Felicitous Geometric Algebra: Carefully demystifying Euler's Identity by Gary Harper

Author:Gary Harper [Harper, Gary]
Language: eng
Format: azw3
Published: 2018-10-28T00:00:00+00:00

Elements for physical space. Dashed lines mean +

You already know that free vector v on top left is the free part of bound v on bottom left. In a similar way, free bivector B on top is the free part of bound B on bottom: its extension with any point in bound B's confining plane generates bound B. To see that, place one end of free B on the point before extending, just as with a free vector. Poof!—that part of the extension vanishes leaving a bound vector extended to a point, a bound bivector. Similarly for trivectors T and T. This process exposes a crucial ambiguity and an apparent incongruity.

The ambiguity is that there are always two distinct ways to generate a bound n-vector: either by extending a bound (n–1)-vector to a point, or by extending a free n-vector to a point. This is crucial because it means that undoing extension—retracting—has two options: a free result or a bound one. So which does it choose?

Those options seem to induce the incongruity that the same dimension arises from arguments with different dimensions. This is superficial—those arguments in fact do have the same numeric dimension when you carefully distinguish free from bound, like so:

Whereas a free n-vector has dimension {n}, just as in the conventional free algebra, a bound n-vector has dimension {n+1}, already seen in extensions up from the floor. At the floor, a bound 0-vector is a point, dimension {1}; a free 0-vector is a scalar, dimension {0}. Going up, a bound 1-vector is a line segment, dimension {2}; a free 1-vector is a roving bundle of points, dimension {1}, and on up.

In short, the n in an n-vector, free or bound, tells its spatial expanse. To discover its numeric dimension, you need to know whether that expanse was generated by n+1 fixed points, or by n roving expansive bundles of them.

Said differently, perhaps better, once you know the n-expanse, you need to further discover whether it is filled-in or empty—generated by n+1 points or by n free vectors. Such careful free-versus-bound distinctions dispel—and even somewhat accommodate—our timeworn Euclidean–Cartesian confusion about the dimensions of scalars, points, lines, planes and on up.

So, back to the non-superficial question: ¿Which option does retraction choose? Free result or bound result? It becomes clear when you realize that retraction—in order to eventually combine with extension—must scale like extension does.

This means that retraction should be able to scale either its retractee, or its result, indifferently. For a bound result, these different kinds of scaling would produce things in different locations with different magnitudes. Conversely, a free result has no fixed location to differ from; and for it, the different kinds of scaling simply scale the separation of what is produced, indifferently.

Consequently, retraction must produce a free result. This is somewhat intuitive: undoing extension naturally loses the locus information extension had gained.

However, there is an important subtlety: the just-mentioned contradiction for a bound result is geometric, not algebraic. The exact same algebra, for a free result, fails to generate a contradiction. Here you see semantics disciplining syntax, contrary to mathematical fashion for the last four centuries.

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