# Nonplanar Rectangular Heptagons

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This Demonstration shows two infinite families of nonplanar regular 7-gons in 3-space having a 90° angle at each junction. Each family is parametrized by the points on a circular arc.

Contributed by: Barry Cox and Stan Wagon (July 2013)

(University of Adelaide, Australia, Macalester College, St. Paul, Minnesota, USA)

Open content licensed under CC BY-NC-SA

## Details

The existence of a nonplanar regular 7-gon with right angles at each vertex was discovered by Wildenberg [1]. He showed that the only positive integers for which rectangular regular -gons do not exist are 1, 2, 3 and 5.

This appears to be an unresolved question: is there a rectangular regular 7-gon that does not belong in either of these two families?

For a discussion of regular heptagons with common angle not equal to a right angle, see [2].

,

,

.

Now each vector between points is guaranteed to be a unit vector, except for , so we need only specify that condition and the remaining two right angles as follows:

,

,

.

So when we view this system as having the parameter and use a specific value such as , we find that the solution involves a root of a degree-16 polynomial.

But there appears to not be a general algebraic result, and we can use a numerical technique to obtain two distinct one-dimensional families of solutions. In the first family, where the symmetric instance is known as the *chair*, the angle runs from to and down to and back up to in rough sine-curve form. The second family is based on the *boat* and has running from to , but it has extra dips, so is more complicated than the simplest wave. We cannot get, physically, from a solution in one family to a solution in the other.

References

[3] G. Wildenberg, "Problem 1110, Seven Is Possible," *Mathematics Magazine*, 55(1), 1982 pp. 47–48.

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