Bogglers: Mathematica Redux

DISCOVER Vol. 23 No. 4 (April 2002)
Table of Contents

Bogglers
By Scott Kim

Mathematica Redux
In the late 1950s IBM asked designers Charles and Ray Eames to create an exhibit for the Los Angeles Museum of Science and Industry. The result was Mathematica, a celebration of mathematics that endures to this day as a standout among science exhibits. Now permanently housed at the Boston Museum of Science, Mathematica was intended, in the words of Charles Eames, to enlighten the amateur without embarrassing the specialist. A traveling version of the exhibit is at San Francisco's Exploratorium through May 5. It will then move on to the California Science Center in Los Angeles, where it will be in residence from July 12 through September 3.

LOVE TRIANGLES
A sign in Mathematica reads: "Projective geometry is the study of geometric properties that are preserved by projective transformations." To illustrate, the Eames studio suspended an array of triangles so that they formed different designs when viewed from different angles. Shown at right are two colored triangles suspended inside a cubic frame and the views you would see from the front, side, and top.

Below are four pairs of views. In each case, eight colored triangles are suspended inside a cubic frame. Each triangle is the same color on both sides. Can you draw the third view for each cube? As in the pairs of views shown, all eight triangles are fully visible in the third view. Can you figure out how the triangles are arranged in the cube? Hint: Problems 1 and 2 are closely related, as are problems 3 and 4.

MÖBIUS STRIP

Mathematica features a famous loop with a twist, similar to the one above, which is named after the 19th-century German mathematician and astronomer August Ferdinand Möbius. A red arrow travels along a track that runs down the middle of the giant Möbius strip in the exhibit. Press a button, and the arrow crawls along the track, soon appearing on the underside of the strip. The arrow then continues in the same direction, returning ultimately to its starting point.

1. What would happen if you cut the strip in half along the track line?
2. Imagine that instead of one track down the middle, two tracks run side by side, dividing the strip in thirds, as shown in the figure at right. How many trains would be required to travel all the tracks?
3. What would happen if you cut the strip in thirds along the tracks?
4. Study the unfolded track below. Can you glue the ends together so that a single train could traverse all the tracks on both sides? Note: The train must go straight—it can't make a turn at the intersections. How could you glue the ends together so that two trains are required to traverse all the tracks? How about three trains? Can you connect the ends so that four trains are required to travel all the tracks?

SQUARE DANCE
This curious diagram hangs on a wall in the Mathematica exhibit. Nine squares, all different sizes, fit together to form a rectangle. This is the smallest number of squares that can appear in such a construction. Can you deduce the sizes of the squares in this diagram and in each of the diagrams at right? The squares have been stretched into rectangles to avoid giving away their relative sizes. All dimensions are whole numbers. Hint: Select two adjacent squares and call their edge lengths x and y. Express the sizes of the other squares in terms of x and y. Eventually, you will reach a point where the size of a square can be expressed two different ways, and you can resurrect your long-dormant algebra to solve for x and y. Choose the smallest numbers that work. Some squares in problem 1 are the same size. In the remaining problems, all the squares are different sizes.

Solution

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The original work on the squared rectangle problem was done by William Tutte, C.A.B. Smith, A. H. Stone, and R. L. Brooks in 1938, culminating in the discovery of a squared square (a square divided into squares of all different sizes). It has recently been proved that the smallest possible number of squares in a squared square is 21. For more about the Mathematica exhibition, see www.eamesoffice.com (Eames Office), www.exploratorium.edu (the Exploratorium in San Francisco), and www.mos.org (Boston Museum of Science).