Author Archives: dickgordoncan

About dickgordoncan

Itinerant Theoretical Biologist

Staying on the road with bungee cords.

Richard Gordon

March 17, 2017, Cedar Lake, Ouachita National Forest, Oklahoma, USA

In the race to see who ages faster, we or our now 7 year old travel trailer, the trailer seems to be winning, held together indefinitely by duct tape. We buy the everyman version known as Duck tape because it comes in many colors which show less when we tape something. Duck tape is marvelous to hold together the cover of our air conditioner or a bandage on Fred. (Duct tape, unlike the regular bandage tape meant for use on humans, doesn’t pull out fur when removed, and he can’t bite through it.)

Unlike duck tape however, bungee cords not only come in brighter colors and pleasing patterns, but don’t leave marks when replaced. So we keep a good supply of all sizes on hand, much like our bandage supply for us and the pets.Bungee

Actually, it was our escape artist cat, Klinger (who should have been named Houdini), who taught us the value of bungee cords. He outwitted us for years with his charges through the open door underfoot, flying from the bar counter, sneaking from under my computer lab bench or an exiting dog, opening the screen door slider (just another cat door to him), and when we put on a latch, throwing his body against the door to transiently warp it enough to fly out. But one bungee cord, knotted to sufficient tension, finally defeated him. Now Klinger is a well travelled and very expensive cat, having stayed at the Toronto Feline Hilton en route to rejoin us in Disneyland after one deft escape into the talons of an eagle. Bungee cords are cheaper.

Klinger of course has nothing more to do all day than plot his escapes, awake or in his dreams (he sleeps a lot, except when we do). His latest success was learning how to open a window screen. Being hairless apes, we scratched our heads but finally recalled The Bungee Solution. The metal prairie rose (by our friend Steve McGrew) anchors a bungee cord to the screen now. Enough about our cat. This was supposed to be about our trailer. Anything not secured manages to meander to the opposite end of our trailer unless it’s tied down – via bungee cords. The metal stair, needed on those ungraded hilltop RV sites, and our portable microscope, are battened down.

One essential bungee cord keeps our red, white and blue towels from plunging into the toilet, which must be kept open for, you guessed it: the cat, who is toilet trained. Leaving and returning to Canada is a cold experience due to our government’s 6 month and a day bed check rule, so we keep the trailer bathroom warm for Klinger with a vent pad, held in place by – a bungee cord.

Of course, on rough roads our kitchen drawers always fling open, now kept closed by a cleverly placed bungee cord. Note the counter balance on the sink door, so that the knobs aren’t pulled out. But au contraire, when we’re parked, especially on one of those sites tilting us port, the drawer won’t stay out while we put away the silverware, so another bungee cord comes to the rescue. The tall Sodastream bottles in the door leave us with a narrow shelf, good for cheese and sausage.

The other crash, into the bathroom sink, is now also a fond event from our past.

Medicine cabinet

One bungee cord keeps the computer lock away from the mouse, and doubles to restrain the battery backup from scooting to the floor while en route.

That takes care of the interior of the trailer, for now. Outside a bungee cord holds the power cord up away from wandering ants, though a ring of Vaseline is still sometimes needed. On our roof are four solar panels, protected, when needed, from hailstones by Styrofoam panels, held in place by bungee cords. A too sharp turn once severed the power cord from the trailer to our workhorse pickup truck. The replacement didn’t quite match, and is held in place by bungee cords. Inside the truck’s cap, our travelling garage, bungee cords keep the spare propane tank, bikes and lawn chair from rattling around. On the side of the cap, two bungee cords suspend our pick axe, so it doesn’t crash to our toes. A bungee cord also helps secure our canoe.

The original bungee cord was a 1930s elastic cord for launching a glider. If we outlast our trailer, our earthbound spaceship, perhaps someday it will be replaced it by an airborne trailer towed by an aircar. Undoubtedly it, too, will be held together with bungee cords (and duck tape).


On a Mathematical Limitation to Lawn Mowing

Natalie and I avoid shoveling snow by heading to the Deep South each winter. But now that we’ve acquired a wheelless house in Manitoba (to distinguish it from our wheel house, our trailer, as named by grandson Nick), we are subject to the opposite season’s green scourge, luxuriant growth of grass over the brief summer that, due to long days here up North, is faster by far than my beard growth, which I also prefer to neglect. Now this is great for our tall grass prairie quarter section, with stalks that reach over my head, but in nearby small town Alonsa the one sin no one dare yield to is not mowing one’s lawn.

