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Tony Phillips' Take on Math in the Media A monthly survey of math news |

**December 2001**

- Living and breathing fractals
- More about hats
- Biomolecular finite-state automata
- A new
*Proof* - Mathematica - the show

**Living and breathing fractals.** As a ``News Feature" in the September 27 2001 *Nature*, John Whitfield reports on recent, controversial work on the problem of scaling in living organisms. The authors are James Brown and Brian Enquist (UNM, Albuquerque) and Geoffrey West (Los Alamos); the work has been reported in *Science* (**284** 1677-1679 (1999)) and picked up in the *Scientific American* and the *New York Times*. It follows from elementary geometry that as an organism grows its skin area scales as its ``radius'' squared, while its weight scales as its radius cubed, so the skin area scales as weight to the power `2/3`. But other biological parameters scale differently. In particular the metabolic rate scales as weight to the power `3/4`. Where does that number come from? Enquist, Brown and West analyzed the problem in terms of ``the dynamics of the organism's internal transport of nutrients and other resources,'' as Whitfield explains it. They found that a resource-distribution network (for example, the network of blood vessels in a human body), to maximize the area across which it acts, and to minimize the time and energy needed for transport, must have fractal geometry creating ``a four-dimensional entity,'' and they used this observation to derive the `3/4` metabolism/weight power law. Their theory is disputed by other scientists. Some claim that the `3/4` is an experimental error to start with (should be `2/3`), others that the `3/4` can be deduced without resort to fractals. The voice of reason seems to be that ``simple and general rules governing diverse biological phenomena lie waiting to be discovered.''

**More about hats,** and the ``Hat problem'' in particular, in John Allen Paulos' *Who's counting* ABCNews.com column Put on Your Hats and Codes (ouch!) for November 29 2001.

**Biomolecular finite-state automata.** A six-member team of Israeli scientists has invented a procedure for realizing finite-state automata using enzymes and DNA-type molecules. A finite-state automaton is a theoretical device with a set of states `S0, S1, ..., Sn`, an input channel which can feed in, one at a time, symbols like `a, b, ...` and an end-of-message symbol `t`, and a program: a set of rules of the form: ``If the device is in state `S`, input ``x`' will send it to state `S'`.'' The image below shows a finite-state automaton with two states `Green` and `Blue`. The program illustrated says, in brief, ``change states on input `b`, maintain states on input `a`.'' There is an initial state, marked with an arrow, and an ``Accept'' state marked with a circle. In the image they are both the `Green` state. The automaton starts in the initial state, reads input character by character and changes state according to the program. If when the end-of-message symbol is read the program is in the ``Accept'' state, it accepts the message. Otherwise it does not. The automaton illustrated here is programmed to accept any message with an even number of `b`'s.

A 2-state automaton which accepts only messages with an even number of

The team, Benenson, Paz-Elizur, Adar, Keinan, Livneh and Shapiro, published their work in the November 22 2001 *Nature*. It's all done with molecules: ``The automaton's hardware consists of a restriction nuclease and ligase, the software and input are encoded by double-stranded DNA, and programming amounts to choosing appropriate software molecules.'' One of several challenges is to construct the input-string molecule so that the markers will be peeled off one by one in sequence. ``In our implementation 10^{12} automata sharing the same software run independently and in parallel on inputs (which could, in principle, be distinct) in 120 micro-liter solution at room temperature at a combined rate of 109 transitions per second with a transition fidelity greater than 99.8%, consuming less than 10^{-10} watts.''

**A new Proof** is now available on Broadway. In the October 27 2001

**Mathematica - the show**- is running at the San Francisco Exploratorium through May 5, 2002. The exhibit, first set up in 1961 by the designers Ray and Charles Eames (of the Eames chair and *Powers of Ten*), brings math to concrete and elegant life in a collection of interactive installations. In the *New York Times* for November 4 2001 (``Math Exhibition Finds Common Denominators,'' Sunday travel section) Eric P. Nash emphasizes the sense of play that the ``designing couple'' brought to everything they did. ``... the Eameses created for the exhibition a one-surfaced Möbius strip (a visitor starts a red arrow on a path along the seemingly two-sided band, and discovers that the ribbonlike form has only one surface). Another device demonstrates probability by showing how 30,000 randomly cascading plastic balls align themselves into a standard bell curve. A soap bubble dipping device that produces square bubbles and triangular ones illustrates principles of topology ...'' The show is also described in a web press release from the Exploratorium.

-*Tony Phillips Stony Brook *