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Final Remarks Some Observations: For an idealized model for the the distribution of seeds generated at a central core, a relative angle from one to the next given by the golden ratio f provides the most uniform spacing because of a unique mathematical property of f. The continued fraction expansion of an irrational number carries information about how closely the number can be approximated by rationals of small denominator. Among all irrational numbers f minimizes the rate of approximations by rationals. This is why it optimizes the distribution of seeds from a central source in our simple model. There exists a mathematical proof of this. Both the Rotating Gun and the Farey Series applets have tremendous potential for investigation: The nature of periodicity; the important role of the denominator in a reduced fraction. The meaning of scale. Much research is still going into the biophysical mechanisms that are responsible for the formation of phyllotactic patterns. In the abstract mathematical world, the golden ratio and its properties are strikingly beautiful. It is also pleasing that this transient physical world provides us with some visible representations of this beauty. Thank you.