Mike Mertsock

A recent Simpsons rerun (Sweets and Sour Marge) featured the citizens of Springfield erecting a tall human pyramid. It’s pretty tall, at least ten levels, so I was curious how much weight each person at the base of the had to support. With homer at the apex of the pyramid, the weight had to be high. It turns out that it is a good thing that Ned, who is in the bottom level of the pyramid, is buff.

I wrote a Java program that calculates the average load each person bears at the base of a human pyramid. You specify the average weight per person and the height (number of levels) of the pyramid. It does calculations for five different pyramids: the traditional “flat” pyramid, a square-based pyramid (like the Egyptian pyramids), a tetrahedron, a four-dimensional pyramid (having n³ people at level n), and a “smart” design (explained below).

Pyramid Types

The four pyramids described below, except for the four-dimensional one, clockwise from top left.

Flat (2D) pyramid

This is the traditional pyramid that cheerleaders and kids in phys ed class build. The number of people at the bottom of an n-level flat pyramid (designated for now own as the function f(n)) is f(n) = n. A 4-level flat pyramid has a total of 10 people.

Square pyramid

A basic three-dimensional pyramid in the shape of the Egyptian pyramids — it has a square base. A square pyramid of people would have the same shape, with four people supporting each person above (one person supports the left hand, another supports the right hand, a third supports the left leg…). The people-per-level function is f(n) = n². A 4-level square pyramid has 30 people in it.

Tetrahedron

This is a three-dimensional pyramid with a triangular base. This seems less practical for human pyramids — I can’t think of a good way to support someone at three points. The people-per-level function is f(n) = (n² + n) / 2. A 4-level tetrahedron has 20 people.

Four-dimensional pyramid

This is described by the function f(n) = n³, which is very hard to visualize in three dimensions. This is the four-dimensional equivalent of a square pyramid. I made calculations for this to see the effect of adding an extra dimension.

Smart pyramid design

This design was inspired by an image of a human pyramid in the Hindu festival Krishna Janmaashtami, in which the top three levels have one, then one, then two people. I realized that in these top three levels, assuming an even distribution of weight, nobody has to support more than the weight of one person. So my smart pyramid design uses an algorithm described by f(x) = the minimum number of people needed so that each person supports an average load of two people or less. A four-level smart pyramid has only 5 people.

Measurements

My Java program calculates the number of people per level, the total people per n-level pyramid, and the average load per person at the base of an n-level pyramid. You specify the number of levels and average weight of a person, and the program does the calculations for the given number of levels and also every number down to 1.

All calculations assume an even distribution of weight. I made this choice to simplify the calculations for this initial investigation.

The program

The Java program consists of a main class PyramidCalc and an abstract class Pyramid that the various pyramid algorithms implement. This makes it easy to add new pyramid types to the calculations.

The output for the program is comma-separated values that I imported directly into my spreadsheet software to make some nice tables and graphs.

You can download the program source code, class files, and output (in PDF and OpenOffice Calc formats) are available for download (public domain).

Results

Results discussed here are from the tables and charts PDF file.

As expected, the average load per person quickly becomes unreasonable for the “flat” pyramid. The average load per person at the bottom of a 10-level flat pyramid is 675 pounds, assuming an average weight per person of 150 pounds. The three-dimensional (square and tetrahedron) pyramids surprisingly are not much of an improvement over the flat pyramid, with average loads of 427.5 lb and 450 lb respectively.

As shown in the charts in the results file, the “smart” pyramid is quite different from the other pyramids. By definition, the average load per person is never more than the weight of two people (300 pounds in my calculations). Thus, it has a chance of being a feasible means of constructing human pyramids with more than five levels.

However, the “smart” pyramid has an unexpectedly exponential behavior when measuring the total number of people in the pyramid. It starts out being quite efficient, having half as many people in a five-level pyramid and only 62 in a 10-level pyramid, compared to 55 in a flat pyramid and 385 in a square pyramid. However, the number of people grows explosively after 15 levels, jumping to 27,309 in 25-level pyramid, compared to 325 and 5,525 for flat and square pyramids.

When I put this in a chart (the second chart in the results file), initially with a plain linear scale, the smart pyramid curve made the other curves imperceptible, especially below 15 levels. So I tried a logarithmic scale for the y-axis (number of people), and the smart pyramid curve unexpectedly turned out being an almost straight line. This was unexpected because I had no clear formula for calculating the number of people per level in this pyramid, only a practical, recursive algorithm.

Further research

It’s not clear yet how tall the “smart” human pyramid can feasibly be, because these calculations assume an even distribution of weight. In reality, each person at a particular level of a human pyramid supports more or less weight depending on his or her position and the arrangement of body parts. Therefore, the next step is to introduce a quantitative load distribution model into the calculations to determine the level at which maximum loads exceed reasonable limits. This could in turn result in a reformulation of the smart pyramid model to attempt a better distribution of weight.

It’s easy to visualize the arrangement of the flat and square pyramids, and thus it’s easy to determine a specific formula for intra-level weight distribution. For example, for the flat pyramid, the binomial coefficient C(n, k), times an appropriate constant, is probably an accurate formula for the load on the k-th person at the n-th level (counting from the top) of the pyramid. This is intuitive because a flat human pyramid looks exactly like Pascal’s triangle. The tetrahedron is just weird because it asks for three people to support one person. This is hard to imagine because the number of extremities we tend to use is not divisible by three.

However, the smart pyramid is more complex. First, I only know a recursive formula for f(n) instead of a simple formula as the other pyramids have. Also, a general pattern for the physical arrangement of people in this pyramid is not immediately apparent—this means that there is not yet an algorithm to describe how individuals at each level arrange themselves and support each other.

Such an algorithm is not trivial because of the way the number of people per level grows; indeed it is probably not even meaningful to find a way to arrange people in sufficiently large smart pyramids. The point is that until such an algorithm is formulated, there is no way to accurately determine the load per person at an arbitrary position; the best that can be done is manual arrangement and calculations for small numbers of levels.

This entry was posted on Friday, October 14th, 2005 at 12:38 am and is filed under Computer Science, Fun. You can follow any responses to this entry through the RSS 2.0 feed. You can leave a response, or trackback from your own site.