Incomplete Open Cubes

Examples of Sol LeWitt's Incomplete Open Cubes on display at City Hall Park in New York City.

A cube has six faces, eight vertices, and twelve edges. In his series titled Incomplete Open Cubes, conceptual and minimalist artist Sol LeWitt (1928-2007) chose to work with cubes represented as frameworks.

LeWitt started by removing one edge from an open cube, then two edges, and so on, as an exploration of how many variations of an incomplete open cube exist and what they look like. One key constraint was that the remaining edges had to be joined.

Removing one edge results in just one possible configuration for an incomplete open cube.

By experiment, LeWitt identified 122 unique variations of open cubes with three edges (the minimum number needed to suggest three dimensions) to eleven edges. Nine of these frameworks, rendered in painted aluminum, were recently on display in the exhibition "Sol LeWitt: Structures, 1965-2006" at City Hall Park in New York City.

Three connected edges is the minimum number needed to suggest three dimensions.

In 2000, the San Francisco Museum of Modern Art exhibited all 122 incomplete open cubes. A representation of these variations is shown here.

One interesting mathematical question is whether LeWitt found all the possible variations of incomplete open cubes that met his criteria? Did he miss any? How would you find all the possibilities and prove that no others exist?

Scott Kim wondered the same thing in a "Bogglers" column in the April 2003 issue of Discover. As an "easy" question, he asked, "Can you make six different shapes that each contain four edges? The edges in each shape must all connect to form a single figure. Mirror images of the same shape are considered different, but rotations are not."

One example of a configuration made from four connected edges.

More difficult: How many distinct shapes can you make using five connected edges of a cube? The answer is 14. Can you find them? Is there a formula or some systematic way to determine the number of possibilities, from three to eleven edges.

It turns out that LeWitt did not include examples in which three or four edges are all lie in the same plane. Are other configurations missing?

For another mathematical puzzle concerning a LeWitt artwork, see "Puzzling Lines." For articles about other LeWitt artworks, see "Thirteen Geometric Shapes" and "LeWitt’s Pyramid."

Photos by I. Peterson

Mathematical Morsels I (Solutions)

THE FERRY BOATS

Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the river.

Solution: By the time of their first meeting, the total distance that the two boats have traveled is just the width of the river. It may take one mildly by surprise, however, to realize that, by the time they meet again, the total distance they have traveled is three times the width of the river. Since the speeds are constant, the second meeting occurs after a total time that is three times as long as the time for the first meeting. In getting to the first meeting, ferry A (say) traveled 700 feet. In three times as long, it would go 2100 feet. But, in making the second meeting, A goes all the way across the river and then back 400 feet. Thus the river must be 2100 – 400 = 1700 feet wide.

Source: American Mathematical Monthly, 1940, p. 111, Problem E366, proposed by C.O. Oakley, Haverford College, solved by W.C. Rufus, University of Michigan.

ROLLING A DIE

A normal die bearing the numbers 1, 2, 3, 4, 5, 6 on its faces is thrown repeatedly until the running total first exceeds 12. What is the most likely total that will be obtained?

Solution: Consider the throw before the last one. After the first throw the total must be either 12, 11, 10, 9, 8, or 7. If it is 12, then the final result will be either 13, 14, 15, 16, 17, or 18, with an equal chance for each. Similarly, if the next to last total is 11, the final result is either 13, 14, 15, 16, or 17, with an equal chance for each; and so on. The 13 appears as an equal candidate in every case, and is the only number to do so. Thus the most likely total is 13.

In general, the same argument shows the most likely total that first exceeds the number n (n > 5) is n + 1.

Source: American Mathematical Monthly, 1948, p. 98, Problem E771, proposed by C.C. Carter, Bluffs, Illinois, solved by N.J. Fine, University of Pennsylvania.

RED AND BLUE DOTS

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a segment of their common color; adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, none at the corners. There are 237 black segments. How many blue segments are there?

Solution: There are 19 segments in each of 20 rows, giving 19 x 20 = 380 horizontal segments. There is the same number of vertical segments, giving a total of 760. Since 237 are black, the other 523 are either red or blue.

Let r denote the number of red segments and let us count up the number of times a red dot is the endpoint of a segment. Each black segment has one red end, and each red segment has both ends red, giving a total of 237 + 2r red ends.

But, the 39 red dots on the border are each the end of 3 segments, and each of the remaining 180 red dots in the interior of the array is the end of 4 segments. Thus the total number of times a red dot is the end of a segment is 39(3) + 180(4) = 837. Therefore 237 + 2r = 837, and r = 300.

