Wherever chaos, turbulence, and disorder are found, fractal geometry is at play.
~ John Briggs and F. David Peat, Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness (1989, p. 95)
Over the years my thoughts about the mathematics of nature have had a decidedly biological bent. Curiously, their roots lie mostly in the early days of the personal computer. Inevitably, given the confluence in those years of accessible computing power and the proselytizing of Benoit Mandelbrot, I tried my hand at programs that made my screen into a psychedelic painting of wild and colorful waves surrounding a bizarre, horizontally symmetrical, black “bug” – the Mandelbrot set, generated by a wonderfully simple mathematical formula.
The black figure in the center of the image below is the Mandelbrot set (numbers falling within that black bug are in the set). (This is a image from a great program called Fractile Plus that runs on the iPhone. This program is amazingly more sophisticated than anything I ever put together to generate the set.)
There are many explanations of the Mandelbrot set on the web. A somewhat accessible one that is good on the mathematics behind it comes from Yale University (though it’s hard to tell what this site really is).
The Mandelbrot set is a fractal. Its essential attribute is that it exhibits “self-similarity,” that is, examining a fractal at ever great levels of magnification, will reveal endlessly repeating patterns. In this instance, embedded in the hallucinatory waves of color are an infinite number of ever smaller, connected versions of the black bug. The image below centers on one of the buds of the Mandelbrot set; notice that tethered to that bud are several little replicas adrift in the waves, albeit with a lifeline tying them to the set.
Mandelbrot and other mathematicians have made various claims about what is to be learned from fractals and fractal geometry. One of their central assertions concerns the ability of fractal geometry to represent natural objects. Here’s Benoit Mandelbrot:
Why is geometry often described as “cold” and “dry?” One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line.
More generally, I claim that many patterns of Nature are so irregular and fragmented, that, compared with Euclid – a term used in this work to denote all standard geometry – Nature exhibits not simply a higher degree but an altogether different level of complexity. The number of distinct scales of length of natural patterns is for all practical purposes infinite.
. . . . I conceived and developed a new geometry of nature and implemented its use in a number of diverse fields. It describes many of the irregular and fragmented patterns around us, and leads to full-fledged theories, by identifying a family of shapes I call fractals.
~ The Fractal Geometry of Nature (1983, p. 1)He was not claiming that all patterns in nature challenge Euclidian geometry (i.e., do not exhibit the traditional circles, cones, etc.), only that many do. Nevertheless, I was so thoroughly converted by Mandelbrot (for better or worse) that I am utterly amazed when natural objects, particularly those secreted or shaped by living organisms (other than human beings), are distinctly Euclidian in their geometry, such as the hexagonal cells in a honeycomb or the star shape of the carambola (star fruit).
So, it was that, in recent weeks, as I worked my way through some fossil material from the Oxford Clay in Great Britain (this Jurassic material is roughly 160 million years old), I was repeatedly surprised by the geometry of the crinoid ossicles I was finding. Frankly, it was more than that, I had a welcome sense of relief when these particular ossicles appeared.
Crinoids (common name is sea lilies) are marine invertebrates that have been around for hundreds of millions of years. The animal is a filter feeder and lives within a calcite skeleton made up of many individual plates. The crinoids represented in the fossil record were primarily sessile, anchored to the sea floor by a stem of varying length; modern crinoids are mostly free swimming.
The plates or ossicles that constitute the crinoid skeleton are what the fossil record offers up, sometimes as articulated structures but generally as individual disks. Pictured below is an articulated portion of an Eretmocrinus tentor from the Mississippian Epoch (359 to 318 million years ago). The stem comes up from the bottom, is twisted and buried in matrix. It joins the crinoid head or calyx from which arms protrude.
The ossicles I found in this material fall across a continuum of shapes, moving from these distinctly star-shaped (five points) to fairly mundane pentagons. The crinoid body morphology has “pentaradiate” symmetry (that is, it is organized in five rays or along five axes), but that doesn’t necessarily extend to the individual ossicles making up these animals’ skeletons. As already noted, ossicles can be circular, square, or other shapes.
As attractive as the overall shape of these particular ossicles might be, there’s another extraordinarily appealing aspect of these plates – the patterns that adorn their surfaces. These grooves and other surface features are collectively known as crenulae.
Despite their aesthetic appeal, the crenulae are decidedly functional. The crenulae on each face of a stem ossicle interlock with the crenulae of adjoining plates, thereby providing both rigidity to the stem as well as some degree of flexibility. Further, the patterns on pentastellate-shaped ossicle offer the crinoid species that sport them a distinct advantage in coping with water-borne shear stresses on the stem column. William I. Ausich et al., in Crinoid Form and Function (Hans Hess, et al., Fossil Crinoids, 1999) posit that the patterns of star-shaped crenulae significantly increase the number of crenulae that will be lined up at right angles to any shear stress visited on the column, helping to keep the stem intact (p. 12).
Identifications of the crinoid species responsible for these ossicles are . . . not very easy. Fossils of the Oxford Clay, edited by David M. Martill and John D. Hudson (The Palaeontological Association, London, 1991) is limited in its treatment of crinoids. The sole crinoid described is Isocrinus fisheri (Forbes) – the ossicle in the first picture above appears to be from a member of that species. Perhaps this scant attention to Oxford Clay crinoids isn’t surprising. E.R. Matheau-Raven, in an article titled UK Fossil Research: Jurassic Crinoids of the Oxford Clay (Deposits Magazine, Issue 1, October, 2004), noted that up until then, I. fisheri was the only crinoid species attributed to the Oxford Clay. In this article, he went on to posit that perhaps four more species appear in this formation. Unfortunately, none of the four seems to match up with those shown above. Matheau-Raven does note that “[o]ssicle morphology changes up the stem but does so in a set pattern consistent with individual species.” That would seem to mean that ossicles of different shapes can come from the same species (not encouraging for easy identification).
My struggle with their identity aside, I am very taken by their beauty. I welcome them. Perhaps it’s because the Euclidian nature of their shapes emerges with a satisfying “bang” from the chaotic background of the matrix under the microscope. They announce their presence in no uncertain terms. Or, perhaps it’s reassuring to find such “traditional” shapes, so regular and orderly, in what seems increasingly like a fractal world.
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Interesting.
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