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

By Thomas F. Banchoff

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

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Additional info for Beyond the Third Dimension: Geometry, Computer Graphics, and Higher Dimensions

Example text

The difficulty seems to he related to the SLICING AND CONTOURS 51 three-dimensional equivalent of the optical illusion that makes two lines of equal length seem different if we put arrows on the ends. The presence of the longest sides makes the square faces appear to be rectangles with unequal sides. Slicing Cylinders > Friedrich Froebel's third figure was a circular cylinder. Once we understand the slicing sequences for balls and cubes, it is easy to imagine two of the sequences for the cylinder.

The slice containing the square separates the tetrahedron into two parts of exactly the same shape. You can make a paper model of tin* two pieces of this decomposition by folding up the pattern at the side of this page. Many people find it difficult to put these two identical pieces together to form a triangular pyramid. Even when they place the two square faces together, most often they hold the pieces so that the longest edges arc parallel instead of perpendicular as they should be. The difficulty seems to he related to the SLICING AND CONTOURS 51 three-dimensional equivalent of the optical illusion that makes two lines of equal length seem different if we put arrows on the ends.

Patterns of hexagons and triangles and squares are quite important in the mathematical subject called tiling. Consider what happens if we stack a large number of cubes together to form a large cube. If we slice the large cube by a flat plane, then in each of the cubes hit by the plane, we get a slice in the shape of a polygon. All of these separate pieces will fit together to cover a portion of the slicing plane. If this plane is parallel to one of the faces of the large cube, then all polygonal slices will be squares, and these squares fit together to form a "tiling" pattern in the plane.

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