The first year of sessile life we hired a fellow with his ride-on to mow our lawn. He was delayed, and the grass, not understanding the situation (despite its undoubtedly self-centered intelligence: Mancuso & Viola 2015) grew beyond its legal height. I was summoned and reported for my imminent handcuffing and arrest. I was told sternly that if I don’t cut my lawn in a timely fashion, the local government would do it and charge me $16/hour. I said “Great”, as I was already paying $20/hour, and they immediately backed off. So much for Justice. Nevertheless, in the name of civil peace, realizing that our community relations should not be left to a busy intermediary, we bought our own lawn mower.

Now despite my lifelong interest in local/global interactions (Gordon, 1966; Portegys et al., 2016), the way I mow grass is strictly local. I mow a line, and then I follow that line, then go back following that line, etc. I don’t look where I’m going. Of course, with laser guided tractors which can hoe a straight line to an accuracy of 0.6 centimeter over a track length of 220 meters (van Zuydam, Sonneveld & Naber, 1995), my approach is antiquated. But I do it deliberately, to amuse myself with a mathematical puzzle, which today I realized I can try to formulate precisely and share with you. What else is there to think about while chopping off the heads of dandelions and developing green toes, if not a green thumb (due to mowing with open toed slippers, not recommended)?


Given a semi-infinite plane (which we can approximate by a strip infinite in one dimension, with periodic boundary conditions in the other direction), we start aligned with its straight edge at, say, x=0 and mow a strip of unit width. Then we do it again. If perfection attained, sequential curves could be designated by y(i,x) = i, i.e., we would have no excuse to stop mowing until the job is done. The excuse lies in our own imperfection.

SAM_7653So we need a function to represent my inability to walk a straight line. Now, blindfolded we walk in circles as small as 20 meters in diameter (Souman et al., 2009), which would be great for limiting the duration of mowing, though I would then chop through the electric cord tethering our mower. While this fundamental result, attributed to “accumulating noise in the sensorimotor system”, has been cited 53 times already, we must look elsewhere for a function to represent noise in the mowing trajectory. For this I turn to boids.

Boids are idealizations of flocking birds and schooling fish. I actually did the first computer simulation of such “swarms”, back in the mid-1960s, while I was a graduate student corresponding with and then visiting the master of schooling fish, Charles Breder (Breder, 1929, 1951, 1954, 1965, 1967) at the Mote Marine Lab in Florida where he retired. This was 2 decades before the first boids simulation in 1986 (Reynolds, 2001). Unfortunately I didn’t think much of the result, because I placed the “fish” into a circular mill, which slowed down as they swam. Breder thought this was realistic, from his personal observations of milling fish. However, I simulated only 300 fish in a plane, on a mainframe computer so slow in those days that the “fish” didn’t get far during the computer time I could command, but a fraction of a turn. I couldn’t tell if the mill was stable, even though we knew that ants would follow each other in a mill unto their death (Schneirla, 1944).  (That’s what local rules will get you! So much for emergence.) So we didn’t publish it. Nowadays whole murmurations of hundreds of thousands of boids in full 3D can be simulated with ease (Ikegami, 2015), and milling is old hat mathematically (Lukeman et al., 2009; Calovi et al., 2014).

The relation between boids and lawn mowing is that a boid aligns with the average direction of its near neighbors, while I align with my former self, at least insofar as my nearby previous track across the grass is what I use to estimate my next direction, moment by moment. So-called “error” of alignment for boids has been discussed (Watson, John & Crowther, 2003) but not its physical and/or mental source. But we may not have to have our heads examined (except as to why we mow grass in the first place). A simple trigonometric error analysis shows that if boids make small errors in the vectorial direction of their motion, their net random motion perpendicular to the mean direction of motion is much larger than that along the direction of motion (Toner & Tu, 1998). Thus the wavy curvature of my lawn mowing will amplify, until my mowing path closes upon and crosses itself and my need to mow ceases (invoking my local-only rule and my goal of death to lawnmowing). This is what mathematics is for: justifying as little mowing as I can get away with. The only thing left to do is calculate how much alcohol I would have to consume so that my error and thus the curvature reaches this closing point before my (finite) lawn is completely mowed. For math aficionados, note that local lawnmowing is an example of a stochastic wave in an active medium, but a peculiar one, as propagation is in finite steps, opening up great new insights into discrete aspects of the continuum. I rest my case and my lawn mower, and leave it for the ambitious computer programmer and/or mathematician to work out the details, while I lounge on my lawn chair. RAASAM_7651

