James Gleick

Chaos: Making a New Science

Chapter 4. A Geometry of Nature

Benoit Mandelbrot (1960s). Contributed ideas on fractional dimension.

Born in Warsaw 1924. Lithuanian Jewish family.

Father clothing wholesaler, mother dentist.

Moved to Paris in 1936 (!!!)

Uncle Szolem Mandelborjt, mathematician in Paris, founter of Bourbaki.

• Bourbaki: post-WWI formalistic mathematical reaction to loose "intuititionism" of Poincare.
• Named for 19th century french general.
• Fixed size, forced retirement at 50. Members were assigned to write monographs in areas they didn't know.
• Approach came to dominate French mathematics in style and notation.

Apprentice toolmaker.

Sporadic education.

Liberation of Paris.

Passed month-long oral and written exam for elite Ecole Normale and Ecole Polytechnique. Relied on geometrical intuition to solve analytical problems. Didn't fit Bourbaki mold.

Acquired knowledge of history of math and science.

Mathematical linguistics: Distribution of words in texts.

Investigations into game theory.

Economics: scaling regularities in economies of large and small cities.

• Invited by economist Houthakker to speak. Houthakker's blackboard had diagram of cotton prices that looked similar.
• Standard models: long run trends driven by macroeconomic causes, short run noise.
• Ratio of small changes to large was not as high as Houthakker expected on this model.
• Mandelbrot was looking for common patterns across scales.
• Cotton data: 60 years, centralized market in New York.
• Amazing result: pattern the same for daily changes as for yearly changes.

At IBM: study of distribution of noise in telephone lines.

• Discovered that each interval contained a noise-free subinterval. Analogous to cantor 1/3 set.
• Recommended redundancy instead of increasing signal/noise ratio.

Study of water level of Nile river. Quantity can change discretely, but trends can be extended through time (Cantor-like behavior)

Classical geomterty: too smooth to capture nature. Infinitely embedded pits and roughtness are the essence of nature, rather than accidents to be ignored (note figure/ground issue again).

Coastline study:

• English Scientist Lewis F. Richardson. Coastline length is a nontrivial question. Twenty percent differences in published lengths.
• Mandelbrot: coastlines are infinitely long. Measuring with a smaller and smaller ruler yields ever higher estimates that don't converge.
• Switch attention from length to dimension.
• Infinitely kinky coastline has fractional dimensino between 1 and 2. "Partly" fills space.
• Example: Koch snowflake curve:
• Triangle, triangles on sides, triangles on their sides, etc.
• Encloses finite space (fits in superscribed circle about first triangle)
• but boundary has infinite length 3 X 4/3 X 4/3 X ...
• Box-counting dimension: D(S) = lim_z ---> 0 [ln #(S, z)] / [ln 1/z], where #(S, z)] is the number of cubes of side-length z required to cover the set S.
• Examples:
• D(point) = lim_z ---> 0 [ln 1] / [ln 1/z] = 0.
• D(line) = lim_z ---> 0 [ln length/z] / [ln 1/z] =
• = lim_z ---> 0 [ln length - ln z] / [ln 1 - ln z] =
• = lim_z ---> 0 [k - ln z] / [0 - ln z] = 1, because -ln z goes to infinity as z goes to 0, so the k matters less and less.
• D(plane figure) = lim_z ---> 0 [area/z^2] / [ln 1/z] =
• = lim_z ---> 0 [ln area - ln z^2] / [ln 1 - ln z] =
• = lim_z ---> 0 [ln area - ln z^2] / [ln 1 - ln z^2] =
• = lim_z ---> 0 [ln area - 2 ln z] / [ln 1 - ln z] =
• = lim_z ---> 0 [k - 2 ln z] / [0 - ln z] = 2, because as before the constant k matters less and less as -ln z becomes arbitrarily large.
• D(cantor set): We can compute the limit over any sequence of nonzero rational box side lengths converging to zero. For the cantor 1/3 set, it is convenient to choose z_i = (1/3)^i. Then the number #(cantor set, z_i) of boxes of size z_i required to cover the cantor set is 2^i (1 of length 1 for the whole unit interval, 2 of length 1/3 for the two outside thirds, four of length 1/9 for the outer thirds of the outer thirds, etc.) Thus we have:
• D(cantor set) = lim_i ---> infty [ln 2^i] / [ln 1/(1/3)^i] =
• = lim_i ---> infty [ln 2^i] / [ln 1/(1/3)^i] =
• = lim_i ---> infty [ln 2^i] / [ln 3^i] =
• = lim_i ---> infty i ln 2 / i ln 3 =
• = ln 2 / ln 3 = approximately .63.
• Sierpinski carpet (two dimensional version of cantor set) Square, divide into 9 squares, knock out all but corners, divide those into nine squares, knock out all but corners, etc. The analysis is like that for the cantor set except that #(carpet, z_i) = 2^2i instead of 2^i. Then we have:
• D(carpet) = lim_i ---> infty [ln 2^2i] / [ln 1/(1/3)^i] =
• = lim_i ---> infty [ln 2^2i] / [ln 3^i] =
• = lim_i ---> infty 2i ln 2 / i ln 3
• = 2 * ln 2 / ln 3 =
• = 2 D(cantor set) =
• = approximately 2 * .63 = 1.26.
• Menger sponge: 3 dimensional version of Cantor set. Start with corners of unit cube, their corners, their corners, etc. What's the answer?
• D(Koch curve) = 1.2618.
• Fractal is set with fractional dimension. Examples: Cantor set, Sierpinski carpet, Koch curve.
• Computers at IBM allowed Mandelbrot to draw arbitrarily complex fractals generated by simple rules.
• Computer plotting overcame the traditional geometrical bias against pictures and imagination.
• Self-similarity at all scales: as with cotton prices an noise.

