The Visual Guide to Extra Dimensions: Visualizing The Fourth Dimension, Higher-Dimensional Polytopes, And Curved Hypersurfaces by McMullen Chris

Author:McMullen, Chris [Desconhecido]
Language: eng
Format: epub, azw3
Publisher: Amazon Publishing
Published: 2014-11-03T02:00:00+00:00

In the notation xn, which reads x raised to the power of n, n is called an exponent. Mathematically, xn means to multiply n x ’s together. For example, 34 =3·3·3·3 = 81. If n = 2, x is said to be squared; and if n = 3, x is said to be cubed. It is easy to see that xmxn = xm+n. This works for negative exponents, provided that x–n = 1/xn. Since xmx–m = x0 must be 1, any number raised to the zero power is unity: x0 = 1.

An N -dimensional cube has N sets of 2N–1 parallel edges – i.e. N(2N–1) edges all together. A point has no edges in accordance with 0(2–1) = 0, a line segment is its own edge as 1(20) = 1, a square has 2(21) = 4 edges, a cube has 3(22) = 12 edges, a tesseract has 4(23) = 32 edges, a 5D hypercube has 5(24) = 80 edges, and so on. Another way to put it is: A square has 2 pairs of parallel edges, a cube has 3 sets of 4 parallel edges, a tesseract has 4 sets of 8 parallel edges, a 5D hypercube has 5 sets of 16 parallel edges, etc.

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