The number of blue segments, then, is 523 – 300 = 223.

Source: American Mathematical Monthly, 1972, p. 303, Problem E2344, proposed by Jordi Dou, Barcelona, Spain.

A PERFECT 4^TH POWER

Prove that the product of 8 consecutive natural numbers is never a perfect fourth power.

Solution: Let x denote the least of 8 consecutive natural numbers. Then their product P may be written

P = [x(x + 7)][(x + 1)(x + 6)][x + 2)(x + 5)][(x + 3)(x + 4)] = (x² + 7x)(x² + 7x + 6)(x² + 7x + 10)(x² + 7x + 12).

Letting x² + 7x + 6 = a, we have

P = (a – 6)(a)(a + 4)(a + 6) = (a² – 36)(a² + 4a) = a⁴ + 4a(a + 3)(a – 12).

Since a = x² + 7x + 6 and x ≥ 1, we have a ≥ 14 and a – 12 is positive.

Hence P > a⁴.

However, P = a⁴ + 4a³ – 36a² – 144a reveals that P is less than (a + 1)4 = a⁴ + 4a³ + 6a² + 4a + 1.

Hence a⁴ < P < (a + 1)⁴, showing that P always falls between consecutive fourth powers and never coincides with one.

Source: American Mathematical Monthly, 1936, Problem 3703, proposed by Victor Thébault, Le Mans, France, solved by the Mathematics Club of the New Jersey College for Women, New Brunswick, New Jersey.

Mathematical Morsels I

The American Mathematical Monthly has a long tradition of publishing problems, going all the way back to its first issue in 1894.

In a letter that appeared in the debut issue, Monthly coeditors B.F. Finkel and J.M. Colaw argued the value of posing and solving mathematical problems.

"While realizing that the solution of problems is one of the lowest forms of Mathematical research . . . its educational value cannot be over estimated," they wrote. "It is the ladder by which the mind ascends into the higher fields of original research and investigation. Many dormant minds have been aroused into activity through the mastery of a single problem."

Readers of the Monthly continue to look forward to fresh doses of perplexity and ingenuity with the arrival of each new issue, and the problems sections of past issues remain a treasure house of mathematical gems to revisit and ponder anew.

Several decades ago, Ross Honsberger (University of Waterloo) chose scores of "elementary" problems, originally posed in the Monthly, to appear in a volume titled Mathematical Morsels (Mathematical Association of America, 1978). He wanted to illustrate that "all kinds of simple notions are full of ingenuity."

"Mathematics abounds in bright ideas," Honsberger wrote. "No matter how long and hard one pursues her, mathematics never seems to run out of exciting surprises. And by no means are these gems to be found only in difficult work at an advanced level."

Here are four classic problems from this selection for you to try.

THE FERRY BOATS

Two ferry boats ply back and forth across a river with constant speeds, turning at the banks without loss of time. They leave opposite shores at the same instant, meet for the first rime some 700 feet from one shore, continue on their way to the banks, return and meet for the second time 400 feet from the opposite shore. [Without using pencil and paper] determine the width of the river.

Source: American Mathematical Monthly, 1940, p. 111, Problem E366, proposed by C.O. Oakley, Haverford College, solved by W.C. Rufus, University of Michigan.

ROLLING A DIE

A normal die bearing the numbers 1, 2, 3, 4, 5, 6 on its faces is thrown repeatedly until the running total first exceeds 12. What is the most likely total that will be obtained?

Source: American Mathematical Monthly, 1948, p. 98, Problem E771, proposed by C.C. Carter, Bluffs, Illinois, solved by N.J. Fine, University of Pennsylvania.

RED AND BLUE DOTS

Consider a square array of dots, colored red or blue, with 20 rows and 20 columns. Whenever two dots of the same color are adjacent in the same row or column, they are joined by a segment of their common color; adjacent dots of unlike colors are joined by a black segment. There are 219 red dots, 39 of them on the border of the array, none at the corners. There are 237 black segments. How many blue segments are there?

Source: American Mathematical Monthly, 1972, p. 303, Problem E2344, proposed by Jordi Dou, Barcelona, Spain.

A PERFECT 4^TH POWER

Prove that the product of 8 consecutive natural numbers is never a perfect fourth power.

Source: American Mathematical Monthly, 1936, Problem 3703, proposed by Victor Thébault, Le Mans, France, solved by the Mathematics Club of the New Jersey College for Women, New Brunswick, New Jersey.

Reference:

Honsberger, R. 1978. Mathematical Morsels. Mathematical Association of America.

SOLUTIONS

A Tetrahedral Forest

A neon-framed model of a regular tetrahedron hangs in an entrance to the Milan Central Train Station (Stazione di Milano Centrale).

