Buffon's Needling Ants

The classic probability experiment known as Buffon's needle produces a statistical estimate of the value of pi (π), the ratio of a circle's circumference to its diameter.

The experiment consists of randomly dropping a needle over and over again onto a wooden floor made up of parallel planks. If the needle's length is no greater than the width of the boards, the probability of the needle meeting or crossing a seam between boards is twice the needle's length, l, divided by the plank width, d, times pi: 2l/dπ.

The idea of estimating pi by randomly casting a needle onto an infinite plane ruled with parallel lines was first proposed by the naturalist and mathematician Georges-Louis Leclerc, Comte de Buffon (1707-1788). He himself apparently tried to measure pi by throwing sticklike loaves of French bread over his shoulder onto a tiled floor and counting the number of times the loaves fell across the lines between the tiles.

In 1901, the Italian mathematician Mario Lazzarini claimed to have tossed a needle 3,408 times and obtained a value of pi equal to 355/113, or 3.1415929, which differs from the exact value by less than 0.0000003. How he managed to ensure truly random needle casting and got his remarkably accurate pi estimate isn't clear, though mathematicians have argued that cheating must have been involved.

Subsequent experiments by other investigators typically produced less accurate values of pi. In recent years, computer simulations have taken over—with results modulated by quirks of the random number generators involved in the computations.

In the year 2000, researchers reported that an ant species appears to use a Buffon's needle algorithm to measure the size of potential nest sites. Eamonn B. Mallon and Nigel R. Franks described their findings in the article "Ants estimate area using Buffon's needle," published in the Proceedings of the Royal Society, London (B).

Ants of the species Leptothorax albipennis inhabit small, flat crevices in rocks. A colony typically consists of a single queen, her brood, and 50 to 100 workers. When a nest happens to get destroyed, the colony sends out scouts to assess potential new nest sites.

Given a choice, a colony's preference is for a nest of a certain "standard" size (related to the number of ants in the colony), which suggests that these ants can measure area. How do they do it?

Mallon and Franks collected ants from areas near the Dorset coast of England and cultured them in the laboratory. They then transferred colonies to large, square Petri dishes and offered the colonies choices of various cavity habitats, made from pairs of microscope slides with cardboard walls spanning the narrow gap between glass floor and glass ceiling.

"We used such microscope slide nests with nest cavities of different sizes, shapes, and configurations in order to examine preferences," the researchers noted.

Experiments involving individually marked workers demonstrated that a scout generally spends about 2 minutes scurrying within any particular candidate cavity. Moreover, scouts typically end up making two visits to an acceptable nest site before recruiting followers.

When a scout initially explores a potential nest site, it lays down a pheromone-laced track. On its second visit, it follows a different track, repeatedly crossing its original path.

Mallon and Franks argued that a scout can obtain an estimate of the potential nest's area by detecting the number of intersections between the first and second set of tracks. In effect, an ant scout applies a variant of Buffon's needle theorem: The estimated area, A, of a flat surface is inversely proportional to the number of intersections, N, between two sets of lines, of total lengths S and L, randomly scattered across the surface, or A = π.

"There is evidence that individual scouts recognize and respond to intersections between their second visit path and their first visit path," the researchers said. "Scouts briefly but significantly slowed down during their second visit when they intersected their first visit path."

Additional observations bolster the plausibility of the claim that these ants assess nest size using a Buffon's needle algorithm. Moreover, the method is relatively insensitive to the shape of the area to be assessed and to the exact pattern of the tracks (as long as the tracks are not concentrated within just one region). In addition, it will work in complete darkness.

"Our findings, that individual ants can make accurate assessments of nest areas based on a rule of thumb, show in a unique way how animals use robust algorithms to make well-informed quantitative decisions," Mallon and Franks concluded. The results demonstrated how information gathering by individual workers can contribute to crucial collective decisions.

Originally posted May 15, 2000

Elevator Buttons and Stone Steps

Human activity can leave telltale marks on its surroundings. These marks, in turn, can provide clues about the nature of the activity that created them or about the setting itself. See, for example, "Statistical Wear" and "An Irresistible Edge."

