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patterns in nature

Patterns in Nature

Related to Software Design Patterns

Peter JohanssonUndergraduate studentpetjo450@student.liu.seAmanuens at AIICS,


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

Philip Ball: The Self-made Tapestry

Ian Stewart & Martin Golubitsky: Fearful Symmetry

patterns in nature4
Patterns in Nature
  • Bubbles
  • Waves
  • Bodies
  • Branches
  • Breakdowns
  • Fluids
  • Grains
  • Communities

This is actually the table of contents from The Self-made Tapestry.

patterns in nature5
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?)

How did the Zebra get its stripes?

activator inhibitor model


Activator A





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.

activator inhibitor
  • Examples of activator-inhibitor systems. Light areas are dominated by one compound, dark areas by another.
activator inhibitor9
  • Patterns formed by activator-inhibitor systems depend on the size of the system.
activator inhibitor10
  • 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
activator inhibitor11
  • This process may also explain the marks of other animals.
  • “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?
what is symmetry
What is Symmetry?

A symmetry of an object is a transformation that leaves it apparently unchanged.

Rotation Reflection

symmetry groups
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

symmetry groups15
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)
symmetries in nature
Symmetries in Nature
  • Animal bodies
  • Crystals
  • Soap bubbles
  • Flowers (e.g. Sunflowers)
more or less symmetry
More or less symmetry?

Infinite number of rotations and reflections

24 rotations and 12 reflections

symmetry breaking
  • A falling drop of milk has circular symmetry…
  • …but after impact a ”crown” rises that only has 24 possible rotations.
where does it go
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.
where does it go20
Where does it go?
  • A symmetric cause produces one from a symmetrically related set of effects.
  • Symmetry is actually not broken, rather shared.
what is a pattern
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.
  • 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