Size Matters: The Square-Cube Law and Really Big Feet

Size is a trait so strongly associated with Bigfoot it has become embedded in its mythology. After all, “Big” is half of the name. But just how Big are we talking about here?

Short of weighing one in a zoo, there are a few mathematical ways we could guess the answer to this question, and it starts, conveniently, back at the other half of the name: the Foot.

The hairy subject seen in the Patterson-Gimlin Film in 1967 is often the focus of any debate, analysis, or controversy surrounding the film, but it is often underappreciated (or not known) that it also left a long trackway embedded in the soft silty riverbank. This physical record not only serves as corroborating evidence that something very heavy did indeed walk across the riverbank on October 20, 1967, but also allows us to theorize about the body dimensions of the subject with more concrete numerical methods than otherwise would be possible.

First up is the height. There’s no hard and fast way to settle this once and for all as the height of the subject seen in the film is subject to a few variables that cannot be known for certain. Bill Munns has some analysis up using a trigonometric calculation of the subject height as a function of the distance from the camera to the target, the height of the subject in frame, and the focal length of the camera. Unfortunately, all three of these variables are not known for certain; the distance from Roger to the film subject could only be known by measuring the distance from Roger’s footprints to Patty’s, and those are long gone. The exact lens on the camera is also not known, and the exact height of the subject in frame is subject to perspective shortening. That is, the subject is not perfectly upright and perpendicular to the plane of the camera, and any bend or hunch away from the axis of the camera will make the subject appear shorter than reality.

We can use the footprint to take an inconclusive but reasonable guess at the true upright height of the subject; According to an article in the Journal of Forensic Sciences, 1991, a human foot is approximately 15% the length of the owner’s standing height. As the great philosophers Meldrum and Barackman often say, a track is not an image of the foot, but rather the damage the foot did to the substrate. Nevertheless, the 14.5 inch track left in the clay-like riverbank was about as crisp and clear as you can get a footprint, and it’s certainly a reasonable measure within a margin of probably less than an inch. To make a sports analogy, you might have gotten 2 yards, or you might have gotten 3, but you sure as hell didn’t get 5.

A 14.5″ foot length, at 15% of the standing height, would mean the owner stands just a hair over 8 feet tall (96 inches).

[latex] \frac{14.5}{h} = \frac{15}{100} [/latex]

 

[latex] h = 14.5 * \frac{100}{15} = 96.67 [/latex] inches

This 15% ratio isn’t precise, as Patty doesn’t have human dimensions and even among the known great apes there is variation in this ratio. In Wang et al. (2004), the authors noted a 14.3% foot – to- height ratio for humans, and a 20.8% ratio for orangutans, a 17.6% for gorillas, and a 16.8% ratio for chimpanzees, based on skeletal measurements. So, although 15% is not absolute, we know this ratio is probably not as low as 10% nor as clownishly high as 25%. A foot that is a quarter the length of the entire standing height would appear close to the length of the femur in the film, and we can pretty clearly (using the scientific tool of ballpark-eyeballing) see in the film that is not the case. Assuming 15% is an overestimate would push the standing height of Patty further over the 8 feet mark, but if we assume an upper bound of 20%, then Patty stands at 72.5 inches, or 6 feet and one half inch. A 2 foot margin of error is probably not a very satisfying answer, but given that the Patty track has an aligned big toe like a human foot and not like the grasping divergent big toe of an orangutan, the ratio is probably more safely towards the 15-18% estimate considering the function of the foot is much more similar to a terrestrial organism such as a human or a gorilla. An 18% foot-to-height ratio, higher than every extant hominoid except the arboreal orangutan, would still produce a standing height of 80.5 inches, or a little over 6 foot 8 as a conservative floor. As the foot has a non-divergent hallux (big toe) and the subject moves bipedally, the ratio is likely more similar to that of a human (14-15%) and less similar to that of a chimp or gorilla (17%). A 16-17% ratio tightens this estimate to 7’1″ – 7’6″.

A slightly tighter estimate of 6′ 8″ to 8’1″ is still rather imprecise, but it certainly puts the subjected observed in the film between the ranges of very tall and obscenely, insanely tall. Although this is absurdly large for a human, or any known ape, keep in mind that in terms of absolute size for large land mammals, it’s not actually that extraordinary, even for North America. The fascination with the huge size of these animals is a potent part of its marvel and mythology, but as scientists it is important to note how believable and even normal this size of creature actually is.

