"Four-dimensional models are useful for thinking about and finding new relationships and phenomena," said Ocneanu. "The process is actually quite simple -- think in one dimension less." To explain this concept, he points to a map. While the Earth is a three-dimensional object, its surface can be represented on a flat two-dimensional map.
Ocneanu's sculpture similarly maps the four-dimensional solid into a space perceptible to the human observer. His process, radial stereography, presents a new way of making this projection. He explained the process by analogy to mapping a globe of the Earth onto a flat surface.
"We place a light bulb at the north pole of the Earth and we project onto a sheet of paper placed underneath it," he said. "The southern hemisphere, away from the north pole, will remain quite small, while the northern hemisphere, near the projection pole, will become very big and north pole itself will be sent toward infinity."
The technique can be used to make a two-dimensional projection of a cube by first mapping the cube radially onto the surface of a globe. Ocneanu explained, "The edges of our cube become circles on the map, just like straight highways are slightly curved on maps of the Earth. Its angles, however, remain true in this projection, so the map retains the key aspects of the symmetry of the original cube, unlike a photograph of a cube."
When the same technique is applied to project a four-dimensional solid into three dimensions, the inner part of the projection -- equivalent to the south pole on the map -- has smaller, undistorted faces, while the outer part extends toward infinity. Linear edges of the solid become circles in the projection.
