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.


Snapshots



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