Breder, C.M. (1929). Certain effects in the habits of schooling fishes, as based on the observation of Jenkinsia. Amer Mus Novitates 382, 1-5.

Breder, C.M. (1951). Studies on the structure of the fish school. Bulletin of the American Museum of Natural History 98(1), 1-28.

Breder, C.M. (1954). Equations descriptive of fish schools and other animal aggregations. Ecology 35(3), 361-370.

Breder, C.M. (1965). Vortices and fish schools. Zoologica New York 50(2), 97-114.

Breder, C.M. (1967). On survival value of fish schools. Zoologica-New York 52(2), 25.

Calovi, D.S., U. Lopez, S. Ngo, C. Sire, H. Chaté & G. Theraulaz (2014). Swarming, schooling, milling: phase diagram of a data-driven fish school model. New J. Phys. 16, #015026.

Gordon, R. (1966). On stochastic growth and form. Proceedings of the National Academy of Sciences USA 56(5), 1497-1504.

Ikegami, T. (2015). A dynamics of large scale swarms.

Lukeman, R., Y.X. Li & L. Edelstein-Keshet (2009). A conceptual model for milling formations in biological aggregates. Bulletin of Mathematical Biology 71(2), 352-382.

Mancuso, S. & A. Viola (2015). Brilliant Green: The Surprising History and Science of Plant Intelligence, Island Press.

Portegys, T., G. Pascualy, R. Gordon, S. McGrew & B. Alicea (2016). Morphozoic, cellular automata with nested neighborhoods as a metamorphic representation of morphogenesis [invited]. In: Multi-Agent Based Simulations Applied to Biological and Environmental Systems. Ed.: D.F. Adamatti, IGI Global: Submitted.

Reynolds, C. (2001). Boids: Background and Update.

Schneirla, T.C. (1944). A unique case of circular milling in ants, considered in relation to trail following and the general problem of orientation. Amer Mus Novitates(1253), 1-26.

Souman, J.L., I. Frissen, M.N. Sreenivasa & M.O. Ernst (2009). Walking straight into circles. Current Biology 19(18), 1538-1542.

Toner, J. & Y.H. Tu (1998). Flocks, herds, and schools: A quantitative theory of flocking. Physical Review E 58(4), 4828-4858.

van Zuydam, R.P., C. Sonneveld & H. Naber (1995). Weed control in sugar beet by precision guided implements. Crop Prot. 14(4), 335-340.

Watson, N.R., N.W. John & W.J. Crowther (2003). Simulation of unmanned air vehicle flocking. In:  Theory and Practice of Computer Graphics, Proceedings. Ed.: M.W. Jones: 130-137.




Near Misses: Paths not Crossed with Richard Bellman

World Scientific Publishing recently had a sale of electronic books, in which I came across and downloaded:

Bellman, Richard (1984). Eye of the Hurricane: An Autobiography,  World Scientific. Web:;

for US$9.90. I had heard that Bellman had a reputation of meeting someone, having a chat, and sending them a manuscript to co-author the next day. In this way he was the applied math complement to Paul Erdös, about whom I wrote:

Gordon, R. (2011). Cosmic Embryo #1: My Erdös Number Is 2i.

While Bellman doesn’t discuss this story, he did love to travel, and much of the book is about the places he has been, even including in some cases the addresses of hotels he liked. He was indeed prolific: “Over the course of his career he published 619 papers and 39 books. During the last 11 years of his life [1920-1984] he published over 100 papers despite suffering from crippling complications of brain surgery” ( Whoever added his CV to the end of the autobiography upped it to 620 papers and 40 books. While it was written in 1978, his autobiography seems to have been published after his death in 1984. He doesn’t even mention his medical condition in the book.