Recently, I started paying attention to wear caused by finger contact in the vicinity of elevator buttons. Shown below is a set of buttons for a hotel elevator. What can you tell about the setting just from the scuff marks?

The wear pattern suggests that this particular set of buttons is most likely located on the lobby floor; many more people have pressed (or tried to press and missed) the "up" button than the "down" button. The curious tail toward the right indicates that people tended to come from the right, presumably making contact with the brass plate prematurely.

Indeed, these buttons are on the lobby floor of a hotel in Toronto, and the only entrance to this bank of elevators is from the right, with no exit to the left.

You'll see similar wear marks in the example below, from a hotel in Austin, though the distribution isn't quite as strikingly asymmetric.

Wear marks appear in all sorts of settings. Foot traffic can be responsible for some of the more striking examples, particularly when the marks appear on stone, as seen in the photo below of a stone staircase in Wells Cathedral in Great Britain.

What can you say about the traffic patterns that these worn steps reveal?

Photos by I. Peterson

An Irresistible Edge

Based on a trapezoid split into two triangles, the National Gallery of Art's East Building features walls that meet at odd angles, eschewing the right angles of more conventional structures.

One particularly sharp corner, visible to the right of the East Building's H-shaped, west-facing façade, has attracted a lot of attention. There, two walls meet at 19 degrees to form the apex of a narrow triangle. This 19-degree "fin" rises 107 feet from ground to roofline. If you look closely at the corner, you'll see a smudge darkening the lavender-pink marble.

Over the years, so many people have felt the urge to touch the unusually sharp corner that countless hands have deposited their oils on the marble to create a dark stain. It stretches over a span of about two feet, tapering off at its upper and lower ends.

In effect, the stain is a population distribution, representing all the people who have visited and touched the corner, given that most people naturally reach out to touch the wall just below shoulder height.

Normal distribution (bell curve).

The fin's stain is just one of many instances in which human use can leave its mark on objects in the environment. Such usage or wear patterns can often tell you something about the population or behavior responsible for the marks. See, for example, "Statistical Wear."

Statistician Robert W. Jernigan of American University has long studied such patterns and collected images that illustrate statistical ideas. His blog, "Statpics," includes a wide variety of such examples.

One of my favorite examples from my own travels is a stone staircase in Wells Cathedral, England, where centuries of foot traffic have worn characteristic concavities into the steps.

Photos by I. Peterson

Statistical Wear

Marks on objects can provide intriguing statistical glimpses of usage patterns. The darkened leaves of a well-thumbed book may point to favorite passages; the distinctive hollows of oft-traversed steps suggest the characteristic tread of countless feet.

Last year, while I was at East Tennessee State University, I happened to notice a particularly striking example of such "statistical wear" on the door to the men's restroom, just down the hall from my office. Entry to the restroom was by a swinging door, which opened inward with a push.

Countless hands pushing on the door had worn away the brown stain in one particular area, reflecting where men had preferred to place a hand. The result was a roughly circular spot—a two-dimensional statistical distribution—with the most wear in the middle and progressively less wear away from the center.


Countless hands have worn away the stain in one particular spot on the swinging door to a men's restroom. Photos by I. Peterson.

Two factors probably contributed most to the two-dimensional pattern: the height of the individuals pushing on the door and perhaps some preference for how much force to apply (less force would be required to push open the door farther away from the hinges). Curiously, the pattern largely misses a brass plate that was likely supposed to be the target.

What pattern would you expect to see on a nearly identical swinging door to the women's restroom?

The pattern is similar, but it is lower and a little closer to the door's edge, reflecting a lower average height and a greater preference for pushing with less force. As a result, the pattern has a significantly greater overlap with the door's brass plate.


The door to the women's restroom also has a distinctive two-dimensional wear pattern.

Statistician Robert W. Jernigan of American University has been collecting such "visualizations" of statistical concepts for many years, from the pattern created on a brick wall by a leaking downspout to oil stains on a parking lot. His fascinating "Statpics" blog is devoted to images that illustrate statistical ideas.

Jernigan's paper, "A Photographic View of Cumulative Distribution Functions," appeared in the March 2008 Journal of Statistics Education.