December 2001
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 16771679 (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 resourcedistribution 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 fourdimensional 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 finitestate automata. A sixmember team of Israeli scientists has invented a procedure for realizing finitestate automata using enzymes and DNAtype molecules. A finitestate 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 endofmessage 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 finitestate 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 endofmessage 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 2state automaton which accepts only messages with an even number of b's. The team, Benenson, PazElizur, 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 doublestranded DNA, and programming amounts to choosing appropriate software molecules.'' One of several challenges is to construct the inputstring 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 microliter 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 New York Times Bruce Weber reviews the new cast and pronounces: ``Equation Changes, Sum's the Same,'' the title of his piece. ``It doesn't feel like a different play  good news for those who have never seen this beautifully carpentered story about mathematics, decent people and the whimsical wiring of the heart and the brain  but there is enough change for return theatergoers to play a very satisfying game of compare and contrast.'' Proof has been running for a year. 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 onesurfaced Möbius strip (a visitor starts a red arrow on a path along the seemingly twosided 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

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