Many Hands Make Fractals Tactile

But what if we could experience math directly — just as we experience language by speaking it? Some years ago I founded an organization, the Institute for Figuring, dedicated to the proposition that many ideas in math and science could be approached not just through equations and formulas but through concrete, physical activities.

Take fractals, mathematical structures or sets with intermediate dimensionality. Coined by the mathematician Benoit B. Mandelbrot, the term comes from the Latin “fractus,” meaning broken. Instead of having one, two or three dimensions, a fractal will have, say, 1.89 or 2.73 dimensions.

Margaret Wertheim/Institute for Figuring

The Institute for Figuring, founded by Margaret Wertheim, is dedicated to the proposition that many math and sciences ideas could be approached through physical activities. One such idea is fractals: mathematical structures or sets with intermediate dimensionality. Working with Jeannine Mosley, a software engineer, Ms. Wertheim spent the past year teaching people about them by building a giant fractal model out of 48,912 business cards.

This sounds absurd; indeed, in the late 19th century, when mathematicians began to explore such forms, they were flabbergasted, using terms like pathological to describe them.

Fractals possess the strange property of “self-similarity.” Zoom in on any part of a fractal, and each small section will have the same richly complicated structure as the whole.

Clouds and coastlines, with what Dr. Mandelbrot called their highly “wiggled” geometries, exhibit this self-similar scaling (at least to the degree possible in nature, for true fractals with infinite levels of patterning are mathematical ideals).

Over the past year, working with Jeannine Mosely, a software engineer, I have led a project to teach people about these concepts by having them build a giant fractal model out of 48,912 business cards (all man-made models are also approximations). The finished object, known as the Mosely Snowflake Sponge, is on display at the University of Southern California’s Doheny Memorial Library.

Christina Simons/Institute for Figuring

The finished object, known as the Mosely Snowflake Sponge, is on display at the University of Southern California’s Doheny Memorial Library. This image shows a layer of a Level 2 Mosely Snowflake Sponge, which reveals an internal anatomy of crosses and rings.

Hundreds of students across campus participated in the construction, coming from departments as diverse as fine arts, psychology, cinema studies and engineering. High school students, professors, librarians and local artists took part.

Together we folded business cards into cubes and linked thousands of cubes in intricate configurations that reveal the fractal’s self-similar anatomy. Visual symphonies of rings and crosses — then rings of rings, rings of crosses, and crosses of crosses — became apparent to the eye, a physical manifestation of the concept of recursion.

Construction took more than 3,000 hours, and the form we made, held together by nothing more than the folded cards — no glue or Scotch tape — stands as a sculptural monument to a once heretical abstraction.

Christina Simons/Institute for Figuring

Visitors to the institute are encouraged to explore and play. David Orozco, an artist and stay-at-home father, developed an elegant, boxy lattice.

The idea of making models of fractals out of business cards was dreamed up by Dr. Mosely, a leading practitioner of mathematical origami and a specialist in curved origami, which requires a serious knowledge of differential geometry.

From 1996 to 2005, Dr. Mosely spent her leisure time supervising the building of a 66,048-card model of a fractal known as the Menger Sponge, named for its discoverer, Karl Menger, the Austrian mathematician, and for its resemblance to a sea sponge. Imagine a cube riddled with hundreds of square-shaped holes — it is the three-dimensional analog of an important mathematical object known as the Cantor Set.

After she had made the Menger Sponge, Dr. Mosely realized that it was one of a whole family of fractals. Some are trivial, others cannot be made, but one was especially interesting. She named it the Snowflake Sponge for its enigmatic sixfold symmetry.