A Hand-waving Introduction to Geometric Algebra and Applications

Geometric  Algebra (GA, real-valued Clifford Algebras, a.k.a. hypercomplex numbers) gives the only mostly-comprehensible-to-me account not only of higher spatial / temporal dimensions, but of physics in general. I have been studying GA now for over ten years. One of the best things about it is that nearly every paper using GA explains it from first principles before going on to use it for physics or computer science. Most physics papers in other fields seem to take a positive joy in obscure math and impenetrable jargon. I'll try here to give a even less mathematically difficult account of some of GAs implications than most GA papers.

Given a set of n mutually orthogonal basis vectors, one vector for each independent  dimension, a space of 2^n quantities results from considering all possible combinations of these basis vectors multiplied together. For instance taking pairs of vectors from a 5D space gives 10 possible planes of rotation, 4D space 6 planes of rotation, while in 3D there are only 3 independent planes of rotation. (The numbers of other combinations for n dimensions go as the n-th row of Pascals triangle or binomial.)
Sums of all the 2^n elements, each weighted by a different scale factor give "multivectors", which are generalizations of complex numbers.

Each of the basis vectors will have a positive or negative square. (Vectors' squares are always scalars, that is, real numbers.) In conventional relativity the basis vectors squares' signs, also called "signatures" are (+ - - - ) or (+ + + -), with the different sign from the others belonging to time. When plugging into the Pythagorean theorem, the square of time can cancel out the squares of the spatial dimensions, giving a distance of zero when the spatial distance equals the time interval (time multiplied by c to give all units in meters). This happens for anything moving at the speed of light. The zero interval is the amount of perceived or "proper" time for a light wave traveling between any two points. This light-speed type of path is also called a "null geodesic". To the photons of the microwave background, no time has passed since they were emitted, supposedly shortly after the universe began.

Now it is possible and actually quite useful for computer graphics to add a pair of dimensions with signature (+ -) to the usual spatial ones (+ + +).  The sum and difference of the extra dimensions give an alternate basis for these two dimensions, but with the basis vectors squaring to zero (0 0). These "null dimensions" are called "origin" and "infinity".  A projection from this augmented space down to 3D allows many other structures besides points and directions to be represented by vectors in the 5D space. For instance, multiplying 3 points gives a circle passing through those points, 4 points gives a sphere. If one of those points is the point at infinity, then the product is a line or a plane respectively. The other advantages of this way of doing things are too many to list here. This "conformal" scheme is actually quite easy to visualize and learn to use without getting into abstruse math by using the free GAViewer visualization software and its tutorials.

One fellow at Intel extended this to having three pairs of extra dimensions, for a total of nine, so that general ellipsoids rather than just spheres could be specified, but the idea has not become popular since each multivector in it has 2^9 = 512 parts. The 32 parts of regular conformal 3D / 5D multivectors is hard enough to convince people to use. The 11 dimensions of superstring theory are not so well defined as conformal dimensions  since seven of the string dimensions are said to be curled up small, "compactified" in some complicated and unspecified fashion.

An interesting thing about the ( +++, +- ) signature algebra is that it is the same as one that has been proposed by José B. Almeida as an extension of the usual 3D+t (+++-) "Minkowsi space" of relativity, augmenting the usual external time (-) with a second sort of time having positive square and describing internal or "proper time", (which in relativity will be measured differently by a moving external observer). But if it is assumed that everything in the universe is about the same age, then they have comparable proper time coordinates, so proper time can be used as a universal coordinate corresponding to the universe's temporal radius. This gives a sort of preferred reference frame for the universe, which is ordinarily considered impossible. In this 5D scheme, not just light but also massive particles follow null geodesics, and from that single assumption can be deduced relativity, quantum mechanics, electromagnetism, and in addition dark matter, the big bang  and the spatial expansion of the universe seem to be illusions.

The math is also easier than the usual warped-space general relativity, instead using flat euclidean space and having light, etc. move more slowly near mass, that is, treating gravitational fields as being regions of higher refractive index than regular space. This is also the case in gauge-theory gravity, (GTG) which also uses Geometric Algebra, though sticking to the usual 4D Minkowski space.  GTG is the only alternative to general relativity that is in agreement with experiment, but GTG also is more consistent, easier, allows dealing with black holes correctly, unlike GR, and is much easier to reconcile with quantum mechanics, which is also  much much easier to visualize using GA. For instance, the behavior of the electron can be described fully by treating it as a point charge moving in a tight helix at light speed around its average path (a "jittery motion", or in German: "zitterbewegung"). The handedness of the helix is the electron spin, the curvature of the helix is the mass, the angle of the particle around the helix is the phase. 

Geometric Algebra is useful in all areas of physics and computer modelling of physics. GA has been successfully applied to robot path planning, electromagnetic field simulation, image processing for object recognition and simulation, signal processing, rigid body dynamics, chained rotations in general and many other applications. It gives very clear, terse and generally applicable, practically useful descriptions in diverse areas using a single notation and body of techniques.