Chkristopher Scholz, Lamont-Doherty Geophysical Observatory.

• Physicists and mathematicians ignored Mandelbrot.
• Earthquakes follow same pattern as personal incomes in a ,arlet economy.
• Scholz remembered Mandelbrot's name from the economics work and in 1978 bought his book Fractals: Form, Chance and Dimension.
• Scholz discovered other scientists applying fractals to melding, branching and shattering. In metallurgy, people used fractals to describe metal surfaces of contact.
• Schizosphere: cracks in earth's crust responsible for groundwater flow, oil flow, and earthquakes.
• Using fractals was thought to be too trendy... "bandwagon".
• Question of writing only for specialists or risking exposure to whole geophysical community.
• "[Fractal theory] gives you mathematical and geometrical tools to describe and make predictions. Once you get over the hump, and you understand the paradigm, you can start actually measuring things and thinking about things in a new way. You see them differently. You have a new vision." (Note self-conscious Kuhnian talk).
• Revolutionary shift of attention away from size and toward invariants across scales. Animals of different sizes are different. Clouds and earthquakes of different sizes are the same.

Anatomy:

• Fractal theories of vascular, urinary, bilary systems.
• Fibers carrying electrical current to the heart muscles. (His-Purkinje network). Heart activationfollows income and earathquake pattern; explained by fractal structure of His-Purkinje network.
• Complex behavior generated by simple recursive structures. Evolutionarily plausible.
• Biology: feathers, tree branches, leaves.
• Universal pattern in morphogenesis.

Mandelbrot finds fame:

• Persistence, bright pictures and braggadocio pay off. Also, hobbyists playing with fractals on personal computers.
• I.B. Cohen's list of self-proclaimed scientific revolutionaries: Robert Symmer, radically mistaken contemporary of Franklin, Marat, Von Liebig, Hamilton, Darwin, Virchow, Cantor, Einstein, Minkowski, Von Laue, Wegener (cont. drift), Compton, Just, Watson (DNA) and Mandelbrot.
• Mathematicians resented self-promotion.
• Personally hounded people to cite his book.
• Followers acknowledge egomania, but viewed it as necessary for the tenacity to keep working in isolation. (Interesting addition to Kuhn's profile of innovators. To surivive outside of a paradigm, one must have high self-esteem).
• Lots of I's in his book.
• Resentment that he left hard drudgery work to others.
• Discovery vs. claim-staking.
• Computer heuristics vs. theorems and proofs. Who gets the credit?
• Mandelbrot felt he had to spend enough time placating the hostile establishment. Then became annoyed when others claimed the ideas were obvious or natural.
• Predictable stages of reaction:
• Who are you and why are you interested in our field?
• How does it relate to what we have been doing and why don't you explain it on the basis of what we know?
• Are you sure it's standard mathematics? (Yes)
• Then why haven't we heard of it? (It's obscure)
• What do peole in these branches of mathematics think about your work? (Applications don't add to the mathematics).
• Enthusiastic acceptance among industrial scientists working with oil, rock, minerals, polymers, reactor safety.
• Hollywood special effects designers.
• Fractal geometry describes boundaries between chaos and order in May's and Lorenz's systems.

Self-similarity

• Ancient idea of microcosm/macrocosm: most natural hypothesis before the large and the small were actually seen.
• Idea died when microscopes and telescopes saw what was really there.
• Reductionism: particles are different from the things they compose.
• Self-similarity is apparent only at higher levels of complexity.
• Explosion of microsopic and telescopic images led to natural search for new analogies.
• Aesthetic motive: simple geometry is boring at small scales, traditional architecture appeals at all scales.
• Back-to-nature: simple geometry was a statement of power of nature (formal gardens). Nowadays wildness is in vogue and fractal geometry reflects natural patterns.

Remained for physics to turn chaos into a science. Fractal geometry is an effect, not a cause and scientists want to know why.