For example, take the grizzly bear. People forget that, despite its severely restricted range these days, it used to live as far south as central Mexico, roaming the shores of the California coast until Western settlers exterminated them in the 19th and 20th centuries. Although they became tragic casualties of colonialism, it’s not exactly hard to see why newcomers to the Wild West saw the grizzly as a threat to humans and livestock alike: adult males can stand 8 to 10 feet tall on their hind legs and weigh upwards of 1000 pounds. This is an underappreciated fact about bears as their shoulder height on all fours is only about 4 feet on average. Patty was probably similar in size to a medium sized grizzly. Bipedalism gives the illusion of greatly increased size, which is a very good reason why many animals employ this move as a threat display.

So how heavy is Patty? If we assume she was the size of an average grizzly, then 400 pounds is a pretty safe starting point, as it is the average weight of an adult male grizzly living in Yellowstone National Park but only a fraction the mass of the fattest of their coastal Alaskan cousins. Bears and apes are built kind of differently, though, and that’s where looking at a more closely related animal may be more informative.

Let’s assume Patty is built more like a gorilla, the largest extant great ape. A silverback gorilla can be 6 feet tall and 500 pounds at the most, although couch potatoes living in zoos can fatten up to over 600 pounds. If Patty is indeed 6’8″, then that makes her 11% taller than the largest gorilla. You might think that also makes her 11% heavier, at 555 pounds, but that’s where this nifty quirk of physics called the Square-Cube Law comes into play. Because we live in a 3-dimensional universe, usually when body size increases in one dimension (say, height), the volume of the animal grows in all three dimensions. Volume is then a cubic function of height, and the surface area of the body is the square.

A 72 inch, 500 pound gorilla scaled to 80 inches would weigh 684 pounds by the Square-Cube Law.

[latex] 1.11^3 * 500 = 1.37 * 500 = 684 [/latex]

 

If Patty was more like 8 feet tall, which may be the maximum height of a large adult male, then the numbers get MUCH larger.

[latex] (\frac{96}{72})^3 * 500 = 1185.2 [/latex]

 

That’s a pretty hefty half-ton animal, but again it would only be as tall and heavy as a large male grizzly bear or bull Elk, animals that share an ecosystem with it while managing to number in the tens of thousands in Washington State alone. So much for the “insufficient resources to exist” argument. As is the case for the foot to height ratio, there is some imprecision inherent in this estimate as even a cursory glance at the PGF makes it obvious that Patty has some very different proportions from a gorilla, but we can be pretty comfortable in asserting that at around 7 feet tall, Patty probably did not weigh less than 500 pounds, nor more than 1000.

The calculations tell a story of an animal that is definitely big and heavy, but pretty on par with other large animals it shares a habitat with, which specifically tend to be mountainous, rainy temperate forests with high annual precipitation. This is consistent with Bergmann’s Rule, which states that animals living in colder climates tend to have increased body size relative to animals living in warmer environments. This phenomenon can conveniently also be explained with the Square-Cube Law, because the volume (and therefore heat generating and insulating mass) of an animal increases with the cube of an animal’s length while the surface area through which it loses heat to the environment only increases as a function of the square. As the animal grows larger, it gains insulation faster than it gains exposed skin, resulting in better energy efficiency in cold climes.

The Square-Cube Law also opens up other lines of questions that are unanswered and worth investigating. For example, it is theorized by Dr Meldrum that the population density of Sasquatch is probably two orders of magnitude less than the black bear population, with around 300 individuals in Washington state compared to 30,000 bears. Bears give birth to 2-4 cubs every 2-4 years. As any human mother can attest, hominoids have obnoxiously long gestation periods, difficult births, and long infancy times relative to other mammals, resulting in very sparse reproduction rates in the wild. Female orangutans giving birth to one baby only once every 8 years. Increased body mass also increases gestation and maturation time, creating increasingly large resource demands of the mother. This is one of the main reasons why the largest land mammal to ever live, Paraceratherium, was only the size of a medium sized dinosaur, whereas the largest sauropods still hatched out of eggs no larger than a football. (Whales conveniently dodge the problem of a multi-ton pregnancy by being effectively weightless in water). This mathematical relationship could further put downward pressure on the reproductive rate, both explaining the relative rarity of encounters with this species and also serving as a warning regarding the perpetual fragility of this already rare population.

 

References:

[1] Giles, E., & Vallandigham, P. H. (1991). Height estimation from foot and shoeprint length. Journal of forensic sciences, 36(4), 1134–1151.

[2] Wang, W. J., & Crompton, R. H. (2004). Analysis of the human and ape foot during bipedal standing with implications for the evolution of the foot. Journal of biomechanics, 37(12), 1831–1836. https://doi.org/10.1016/j.jbiomech.2004.02.036