. A Kindle e-book carefully demystifying Euler's Identity.Felicitous Geometric Algebra. "Playing with Geometric AlgebraIn extension theory there appears a characteristic method of calculation which is inexhaustibly fruitful and consists in subjecting spatial structures (points, lines, and so forth) directly to calculation." So wrote Hermann Grassmann in 1845. His novel ideas are only now beginning to seep into mathematical thought. William Clifford tried to promote them in 1878; Giuseppe Peano tried in 1888; Alfred Whitehead tried in 1898. Several others tried in the early 1900s. David Hestenes tried again in the last half of that century, and again, and again … Only he appreciably succeeded. Unfortunately, he did not actually calculate with points, lines, etc. Instead he calculated with free vectors, free bivectors, etc. A free vector is a composite addition of opposite fixed points, a free bivector is a composite addition of opposite fixed lines, and so on up; as Clifford, Peano and Whitehead had learned from Grassmann. In other words, Hestenes's gain was achieved at the loss of half of Grassmann's algebra, the half.bound The free gain can be preserved without the bound loss; and that is the goal of this book. It augments Grassmann's full algebra with the synthesis achieved by Clifford and polished by Hestenes. The result is an extraordinarily expressive mathematical language, able to articulate not only geometric relations, but also geometric locus. . Hermann Grassmann was curious and creative. Curious in both senses: inquisitive and eccentric; creative in both implications: innovative and unconstrained by convention. His most magnificent creation was Grassmann's Upbringing of his Fertile MindExtension Theory, a geometric algebra that generates higher-dimensional elements by sweeping them successively to various points. This essay gives you the opportunity to stroll along with him from the inception of his ideas in the early 1840s to his final tweaking of them in 1877, months before he died.. After mathematicians finally came up with a symbol for Fixing NothingNothing, they couldn't stop. These various vacuities prevent things from really vanishing, which prevents dimension from being well defined. Even worse, disparate kinds of Nothings inhibit a unification of disparate kinds of numbers. Such numbers are generated by Geometric Algebra, and they must be allowed to really vanish to avoid accumulating numeric garbage that would clog its machinery. This hygiene can be achieved by reverting to the primordial economy of just one Nothing in mathematics. After that is achieved, programming languages can acquire the same hygiene. Strangely, the early C language started with it, then degenerated.. Here is the key to Geometric Algebra: Meandering Toward Geometric Algebrapoints are numbers——and their extension generates lines, planes and on up, geometric numberswhich are also numbers. Grassmann’s path to that idea was so esoteric and meandering that few have untangled it over the last two centuries. Surprisingly, it can easily be untangled by anyone willing to acquire the seldom articulated dimension of a point, not the widely discussed one. There’s the rub: our own path toward that dimension has been as meandering and baffling as Grassmann’s path was. To untangle both paths we shall have to go back several millennia and proceed straight and narrow. This will be quick: we’ll be flying low over the winding river of ideas leading toward formal geometric dimension. |