What what I found uncanny about his autobiography is how many people he names who I also knew, and one he didn’t name, but undoubtedly knew: my own father, Jack Gordon. I deduce this because both played handball at Brighton Beach near the boardwalk to Coney Island, New York, on one-wall courts. Bellman, born in 1920, was 7 months older than my father, who I recall was winning at handball at age 13, on those courts. Maybe he trounced Bellman. While my father focussed on handball all his life and became a USA national champion (Singer, Stuffy (1994). Gordon honored with Kendler Award. Handball 44(1), 18.), Bellman was an all-round jock, claiming to excel at other sports: tennis, table tennis, track, football, basketball, baseball, swimming. He even did some ballet. I can recall those courts, the boardwalk, the hot summer beach on which one could hard boil an egg, building sand castles, the lines of rocks with oysters perpendicular to the beach, out into the water, and Nathan’s hotdog stand. It was there my mother, then Diana Lazaroff, met my father. This book rang of childhood nostalgia for me. I was raised nearby until age 5, when my parents moved to Chicago about 1948.

But our lives were further intertwined. I postdoced with Stanislaw Ulam; he reviewed Ulam’s “A Collection of Mathematical Problems”, and knew him well. Three more misses: “Nixon announced that two billion dollars would be available for cancer research. The experts in the field were to gather in Warrentown, Virginia, a suburb of Washington, to divide up the pie. I was chairman of a committee on the use of mathematical methods. The other members of the committee were, John Jacques, Fred Grodins, Bob Rosen, Monas Berman, and John Hearon…. At Warrentown, we had a good time deciding how we would spend the money. Alas, it was a typical Nixon trick. He posed for TV cameras and gave away pens, but not a penny ever appeared.” I had postdoced with Bob Rosen at the Center for Theoretical Biology at SUNY/Buffalo, worked under John Hearon at the Mathematical Research Branch at NIH, and knew Monas Berman while there. Natalie and I had a strange encounter with Bellman’s former student John Casti at the Third International Workshop, Open Problems of Computational Molecular Biology, Telluride, Colorado, July 11-25, 1993, albeit after Bellman’s death. Casti, guest of honor, left the conference the first evening, when (not knowing who he was) I said to him “we can explain that” in reference to a remark about embryology by the host. Beyond that, the book is full of names of mathematicians and scientists whose work I knew, a slice in time through that culture, written by someone one generation ahead of me, but overlapping. It was quite a journey, watching Bellman’s parallel life.

It was from a couple of Bellman’s math books that I learned about concepts such as differential-delay equations and invariant embedding. The former helped me understand the 30 year cycle in academic hiring, reported going back to the 1800’s in:

Nyhart, L.K. (1995). Biology Takes Form: Animal Morphology and the German Universities, 1800-1900. Chicago,  University of Chicago Press.

Let’s say jobs are available for would-be professors. Lots of students decide to go into the open disciplines. By the time they are trained (the delay), the jobs are being snarfed up. So the next generation of students seek other disciplines. And so it goes, with no one doing long-range, 30 or more year planning, to equalize supply and demand. I suppose we could call the oscillating academic job market an emergent phenomenon! I actually hit one of those peaks, at age 33 in 1977, when I applied for 100 jobs, got a couple of interviews, and no offers. Out of luck, with 300 to 500 younger applicants per job opening at that time, I answered a phone call from Winnipeg asking me to recommend someone for a job there with “How about me?”. And so I ended up at the University of Manitoba.

Like Ulam (who is discussed in my blog on Erdös), Bellman was a mathematician first. For instance, he had a moral compunction to work on the H-bomb, but when his math didn’t prove useful to the project, he dropped out, rather than solve the problem with whatever it took. As with Ulam, we would not have seen eye to eye: “There is a subtle difference between mathematical biologists and theoretical biologists. Mathematical biologists tend to be employed in mathematical departments and to be a bit more interested in math inspired by biology than in the biological problems themselves, and vice versa” (Gordon, R. (1993). Careers in theoretical biology. Carolina Tips 56(3), 9-11,

I was about to wind up this blog by adding a photo of Bellman, but came across something even better, a movie by his grandson:

Bellman, G.L. (2011). The Bellman Equation [movie].;

So rather than color my blog by the movie, I’ll post this first, and enjoy the movie tonight with Natalie.

