Building Blocks
A large paper cube, seemingly composed of symmetrical cutouts, sits on a table at the entrance to the Washington and Lee University Math Department in Robinson Hall. On closer inspection, it turns out to be a compilation of discrete smaller cubes. Is it a 3-D M.C. Escher woodcut? An architectural model of a Moorish temple? No—it’s a Menger Sponge.
And that is a three-dimensional fractal, a mathematical set that exhibits a repeating pattern that displays at every scale. This particular fractal is made by taking a cube and cutting out a square section through the center in each of the three directions. Then each of the resulting smaller cubes is cut out in the same way, and so on until you’ve removed infinitely many pieces. Each Menger Sponge is made from 20 identical, but, smaller Menger Sponges. This results in an object that has zero volume but infinite surface area.
W&L math faculty and students built the Menger Sponge on Oct. 29 and 31, part of a crowd-sourced, worldwide project to create a MegaMenger. Elizabeth Denne and Michael Bush, both assistant professors of math, were W&L’s on-site coordinators: 20 other math departments did simultaneous builds, including neighbors VMI and JMU. “This is a cool and unexpected example of global learning,” said Suzanne Keen, dean of the College. “There were people all around the world collaborating to build a giant fractal.”
The project is pretty simple. All it takes is rectangular cards, all the same size, and many willing hands. As she quickly folded cards and slotted them together, Denne explained, “It takes six cards to make a cube, which is a level-zero Menger Sponge. It then takes 20 level-zeros to make a level-one Menger Sponge. From there, it takes 20 level-ones to make a level-two, and so on. It grows exponentially.” She added, “There’s no glue, it’s just folded business cards that you pop into place. The cubes are really quite robust, and the way the cubes fit together is an engineering marvel.” As a finishing touch, the cubes are covered with a card printed in a repeating pattern, giving the fractal even more visual depth.
The goal of the worldwide project called for each of the 20 sites to build a level-three fractal, which would measure about one and a half meters long. For each site, that equals 8,000 small cubes made out of nearly 50,000 business cards. If all of the 20 level- threes (one million business cards) were combined, the result would make a huge level-four fractal, also known as a MegaMenger.
Denne estimated that, she, her students and her math colleagues could make a level-two Menger Sponge. “I figured it would take 30 to 40 people hours.” At any given time, she had about 10 students working on the cubes, and every math professor participated at some point. “We recruited some students who were doing homework in the math lab to join us for a bit,” said Denne. “They said it was quite meditative and relaxing. I think it gave them a moment in their busy lives to slow down for a moment. You don’t really have to use your mind while you fold the cards, but by handling these cubes, they are getting a sense of spatial orientation. Some students picked up the process faster than others, but everyone came up to speed pretty quickly.”
The build progressed over two separate periods, and the Math Department, which purchased the cards, also provided beverages, cookies and pizza to fuel the students. “Our second build started at around 4:45 on Friday afternoon and finished at 9 p.m.,” said Denne. “We had a blast, and every time I see our finished fractal, it brings a smile to my face.”
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