Defined by four triangular faces, the tetrahedron is the simplest of all polyhedra. Any four points in space that are not in the same plane mark its corners.

Despite its apparent simplicity, a variety of artists have used the tetrahedron as the inspiration for artworks (see, for example, "Three Sentinels"). Part of the visual appeal of these constructions is that a tetrahedron is so angular that its aspect can change abruptly as a viewer moves around to see it from different angles.

One enthusiast of the tetrahedron was Philadelphia-based artist Robinson Fredenthal (1940-2009). His large, angular sculptures take advantage of the tetrahedron's amazing rigidity and its ability to resist an incredible amount of force from the outside.

"I can't think of anything more perfect than a tetrahedron," Fredenthal once remarked. If visitors came from outer space, "I'd hand them a tetrahedron, and they would understand."

Fredenthal manipulated tetrahedra in a variety of ways, creating peculiarly balanced, leaning towers of tetrahedra, and great bridges of these remarkable forms.

Fredenthal's Black Forest (above) is located on the campus of the University of Pennsylvania.

A bridge of tetrahedra forms the basis for Fredenthal's sculpture White Water, found at 5^th and Market Streets in Philadelphia.

Huge tetrahedra representing Fire Water Ice loom tall in a three-part sculpture at 1234 Market Street in Philadelphia.

Afflicted with Parkinson's disease for much of his life, Fredenthal spent his time in geometric exploration, crafting thousands of small-scale cardboard models of variations of simple forms such as cubes and tetrahedra. Most of his models are now in the Penn architectural archives.

Reference:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photos by I. Peterson

Block Patterns in Blue and White

Block quilt pattern: Double Irish Chain.

The Charles Hotel in Cambridge, Mass., prides itself on its extensive collection of early American quilts, many of which are on display throughout the hotel. One particularly striking array is a set of nine blue-and-white quilts by the hotel's grand staircase.

Created in the 1880s and 90s, these hand-crafted quilts feature striking traditional designs based on geometric block patterns. Each block is usually a square or rectangle with a distinctive geometric pattern. Identical blocks are then sewn together to create a quilt.

A quilt's pattern name often gives clues about where and when a quilt was sewn, and it may say something about the interests or preoccupations of a particular quilt's creator.

Corn and Beans.

Evoking tidy garden rows, Churn Dash and Corn and Beans, for example, are interpretations of everyday items and chores. Names such as Geese in Flight, Ocean Waves, and Swallows in Flight reflect the underlying geometry of natural forms.

Geese in Flight.

Drunkard's Path is not only a representation of repeated fragments of a random walk but also social commentary dating to a period of U.S. history when the Temperance Movement was strengthening.

Drunkard's Path.

The characteristic blue (indigo dyes) and white of these quilts and their geometric designs meant they could fit into just about any setting.

Chinese Blocks.

Additional information about these quilts and other artworks in the hotel's collection are available in the walking tour brochure "Art at The Charles" (pdf).

Churn Dash.

Can you identify the block unit in each quilt? How would you characterize the symmetry of the block pattern that serves as the basis of each quilt?

Double Ninepatch.

You can find a variety of teaching aids to lead students through characterizing block patterns: Shape and Space in Geometry: Quilts (Annenberg/CPB Math and Science Project); Shapes, Lines, Angles & Quilts (Franklin Institute); Quilt Geometry (Steven H. Cullinane); Quilt Block Patterns (Math Forum).

Irish Chain.

For a colorful, animated tour of geometric quilt designs, see the National Film Board of Canada's wonderful production "Quilt."

Swallows in Flight.

Photos by I. Peterson

Pythagorean Fractal Tree

Pythagorean Fractal Tree was designed by Koos Verhoeff, cast in bronze by Anton Bakker and  Kevin Gallup, and displayed at the first art and mathematics conference in Albany, New York, in 1992.

Born in Holland in 1927, Verhoeff studied mathematics and computer science. He worked for a time at the Mathematical Center in Amsterdam, where he encountered the Dutch artist M.C. Escher, who often came to the center to research mathematical ideas he applied to his artworks.

Inspired by Escher, Verhoeff ended up pursuing the application of mathematics and computers to art. One of his main interests after his retirement in 1988 was the discovery and development of artistic structures based on geometric principles.

Fractal formations (trees), in which small pieces of a structure echo the appearance of the entire structure, inspired the branched structure shown above.

Reference:

Peterson, I. 2001. Fragments of Infinity: A Kaleidoscope of Math and Art. Wiley.

Photo by I. Peterson