere at infinity in the pattern of a cuboctahedron. Continuing the conformal transformations, we send the ismus point to infinity. Now we have a surface with only two wavy ends with a quadruple point at the center. An immersed polyhedral sphere with 12 vertices used by Aeri and Da for their polyhedral aversion has the same four-fold symmetry. The evolver reduces energy, preserving the symmetry while selectively refining the triangulation, and we arrive back at our symmetrically immersed sphere of least energy, which is our halfway model. Note again the four ears where ismus events are happening.
To describe a sphere reversion, it suffices to simplify the symmetric halfway model until it becomes a round sphere. Moran discovered the simplest of such homotopies. Brach's computer program, the evolver, automatically finds for us the same strategy in this case, as well as for different symmetries. The Minimax version is the first in an infinite sequence, the so-called tobacco poucher versions of increasing rotational symmetry. Those of even symmetry, like this order four example, generalize our Minimax AV verion. Their halfway models reverse orientation by a rotational symmetry.
The halfway models for the aversions of odd symmetry are doubly covered projective planes. The order 3 aversion passes through Bo surface, which is an immersion of the projective plane in three space with a single triple point. The two oppositely oriented sheets compete to show their blue and orange colors, accounting for the twinkling facets. When we push away from the energy saddle, the sheets separate. Boy's propeller-shaped double curve is now covered four times with a little growing cube where the triple point had been. Our ensemble of spheres join in performing 2, 3, 4, and five-fold symmetric [Music] inversions. [Music]