Let a hundred flowers bloom: Mao & Bill Gates


A small bit of tall grass prairie in “Silver Bog” in bloom. Photo by Dick Gordon

I am presently reading the magnum opus of philosopher of science Michael Ruse, Monad to Man: The Concept of Progress in Evolutionary Biology ( He told me when we met recently at Gulf Specimen Marine Laboratory & Aquarium ( that he is working on another book on Progress (or perhaps P/progress, human/biological, as he puts it) and I sent him our chapter from Embryogenesis Explained on Why evolution is progressive. The concept of progress has been a conundrum, ever since the ancient Greek atomists conceived the world as a collection of particles rattling around, bumping into one another and occasionally sticking. The idea was decried as leading to atheism. It is at the root of the much maligned reductionism, which itself may be at the root of much of successful science modelled on mathematics. We start with a set of assumptions and deduce the rest.

Atomism led to the problem of “How can there be anything new in the world?”. In other words, what are the sources of innovation? In social terms, how can we make a better world? Concepts of P/progress are indeed intimately entwined, as Ruse observes.

Yesterday (February 28, 2016) I read Bill Gates’ 2016 annual letter: More energy after hearing him talk about it on CNN. He and a number of lesser billionaires have decided that:

“…we need an energy miracle…. We need a massive amount of research into thousands of new ideas—even ones that might sound a little crazy—if we want to get to zero emissions by the end of this century. New ways to make solar and wind power available to everyone around the clock could be one solution. Some of the crazier inventions I’m excited about are a possible way to use solar energy to produce fuel, much like plants use sunlight to make food for themselves, and batteries the size of swimming pools with huge storage capacity.”

So I tested the waters:

Bill Gates
Breakthrough Energy Coalition
Dear Bill,
​Heard you on CNN this morning. In 2009 I published:
Ramachandra, T.V., D.M. Mahapatra, Karthick B. & R. Gordon (2009). Milking diatoms for sustainable energy: biochemical engineering versus gasoline-secreting diatom solar panels. Industrial & Engineering Chemistry Research 48(19, Complex Materials II special issue, October), 8769-8788.(

and have since gathered an international group of scientists (USA, France, India, Egypt) ​working on various aspects of the project. If we ever get the efficiency of artificial photosynthesis to an acceptable level compared to diatoms, we could then go the next step. For now diatom biofuel solar panels would use live diatoms.

Our primary goal is nothing less than replacing fossil fuels by diatom biofuel. Advantages of diatom biofuel solar panels ​are:

  1. Local, rooftop production of gasoline.
  2. Storable energy for transportation, heating, cooling, cooking, etc., riding through the day/night cycle and wind/no wind that plague electric solar and wind energy. No batteries needed. Gasoline has 44x the energy density of the best batteries.
  3. Estimated 10-200x oil production per unit area compared to seed oil crops.
  4. Retention of the matured gasoline engine technology, including well known methods for safe storage.
  5. No competition with food production (the bane of much ethanol production).
  6. Zero carbon footprint.
  7. Diatom biofuel solar panels may prove to be of low maintenance.
  8. Total energy independence for everyone, disrupting the current geopolitics of oil.


Yours, -Dick Gordon <>​


Now Natalie and I had previously run a workshop explicitly suggesting to Bill Gates how to spend the billions he wanted to use to stop HIV/AIDS, which resulted in a special issue:

Smith?, R.J. & R. Gordon (2009). The OptAIDS project: towards global halting of HIV/AIDS [Preface]. BMC Public Health 9(Suppl. 1: OptAIDS Special Issue), S1 (5 pages). Web:;

We didn’t ask him for any money, just that he send someone to hear us out. I broached the idea with and wrote to his representatives at the 2006 AIDS Conference in Toronto. No one came. I have no idea if he ever heard our request that he send a participant, nor if he read our articles. I had to conclude that he surrounds himself with gatekeepers, who filter out potentially innovative ideas. Sure enough, here is the reply to my present missive:


Breakthrough Energy Coalition<>

Automatic reply: Diatom biofuel solar panels

Thank you for contacting the Breakthrough Energy Coalition.  This is an automatic response acknowledging receipt of your email.

