Back in ancient times, the

Greek mathematicians

Euclid and

Archimedes proved the mathematical existence of

geometry's most fundamental

three-dimensional shapes. Over two thousand years later, David Sterner has added a new one to the list.

Abridged from Plazm Media, issue 24. An incredible magazine. Thanks, guys.

The cube, the cone, the cylinder, the pyramid, the sphere:

These are the building blocks of geometry, "fundamental" shapes sharing certain characteristics, such as horizontal symmetry, and tangency to the cube, and a simplicity so stark they can be reproduced using a mere straight edge and a compass.

Alone incredible, together invincible, the fundamental shapes constitute a set of universal truths underlying the structure of everything in the universe (the Newtonian one, anyway), defining the angles and planes of reality itself. Apart from and above human consciousness, they would exist even if people did not.

Since the age of seventeen, however, David Sterner has viewed this grouping of shapes as incomplete. He realized that some shape must exist amongst those "fundamental" shapes, as simple as a cube and a sphere, but somehow overlooked. His reasoning went something like this: If one arranges the fundamental shapes in ascending order according to their number of surfaces, a strange lacuna emerges. Watch as the family moves from one surface (sphere), to two (cone), to three (cylinder), to five (pyramid), to six (cube). What about four? What about a shape somewhere in between the cylinder and the square-bottomed pyramid? There should be a shape in there.

Well, you might ask, what about the tetrahedron, the four-sided shape built of equilateral triangles? That would seem to satisfy the blank spot. That seems fundamental. But, in fact, according to Sterner, it is not, for while it is simple and four-sided, it isn't, like the other shapes, tangent to a cube, nor is it quite as "easy to construct with a straight-edge and a compass." It's just not part of the group, OK? As Sterner complains, "I hate the tetrahedron!" He gets visibly agitated when its name is uttered.

He found it by accident in 1979, while building an aquarium for some fish out of the plastic bulge of a discarded skylight. He cut the pillow-like shell of plexiglass in half, flipped the two halves back to back, and epoxied them together around the seam.

The shape he ended up with had four surfaces, two of them bulbous (in a yin-yang type fashion), and two of them flat and half-circular (and open). From different angles the shape looked like a square or a circle. It displayed inverse symmetry. It was tangent to a cube. Its three views were easily constructed using a compass and straight edge (only the sphere, cylinder, and cube proved simpler). Euclid and Archimedes would have recognized it. By all measures, it appeared to fill that gap in the family of "fundamental shapes" that Sterner had intuited long before. Sterner named his shape the DOR, or Direct Opposite Reverse, pronouncing it the way we say "door". "All shapes," he explains, "can be defined by three primary views: top, front, and side. The sphere, for example, combines three circles. The cone combines the circle and the triangle. So I took a square and a circle, and said, "What would their offspring be?"

"The DOR", of course.

Upon examination, many suggestive shapes begin to emerge: the weave of a DNA strand, the inverted commas of yin and yang, even the magnetic fields of the Van Allen Belts. The implication of these echoes, to Sterner, is tremendous, indicative of the DOR to move humanity towards a higher and more compassionate plane of consciousness.

"If infinity had a three-dimensional shape, this would be it!"

Update 02.11.02: Further information and an illustration of the DOR may currently be found at http://hometown.aol.com/dorrefractions1/page1.html