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Z . Remarkably, even t h o u g h o u r brains c a n n o t visualize an N-dimensional cube, it is easy to write down the formula for its sides. (This is a c o m m o n feature of working in hyperspace. 1. The length of a diagonal of a cube is given by a three-dimensional version of the Pythagorean Theorem: a + b + c = d . By simply adding more terms to the Pythagorean Theorem, this equation easily generalizes to the diagonal of a hypercube in N dimensions. Thus although higher dimensions cannot be visualized, it is easy to represent N dimensions mathematically.
Finally, when he delivered his oral presentation in 1854, the reception was enthusiastic. In retrospect, this was, without question, o n e of t h e most i m p o r t a n t public lectures in the history of mathematics. Word spread quickly t h r o u g h o u t E u r o p e that Riemann h a d decisively broken o u t of the confines of Euclidean geometry that had ruled mathematics for 2 millennia. News of the lecture soon spread t h r o u g h o u t all the centers of learning in E u r o p e , a n d his contributions to mathematics were being hailed t h r o u g h o u t the academic world.
However, if the lasso is placed a r o u n d the e n t r a n c e of the wormhole, then it c a n n o t be s h r u n k to a point. T h e lasso, in fact, enters the wormhole. Such spaces, where lassos are n o t contractible, are called multiply connected. ) Mathematicians dating back to Georg B e r n h a r d R i e m a n n have studied the properties of multiply c o n n e c t e d spaces in which different regions of space a n d time are spliced together. And physicists, w h o once t h o u g h t this was merely an intellectual exercise, are now seriously studying multiply c o n n e c t e d worlds as a practical model of o u r universe.
Advansed plasma theory by Rosenbluth