Due to the high volume of interest, we are not able to respond to each inquiry individually.  If you have contacted us regarding opportunities for funding, collaboration, or employment, we will keep your information on file.


So much for the support of innovative ideas. Then I read the fine print: “I recently helped launch an effort by more than two dozen private citizens that will complement government research being done by several countries. It’s all aimed at delivering energy miracles.” In the name of innovation, ideas screened by big governments will be passed on to the billionaires, or at least their gatekeepers, who will thereby receive the sifted wisdom of layers and layers of sifting out of (good) ideas. Yes, in my experience it is rare that good ideas, let alone the best ideas, survive such massive bureaucracy. Bill Gates has merely added another layer, a globalized layer, to the suppression of innovation. This is what I meant when I wrote:

Gordon, R. (1993). Grant agencies versus the search for truth. Accountability in Research: Policies and Quality Assurance 2(4), 297-301.

I woke up early this morning realizing I had heard Bill Gates’ words 60 years ago: “Let a hundred flowers bloom; let a hundred schools of thought contend”, espoused by Chairman Mao. The resulting cacophony in China was swiftly followed by a “crackdown… against those who were critical of the regime and its ideology. Those targeted were publicly criticized and condemned to prison labor camps” ( The innovators, the intellectuals, were humiliated, as they were in the subsequent Red Guard movement in China ( We live in a milder time now, at least in places where beheadings and labor camps are no longer in style, new ideas being dismissed with “Automatic reply”.




Biotensegrity everywhere?


Statue from somewhere in Georgia that we saw celebrating a family watching murmuration.

Scientists are always looking for general rules, and then stretch what we know to fit assorted Procrustean beds. For example, the rule that “everything is a fractal”, which means it looks much the same no matter how much you magnify or shrink it, works just fine with mathematical fractals, but does have limits when applied to the real world. Take a branching structure, such as a lung (Glenny, 2011). At the bottom you hit the alveoli (air sacs), and at the other end is the trachea (windpipe), neither of which resembles the intermediate, fractal-like branching structure in between. Another grand unifying scenario is The Emergence of Everything (Morowitz, 2002), as if atoms forming crystals, birds forming murmurations, and galaxies forming clusters were somehow one and the same phenomenon. Maybe so, maybe not, maybe they just share some mathematical properties and we thus think they are somehow projections from Plato’s ideals (Wikipedia, 2016). In Embryogenesis Explained (Gordon & Gordon, 2016) we took a crack at the concept of wholeness, showing it is well supported by quantum mechanics, yet could not extend it readily to the seeming wholeness of embryos. Rob Stone and I settled on what we called the Cybernetic Embryo (Stone & Gordon, 2016), and indeed are now trying to extend it to all of evolution. So I’m equally guilty of attempting to find grandiose patterns in nature. It’s fun, but, as William Bateson said, keep your eyes open and “Treasure your exceptions” (Cock & Forsdyke, 2008).


Knee anatomy. If the knee is treated as tensigrity structure then knee pain can be explained as the structure having some part out of balance and pain can be alleviated by correcting the balance of the assorted parts.  Blausen 0597 Knee Anatomy Slide by BruceBlaus Wikiversity Journal of Medicine DOI:1015347/wjm/2014/010

Last November we spent two weeks with Steve (an orthopaedic surgeon by training) and Olga Levin, and he almost convinced me (as he is convinced) that tensegrity is everywhere, at every level, from molecules to our muscles and bones. Now I had indulged in the concept much earlier, regarding the cell state splitter as a tensegrity apparatus, which I modelled using a Tensegritoy set (pp. 141, 170, 310 in Gordon, 1999). Natalie and I used the Wurfel, a tensegrity toy for toddlers, as a model of how changes in the configuration of the whole genome could explain the changes in gene expression during cell differentiation. We included this concept in the “nuclear state splitter”, which we have elaborated in great detail in Embryogenesis Explained, shedding the Wurfel model in the process. So we were open to Steve’s proselytizing (Levin, 2006) resulting in seeing shaped oil droplets as tensegrity structures in a recent blog (Gordon, 2016), stretching the concept to include polygonal diatoms and protocells at the origin of life.

