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

By Thomas F. Banchoff

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

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CHAPTER 2 n = 3, the diagonal will be the hypotenuse of a right triangle with base Vz and height 1, and by the Pythagorean theorem, the length of the diagonal will be "Wz2 + I2 = Vz + 1 = Vif. Moving to the four-dimensional hypercube, we obtain a new right triangle with base of length Vs and height 1, therefore with hypotenuse "Ws2 + I2 = V4 = Z. Thus the hypercube has a diagonal exactly twice the length of a side. It is easy to see that, in general, the length of the longest diagonal of an n-dimensional cube will be y/ii, and this is quickly proved by mathematical induction: if we already know that the length of the diagonal of an (n - l)-cuhe is Vn — 1, then the diagonal of the »i-cuhe is the hypotenuse of a right triangle with one side of length V h — 1 and the other of length 1.

5. Top: Friedrich Froebel, inventor of kindergarten. Right: FroebeFs geometrical models, from the Milton Bradley catalogue of 1889. the United States, as they had in Prussia and other parts of Europe earlier in the nineteenth century. One of Froebel's first "gifts" to inspire kindergarten children was a display of three basic three-dimensional forms, the sphere, the cylinder, and the cube, suspended by strings. As the objects rotated, children could observe them from different views and learn to appreciate their symmetries and structures.

This tetrahedron expands until it reaches a level con- Slices of thi« hypercuhe starlini; with a square SLICING AND CONTOURS Slices of the hypercube starting with an edge. taining 4 vertices of the hypercube. Then it experiences truncation as the corners of the tetrahedron are cut off, and three-eighths of the way through, the slice is a solid figure having four equilateral triangles and four regular hexagons as faces. Each hexagon is the slice exactly halfway through one of the cubes in the boundary of the hypercube.