Tag Archives: swarms

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. https://carnegiescience.edu/events/lectures/re-conceptualizing-origin-life

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. http://www.red3d.com/cwr/boids/

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.