For the uninitiated, an object consisting of stiff parts held together by elastic parts under tension has “integrity”, i.e., holds itself together. The word “tensegrity” was coined by Buckminster Fuller (Wikipedia, 2015).

(Animation of the simplest of tensegrity structures by Cmglee from Wikipedia Commons)

This week I started on a long hoped for adventure: working towards a realistic tensegrity simulation of the structure of cytoplasm. Last March Steve introduced me to Vytas SunSpiral of NASA, who recently brought in Dorothea Blostein of Guelph University. Both have been developing tensegrity robots for exploring the terrain of bodies in the solar system (SunSpiral & Agogino, 2015), and doing computer simulations (Blostein, 2016) of how they move. We have much to teach each other, and then a lot of work to do to introduce biological phenomena into their software. We hope to reach the point where we can challenge biologists to do appropriate measurements to test how close we come to simulating real cytoplasm. Then on to nuclei, spindle apparatus, whole cells, tissues, and embryogenesis, all seen as a panoply of tensegrity phenomena. Well, maybe.


Shaped droplets, diatoms and the origin of life

A remarkable paper appeared online 09 December 2015:

The authors, materials scientists from Bulgaria and the UK, mused out loud that their discovery that cooled oil droplets become polygonal had something to do with the morphogenesis of living creatures, but didn’t know which ones. I immediately started writing “On polygonal drops and centric diatoms” followed shortly by “The tensegrity origin of life via shaped droplets as protocells”, and some of the authors of “Self-shaping of oil droplets” are joining us as co-authors.

I had long been puzzling over the uncanny, nearly perfect symmetry of some centric diatoms, which I demonstrated by rotating a digital image of a diatom with n sectors by 360/n degrees and subtracting the images, in:

  • Sterrenburg, F.A.S., R. Gordon, M.A. Tiffany & S.S. Nagy (2007). Diatoms: living in a constructal environment. In: Algae and Cyanobacteria in Extreme Environments. Series: Cellular Origin, Life in Extreme Habitats and Astrobiology, Vol. 11. Ed.: J. Seckbach. Dordrecht, The Netherlands, Springer: 141-172.

Here’s a less perfect example than those used in that paper, the diatom Triceratium favus with n = 3, so the rotation is 360/3 = 120o (with kind permission of Stephen S. Nagy of Montana Diatoms):


The subtraction image on the right is black where the match is best. The two published examples, with n = 5 and 11, came out almost totally black. You can try this yourself with any front-on image of a diatom you can find on the Internet, if you have software that allows rotation by any angle. For example, try Word: Format Picture: Size: Rotate and scale, after trimming the picture so that the center of the diatom is in the center of the image. I’d like to see what you get. Please send the original, rotated and difference images to me at:, along with the exact source of the diatom image. Anyone mathematically inclined (and these diatoms instantiate a rotation group) may wish to write a computer program to quantify the degree of symmetry by coding some of the math in:

We in polar climes are all aware of the beautiful, generally hexagonal symmetry of snowflakes, which has it explanation in the crystalline stacking of water molecules in ice. Some can approach triangular, although they are hexagons with edges of different lengths:

Libbrecht2016 triangular.jpg

This is from:

Libbrecht, K.G. (2016). Guide to Snowflakes: Triangular Crystals.

with kind permission of Kenneth G. Libbrecht. More pointy triangular snowflakes may be seen at:

Bentley, W.A. & W.J. Humphreys (1931). Snow Crystals,  McGraw-Hill. (reprinted by Dover Press in 2003).

But diatom shells are not crystalline at all. They are made of amorphous silica, which at higher temperatures would be molten glass. They are frozen in the glassy state. Are diatoms real life cases of the liquid metal robot T-1000 in the movie Terminator 2? That puzzle is why diatom symmetry is uncanny.

So we start the New Year with a newly discovered phenomenon: oil drops that “should” be mere spherical blobs looking like diatoms. I’ll just show one oil triangle here (with permission of Nature Publishing Group), though the polygons go up to 11 sides:


Denkov&2015 Fig2b triangle.jpg

How can a liquid have sharp points like that?

