More thoughts about curvature and symmetry


Did D'Arcy Thompson have astigmatism?

Both the cornea and the lens focus light by means of their curvatures.

"Astigmatism" is a defect in vision which results from one or both of these curvatures varying with direction.

(For example, if the curvature of a person's cornea happens to be slightly larger in the dorso-ventral direction, as compared with the curvature of their cornea in the medio-lateral direction, then their vision will be astigmatic.)

Astigmatism might be caused by tension being stronger in one direction than another, or by Young's modulus differing with direction.

If tension couldn't differ as a function of direction, then astigmatism would not be possible.

Therefore, it would be ironic if D'Arcy Thompson's eyes were astigmatic.


Teleost fish oocytes are spherical, until they get fertilized. What does that tell you about tensions in their surfaces?

If you had a treatment that changed the strengths and directions of tensions in teleost oocyte surfaces, could you change their shapes?

Could you cause oocytes to become more cylindrical?

Could you cause bulges, or flat places, to come into existence?

Within a few minutes after fertilization, a hemispherical outward bulge forms in the area of the animal pole.

What change in tensions would be needed to cause such a bulge?

Suppose that the curvature of the bulge were exactly three times the curvature of the surface of the rest of the embryo. (What would that tell you about differences in curvature?

What if the curvature of the bulge were slightly larger in one direction than another? How would that effect the curvatures of the bulges?

Later in development, the enveloping layer forms. It is hemispherical.

What causes it to have a constant curvature?

Can you invent a way to change the curvature?


When solutions of a chemical rotate polarized light, what can you deduce about asymmetry of its molecular structure?

Is this phenomenon a case of Curie's Principle?

Do you suppose that Pasteur knew Pierre Curie?
Or that he knew about Curie's principle, and that it helped Pasteur to discover the relation between optical rotation and symmetries of crystals?