This is me with Barthelemule. Bart is not my cat though, he was just along for the ride.

This is a histogram of a chaotic attractor with hidden symmetry

I am a mathematical biologist who has worked at the newest research university in the United States ( UC Merced. ) and who is now working at the oldest research university in the US ( Harvard ). My main research focus is modeling plant development. I am collaborating with mathematicians at Smith college and a botanist at Harvard on modeling plant development.

neural net models of cognitive processes and I am collaborating with the philosopher Jeff Yoshimi

Recently I've taught mathematics courses while professor of Mathematics at Miami University . Miami University is not in Florida. The University of Miami is located in Miami Florida but Miami University is in Oxford Ohio . Miami University was founded 1809. This was just a few years after Ohio became a state and long before Florida was a part of the United States. In the almost 200 years since it was founded Miami University has grown into a well respected university both for its teaching and research.

My most recent research has been about Phyllotaxis, the study of plant patterns. Despite their diversity similar patterns are found in many different types of plants. A common eye catching pattern consists of two sets of spirals forming a lattice. This can be seen in the stamens of flowers (e.g. the male Leucadendron Discolor shown at the right), the florets of compound flowers (e.g. the Candula shown below), the scales of pine cones and cycads. Naturally this is due to the fact that many plant organs follow the same pathways during their early stages of development.

Around the turn of the 18th century the well known Astronomer Johanne Kepler noted that the Fibonacci numbers are common in plants. And around 1790 Bonnet pointed out that in spiral phyllotaxis the number of spirals going clockwise and counter-clockwise were frequently two successive Fibonacci numbers. For example the orange Candula shown below has 13 spirals going in one direction and 21 spirals going in the other direction. Click here for close ups.

This phenomenon is common throughout much of the plant kingdom and understanding why has become a interdisciplinary effort. Several ideas have been proposed over the years. Many of them have been based on the notion that the Fibonacci numbers somehow promote the survival of mature plants. Among the suggestions on how Fibonacci numbers could promote survival are: by providing dense packings of seeds, by allowing circulation of air through leaves, and by allowing light to fall on as many leaves as possible.

A more modern view is that a common physical condition occurs in the early stages of developement that constrains the structure of the flower to follow the Fibonacci pattern. The point of the Fibonacci numbers does not lie in some activity performed by a mature plant but in the activity which occurs during developement. The Fibonacci numbers are not seen as doing very much to promote or hinder survival in mature plants. A Candula whose flowers have 12 spirals going in one direction and 22 going in the other direction should have about the same chance of producing viable seeds for the next generation as one whose flowers have 13 spirals going in one direction and 21 in the other direction. Yet the first case is virtually unheard of while the second case is very common.

But what is the nature of this constraint? An important result occurred in 1992 when two French Physicists, Douady and Couder, developed a laboratory model of plant developement. To learn more see the website I helped build:

Phyllotaxis: An Interactive Site for the Mathematical Study of Plant Pattern Formation.

The fractal at the top right of the page is the histogram of an attractor for a discrete dynamical system obtained by numerically integrating the Volterra-Lotka equations. The Volterra-Lotka equations were one of the original mathematical models of predator/prey relations and they are well known among population biologists. These equations display periodic behavior in predator and prey populations. However in making the fractal above the time step of the numerical method has been deliberately made too large for approximating the solutions of the Volterra-Lotka equations. Making the time step too large is a way to custom design fractals. For small values of the time step the numerical schemes do a good job of approximating the periodic solutions of the Volterra-Lotka equations but as the time step increases resonances occur. The periodicity of the resonances can be controlled by appropriately choosing the numerical method. For the fractal above the period of the resonance was seven iterations. As the time step was further increased a sequence of period doubling bifurcations occured which resulted in a fractal attractor for the system.

This fractal has a "hidden symmetry". It can't be rotated or reflected back into itself which is how symmetry is often defined. However the period seven resonance has effectively given us an attractor which can be broken up into seven topologically equivalent pieces. This means that each piece differs from the others only by a stretching or compression. No cutting or gluing of the pieces is necessary. Each of these seven pieces individually resembles blue syrup being poured from a bottle.

Although the symmetry of this fractal is "hidden" the type of symmetry that it displays is not really new. We can stretch or compress some regions of the plane to turn this fractal into a figure which has rotational symmetry. This is the case for any planar fractal.

However figures exist in three dimensions which have a "hidden symmetry" that is unlike any geometric symmetry that exists in three dimensions. A good example is an embedding of Klein's quartic surface in three dimensions. This surface forms a three holed torus that is made up of twentyfour heptagons.

There are no really good pictures of this object on the web. The following websites are about the best that is available. Unfortunately it is very difficult to see the "hidden symmetry" from these pictures.

helaman ferguson's online scupture galley

Eric Weisstein's World of Mathematics - Klein's Quartic

University of Siegen - Department of Mathematic's Research on Pure and Applied Geometry - Polyhedral Models of Klein's Quartic

An entire book has been written on this object: "The Eightfold Way: The Beauty of Klein's Quartic" which contains better pictures.

I've illustrated a book on the use of critical curves in discrete two dimensional Dynamical Systems entitled "Chaos in Disctrete Dynamical Systems". by Ralph Abraham, Laura Gardini, and Mira. The critical curves are used to construct Chaotic areas in the plane for a Dynamical System. Their effect on the histograms of chaotic attractors is often appearent. Click on the title for more information about the book.

My college career began by getting a perfect math score on the SAT and a perfect physics score on the ACT test. During my undegraduate days at UCSC I received the Hewlett and Duncan Scholarships which helped me devote more time to school. In June 1993 I received a Bachelors with Honors in Mathematics from UCSC. During grad school I had a GAANN fellowship. In December 1999 I received my doctorate in mathematics from UCSC. During the fall 2000 semester I taught a graduate course in dynamical systems at Humboldt State Univeristy. In spring of 2002 I taught an upper division course on the foundations of geometry.

A nonexhaustive list in no particular order of things which I like to do or learn about:

Dynamical systems, computer graphics, crystallography, swimming, phyllotaxis, gaia, physical organic chemistry, c programming, relativity, morphogenesis, history of philosophy, evolution, theories of perception, algebraic topology, space travel, hiking, beaches, writing, mountains, symmetry, board games, fractals, earth history, non-equilibrium thermodynamics, redwoods in the summer...

Press buttons for more information.

Phyllotaxis: An Interactive Site for the Mathematical Study of Plant Pattern Formation.

Mathematical Phyllotaxis Gallery

My PhD thesis in gzipped postscript format

Geometry Sites on the Web

The discriminant locus of the reduced fourth degree polynomial sitting on a table

Scott Hotton's Fractal Gallery

University of California at Santa Cruz

Humboldt State University

Miami University

Copyright © 2000, 2001, 2002, 2003 Scott Hotton