Connections rattled in my brain. Denkov et al. suggest that the oil molecules line up at the perimeter, forming plastic-like bundles as cooling proceeds. Those bundles could be stiff, and prevent the drop from curving due to its surface tension. But then stiff rods confined by tension means that shaped droplets are tensegrity structures. But this is precisely what Steve Levin and I were complaining about the presentations at the origin of life conference we attended together last November: protocells, the blobs that supposedly led to life, had no postulated structure. Two problems solved at once! Diatoms and protocells are and might have been tensegrity shaped droplets. Martin Hanczyc’s oil droplet protocells might be polygonal under some conditions, and Vadim Annekov’s molecular dynamics simulations of diatom shell morphogenesis interacting with cytokeleton (in progress) may be enhanced. Not quite as good as the kids’ book “Seven in One Blow“, but a very satisfying pair of results.

And by the way, this is why theoretical biologists should be regarded as highly as theoretical physicists, although in general we don’t get no respect.

The Bagnold Dunes on Mars


Dick Gordon on a trip to revisit the Oregon coast in February 2013, with his back to the sea. The logs were gone. (Natalie Gordon)

When I was a graduate student in the Chemical Physics program at the University of Oregon (1963-67), I would occasionally take a break and drive with friends from Eugene to the Pacific coast. At that time huge logs that drifted in would be piled on the beach, and sometimes after we slept in an abandoned cabin up the steep cliffs, the next morning the logs would be seen totally rearranged. One does not turn one’s back on the sea.


Tide pool with typical huge anemones on the Oregon Coast near Gold Beach. Dick put his glove down for perspective. (Natalie Gordon)

Besides the tidal pools in the rocks with their anemones, snails and small fish, I was fascinated by the sand itself. For with each retreating wave dendritic patterns of darker grains atop the lighter toned majority were made. As I was (initially secretly) doing my first paper on morphogenesis:

I became fascinated by all pattern mechanisms, from rippling clouds to river deltas, and in sand. At that time Jack Carmichael was visiting my mentor, statistical mechanic Terrell Hill, working on the basic mechanism of column chromatography. That’s a strange name, because in most chromatography techniques then and now, one sees no colors. But here’s the origin of the word from an online dictionary:

  • 1930s: from German Chromatographie. The name alludes to the earliest separations when the result was displayed as a number of colored bands or spots.

The sand was doing real color chromatography, on itself.

Jack moved on, and after my first postdoc he invited me to spend the summer of 1968 with him at the Department of Polymer Science & Engineering at the University of Massachusetts in Amherst. It was a great, if hot summer, because I also met Ryan Drum there, and launched my career in diatoms. But that is another story.

Jack, his student Frank Isackson and I built a plexiglass, 6 foot long, one dimensional flume. It was 8 inches tall, and just wide enough to hold white pellet gun plastic balls so we could see every one. We ran water through it from one end to the other, slow enough so the balls were not dislodged. Then we would add one ball and photograph its bouncing motion (called saltation) as it made its way driven by the current, using a strobe light to record its motion.


I had done a lot of reading about how sand moves when driven by wind and water. Much of this literature was by Sir Ralph Bagnold, and I recall reading everything he wrote on the subject. While I was visiting Lewis Wolpert in London, UK in 1969, I took a train north to meet Bagnold at his country home, where he had retired, and spent a pleasant afternoon with him. He told me how he got interested in the motion of sand while in the English foreign legion in North Africa during World War II. He spoke of saltations so high during night sandstorms that one could see nothing horizontally, but could look up and see stars. I formulated the concept that it is important to meet the grand old men and women of science while they are still with us, and have frequently done so.


The sand on the Oregon coast arranged into small dunes by the wind from the Pacific ocean. If you were to put your eye at ground level and look across sand in wind, you could watch individual grains saltating. (Dick Gordon)

I did a computer simulation of the bouncing balls, and we published the experimental and computer results in my one and only sandpaper:

acknowledging Bagnold too dryly “for discussions”. A couple of days ago I read that the Mars lander is now exploring the Bagnold Dunes  on Mars, a fitting tribute to a life well spent on shifting sands.


Bagnold Dunes on Mars courtesy of NASA/JPL-Caltech/MSSS. This image, captured by NASA’s Mars Rover Curiosity on Sept 25, 2015 shows the dark Bagnold sand dunes in the middle distance.



The Oregon Coast February 2013 (Dick Gordon)