Patterns in Nature

1. Patterns in Nature Related to Software Design Patterns Peter JohanssonUndergraduate studentpetjo450@student.liu.seAmanuens at AIICS, IDApejoh@ida.liu.se

2. Introduction We have traced the origin of Software Design Patterns back to Alexander’s work on architecture. Can we learn even more by studying pattern formation in nature? • Patterns in nature • Symmetry • Symmetry-breaking • Patterns revisited

3. Books Philip Ball: The Self-made Tapestry Ian Stewart & Martin Golubitsky: Fearful Symmetry

4. Patterns in Nature • Bubbles • Waves • Bodies • Branches • Breakdowns • Fluids • Grains • Communities This is actually the table of contents from The Self-made Tapestry.

5. Patterns in Nature • Patterns are result of naturally occuring processes • One of the purposes of natural science is to build models of these processes • How can complexity arise from simplicity? • (Is this the inverse of Computer Science?)

6. Example How did the Zebra get its stripes?

7. Autocatalysis Activator A Diffusion Degradation + - Inhibitor B Activator-inhibitor model • A generates more of itself and activates B • B inhibits formation of A • A and B diffuses at different rates First proposed by Alan Turing in 1952.

8. Activator-inhibitor • Examples of activator-inhibitor systems. Light areas are dominated by one compound, dark areas by another.

9. Activator-inhibitor • Patterns formed by activator-inhibitor systems depend on the size of the system.

10. Activator-inhibitor • It is though that this process takes place in the embryo and thus forms a pre-pattern • It remains to be shown that this really is the process that gives the Zebra its stripes

11. Activator-inhibitor • This process may also explain the marks of other animals.

12. Symmetry • “Pattern” is a very loose term • Instead we look at a more formal property that objects in the world can have: symmetry • We mean “symmetry” in the mathematical sense • How do patterns relate to symmetry?

13. What is Symmetry? A symmetry of an object is a transformation that leaves it apparently unchanged. Rotation Reflection

14. Symmetry Groups • Closure: For all a, bG, the set G is closed under composition, i.e. ab, baG. • Associativity: For all a, b, cG, the composition is associative, i.e. (ab)c = a(bc) • Identity: For all aG there exists an element eG such that ae = a = ea. • Inverses: For each aG there exists an a-1G such that aa-1 = e = a-1a. A group is a nonempty set G with a law of composition satisfying these axioms

15. Symmetry Groups • Most common symmetry group: the group of rigid motions in two- and three-dimensional space (translation, reflection, rotation) • Time symmetry (e.g. periodic systems like the Earth and the Sun)

16. Symmetries in Nature • Animal bodies • Crystals • Soap bubbles • Flowers (e.g. Sunflowers)

17. More or less symmetry? Infinite number of rotations and reflections 24 rotations and 12 reflections

18. Symmetry-breaking • A falling drop of milk has circular symmetry… • …but after impact a ”crown” rises that only has 24 possible rotations.

19. Where does it go? • Falling drop: O(2) symmetryCrown: D24 symmetry • If we rotate the crown an arbitrary angle we get another crown with spikes in different places. • This ”new” system also has D24 symmetry. • Symmetry-breaking imposes an equivalence relation. It divides the symmetry group into subgroups.

20. Where does it go? • A symmetric cause produces one from a symmetrically related set of effects. • Symmetry is actually not broken, rather shared.

21. What is a pattern? • A pattern is the result of symmetry-breaking. • A system with lesser degree of symmetry is perceived (by humans) as having a pattern. • Example: A circle or a clear surface is not perceived as symmetric, even though it is the most symmetric thing nature can produce.

22. Summary • Symmetrya formally defined property of object • Symmetry-breakinga process in which an applied force breaks the symmetry of the system • Patternan informal property of systems with broken symmetry