Tikalon Blog by Dev Gualtieri

Jumping Beans

April 17, 2023

I'm a member of the baby boomer generation, and comic books were a principal source of my childhood entertainment. Comic books at that time typically cost ten cents, which is slightly more than a dollar in today's money. I wasn't able to pay that much, but there was an option. In those days, before ubiquitous supermarkets, neighborhood grocery stores were common; and, there was one located across the street from our apartment. It was owned by a family of industrious Lebanese immigrants who would buy used comic books for a penny and resell them for two cents. That's how I was able to read science fiction comics such as Mystery in Space with the appearance of Space Cabbie.

One feature of 1950s comic books was the ads for novelty items. These were for such things as ant farms and Sea Monkeys. The Sea Monkeys were sold as the dormant eggs of the brine shrimp, Artemia, along with a nutrient mixture that allowed the eggs to hatch when mixed with water. Sea Monkeys were so named from the supposed resemblance of the shrimp tails to monkey tails, and the ads had drawings of humanoid creatures quite unlike these animals.

The brine shrimp, Artemia monica. These are usually less than a centimeter in length, and about 4 millimeters in width.

(Wikimedia Commons image by djpmapleferryman.)

One novelty item with some physics behind its operation was X-ray eyeglasses. While the ads showed these as a way to view the bones in your hand and see through clothing, the small print advised that what was seen was an illusion. The illusion was caused by the diffraction of light passing through a small hole, through the vanes of a feather, or an actual diffraction grating. Eyeglasses using the small hole method were first patented in 1906.[1] In 1971, Harold von Braunhut (1926-2003), the inventor of the Amazing Sea-Monkeys, patented eyeglasses incorporating plastic lenses processed to achieve a stress-induced birefringence to produce the X-ray illusion.[2]

The ultimate goal of marketing is to make money from sales of an inexpensive and low quality item. Today's prime example would be chicken wings; and, in the comic books, it was Mexican jumping beans. These are seed pods of the Mexican tree, Sebastiania pavoniana, infected with larva of the small moth, Cydia saltitans. Larvae of this moth bore into the immature green seed pods, and they devour the seeds from the inside. The larvae abruptly curl and uncurl inside the seed pod causing the beans to jump.

A jumping bean of the Mexican tree, Sebastiania pavoniana, along with the larva of the small moth, Cydia saltitans, tht infects it.

A jumping bean of the Mexican tree, Sebastiania pavoniana, along with the larva of the small moth, Cydia saltitans, that infects it.

The larvae escape through an opening at the end of the seed pods.

(Wikimedia Commons image, modified, by NobbiP.)

These jumping beans have a long shelf-life that allowed mail order sales from comic book ads, and they jump several times a minute when held in a hand and are thereby heated. Since these beans jump more at higher temperatures, Devon McKee, presently at the University of California, Santa Cruz, and A. Pasha Tabatabai from the Department of Physics of Seattle University (Seattle, Washington) thought that this process was a way in which the larvae are able to relocate from a potentially lethal sunny spot to tree shade.[3-4]

As the researchers state, organisms have strategies for motion that ensure their survival.[3] The larvae inside the seed pods execute their jumps blindly, and McKee and Tabatabai found that these jumps in random directions having no correlation to previous jumps are a random walk.[4] Given enough time, the jumping beans will always find their way from sun to shade.[4]

The researchers used image analysis to determine the quantitative attributes of the temperature-induced jumping, and they developed a computer simulation of the motion of the beans.[3] While a random walk is a slow method of travel, such a walk will visit every point in the plane if given enough time; so, the larvae will find the shade that they need.[3-4]

While their paper in Physical Review E is unfortunately paywalled, the idea is simple enough for me to program my own computer simulation, the C source code for which can be found here. This program utilizes two very useful Linux applications; namely, Gnuplot, for generating images of the instantaneous locations of the jumping beans, and ffmpeg, which allows stitching these images together as frames in a video. The program source code is commented to describe how these applications are used.

In my simulation, there's a large shade tree at the center of a field. The temperature of the field is 40°C, and this tapers to 20°C directly under the tree. A surface plot of the temperature distribution is shown in the following figure. The jumping beans are placed at random positions on the field outside the influence of the tree.

Temperatures within the simulated field.

The field is 1000 by 1000, and a shade tree with branches extending to a circular radius of 200 is placed at (500,500). The temperature tapers from 40°C to 20°C towards the center of the tree.

(The surface plot is an edited Gnumeric image. Click for larger image.)

When the rate of a process depends on temperature, we immediately think of the Arrhenius equation,

in which E_a is an activation energy, k_B is the Boltzmann constant, T is the temperature, A is a constant, and k is the rate. While we could find values of these to fit our jumping bean problem, I instead used the Arrhenius rule-of-thumb that rates double for every 5°C temperature increase, and I set the jump rate in the field at 8, decreasing by factors of two to zero at the center of the tree. You can see the starting positions of twenty beans and their positions after 20,000 jumps in the following figure. A video of the jumping process is also provided.

The starting positions of twenty beans and their positions after 20,000 jumps

The starting positions of twenty beans (left) and their positions after 20,000 jumps in the simulation (right). In this case, twelve of the twenty beans found shade under the central tree, and the eight others may have found a shade tree outside this patch. (Click for larger image.)

Video is released under a Creative Commons License. Right click to download MP4 video file.

Jumping Beans

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