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 might a two-dimensional universe be like? How do we even try and photograph items of 4, 5 - 6 dimensions? Such are the questions tested during this textual content.

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On each of the 3 edges of that triangle, we erect a triangle one-third as large. We then erect a triangle one-ninth as large as the original on each of the 12 edges of the previous figure. Each of these edges is thus replaced by four smaller edges one-third as long as the original. It follows that the total perimeter is multiplied by 4/3 at each stage. Take this process to its limit and we have Koch's snowflake. Five steps in the creation of the Koch snowflake. 35 *■ • V? : JJS^sSS ■ r> Efl SLICING and CONTOURS A botanist analyzes a flower bud by embedding it in a plastic cube, then slicing the cube into thin sheets, which she mounts in glass slides.

The final edge is perpendicular in particular to the longest diagonal of the previous cube, and these two segments form the two sides of a right triangle having the longest diagonal of the new cube as its hypotenuse. This suggests that we can find the length of the longest diagonal by using one of the most famous of all theorems in plane geometry, the Pythagorean theorem. This theorem states that in a right triangle having sides of length p and q, the length of the hypotenuse is Vp2 + q2. The decomposition we found in interpreting the binomial theorem leads to a proof of this theorem by showing that the area of a square on the hypotenuse of a right triangle is the sum of the areas of the squares on these sides.

When the cube is almost fully submerged, we see another triangular slice, which shrinks down to a single point, the point of suspension. You might want to try to imagine what we get in the middle. When the water level has covered exactly one half of the cube, what is the form of the slice? Many people are surprised at the answer. Halfway through the cube, perpendicular to the long diagonal, we get a perfect regular hexagon, with all six sides of equal length and all angles equal. The answer is reasonable.

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