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A perfect parallelepiped does exist, and Cliff Reiter, professor of mathematics, and mathematics major Jorge Sawyer ’10 (Lodi, N.J.) found it.
In fact, the duo has found not one but 12 distinct perfect parallelepipeds, an object widely searched for by number theorists and geometers.
A parallelepiped is a skew box. Like an ordinary box, it has parallel faces, but the angles need not be 90 degrees.
A long-standing problem is whether there is a “perfect box,” usually called a “perfect cuboid.” A perfect cuboid would be a box whose edge lengths, face diagonal lengths, and body diagonal are all positive integers. It is unknown whether a perfect cuboid exists.
In the early 1980s, the question was posed as to whether the problem could be solved if the angles need not be right angles. That is, is there a perfect parallelepiped – a skew box with edges, minor and major face diagonals, and all four body diagonals having positive integer length?
Reiter set out to find such a perfect parallelepiped. Over the course of the past few summers, he worked on the project with EXCEL scholars Jordan Tirrell ’08 and Daniel D’Argenio ’10 (Yardley, Pa.). Finally, he and Sawyer solved the problem.
Sawyer performed searches for 2-D perfect parallelograms and developed formulas based upon a simple representation of the parallelograms. Reiter then wrote an algorithm that put together triples of Sawyer’s perfect parallelograms and used the formulas to sift through millions of cases. Using this method, the pair found some examples of perfect parallelepipeds.
“I was surprised that solutions were found within minutes. In the first few minutes there were 16 proposed solutions that satisfied the necessary conditions. Jorge worked to show one of those was sufficient. That is, a real live perfect parallelepiped. We checked each other’s work and had a solution,” Reiter says.
The discovery makes the perfect cuboid question more intriguing since if there were no perfect parallelepipeds, then there could not be a perfect cuboid. However, Reiter does not intend to turn his attention to looking for a perfect cuboid. “I think better understanding the distribution of perfect parallelograms is more fundamental and interesting,” he says.
Reiter and Sawyer submitted the paper summarizing their findings to a peer-reviewed journal.