Symmetry

1. Symmetry

2. How do we define symmetry • An object invariant under a transformation • Rotation • Translation • Time Lapse • Reflection • How many symmetries does a cube have? 48

3. Curie’s Principle • If certain causes produce certain effects, then the symmetries of the causes reappear in the effects produces • If certain effects reveal a certain asymmetry, then this asymmetry will be reflected in the causes that give rise to them

4. An airplane • An airplane has bilateral symmetry • Does half a plane against a wall reproduce the flow past the airplane?

5. Bifurcation • Symmetry “breaks” by creating an assortment of anti-symmetric states which sum to a symmetric state

6. Airplane Flying • Has 2 steady states just off direct • Switches between the two due to turbulence or other “imperfections” • The overall “math” is symmetric, though there are anti-symmetric solutions

7. Symmetry in Biology

8. Start as spherically symmetric egg Lose symmetry with division Regain symmetry with blastula Lose symmetry as a sphere with a dimple Gastrulation

9. Causes Difference of Pressure creates inward pressure Cell-Cell movement Location selection Breaking due to pressure will occur where walls are weaker Cell-Cell movement could be caused by selectively changing cell-cell adhesion Breaking of Symmetry

10. What actually happens? • Cells change their adhesion properties and move via “random” motion • Axis is selected via signaling molecules established at the time of fertilization

11. Conclusions • Small perturbations away from symmetry can cause substantial effects • Biology often just “pushes” an organism in the correct direction