Beyond the Third Dimension: Geometry, Computer Graphics, and by Thomas F. Banchoff

By Thomas F. Banchoff

This paintings investigates methods of picturing and realizing dimensions under and above our personal. What could a two-dimensional universe be like? How will we even try and photo gadgets of 4, 5 or 6 dimensions? Such are the questions tested during this textual content.

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The rectangle continues to expand until its side length equals the diameter of the circular disc. Then the sequence reverses and the rectangle shrinks back to a single edge. More complicated are the slices of a cylinder suspended from a point on its rim. Half way through, the slice of the circular cylinder is an ellipse. Slicing sequences give us a way to think about an analogue of the cylinder in the fourth dimension. We can imagine a figure that in one guise would appear as "a ball for a while," and in another < Arrows at the ends of equal-length segments produce the illusion that one is longer.

People understood and used the idea of time as a fourth coordinate in something as prosaic as making an appointment in a place like New York City. "I'll meet you at the corner of Seventh Avenue and Fourth Street on the fifth floor" might be entered in the calendar as (7, 4, 5). but the appointment would not be complete without that fourth time coordinate, "at ten o'clock," which could be recorded as (7, 4, 5. 10). This fourdimensional system of "three-space, one-time" is immensely useful in modern physics, not just as a bookkeeping device but as a rich mathematical structure.

Computer axial tomography provides X-ray pictures of slices perpendicular to a patient's spine. More recent techniques produce magnetic resonance imaging of a head's interior structure by showing slices not just in the axial direction, perpendicular to the spine, but also from ear to ear, the "sagittal" slices, or from the tip of the nose to the back of the head, the "coronal" slices (see images on the next page). These images show both the bone and the tissue, different shades and textures indicating the densities of various parts.

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