This is an analog electronic circuit constructed with Paul Falstad's Circuit Simulator intended to highlight two peculiar features of Geometric Algebra vector multiplication: That vectors are not expressed in angles in either degrees or radians, but by their (x,y) components as on a cartesian plot, and second, the many reversals in polarity that occur as the vector rotates in a full circle about the origin. First x flips to negative, next y flips too, then x flips back to positive, and y flips positive too. This is highly suggestive of a phasic cyclic system like the plot on an oscilloscope, where a cyclic waveform traces out a spatial pattern on the scope.
At the upper-left is a sawtooth waveform generator that feeds to an op-amp to amplfiy the signal, the sawtooth is plotted at the lower-left. That signal is split into X and Y components, each of which can be independently modulated by the X and Y rheostats, or variable resistors, which are controlled by sliders. The modulated X and Y sawtooth waves are plotted at the lower center. At the lower-right the X and Y sawtooths are plotted against each other to produce a "flying spot" that cyclically traces out the vector from root to tip. Modulation of the X slider (shown) tilts the vector to different angles.
Here is a movie screen capture running the simulation.
Clifford Analogue Movie
The source code for the circuit used in this simulation can be simply cut and pasted into Paul Falstad's Circuit Simulator to play around with for yourself.
The significance, for those of us who believe that mathematics is an artifact of how our brain works, is that Clifford vector multiplication suggests a phasic cyclic principle in the brain that is used to paint out our cognitive thoughts, or mental images, presumably evolved and adapted from a similar operational principle in perception, the "projection mechanism" that paints out the "picture" of the world that we experience around us, whether in veridical perception, or in dreams and hallucinations.
Imagine that the vector in the simulation is plotted continuously as it rotates continuously round and round. This would be achieved by a cyclic oscillation of the X and Y sliders in quadrature, i.e. with one following the sine and the other its cosine, lagging the sine by π degrees, or a quarter cycle.
The rotation would of course not be continuous, but would occur in intervals, so the vector would look like the second hand of a clock, jumping around the circle in equal angular intervals, and flashing radially outward from the center at each step.
Now imagine the rotation getting faster and faster, making larger and larger jumps, until it jumps a full circle with each jump, thus flickering stationary at one angle, "running on the spot" so to speak, refreshing itself over and over again at the same angle like a "flying spot" on the oscilloscope. By small adjustments of the phase of this cyclic signal relative to the vector's own refresh cycle, the dynamic vector can be made to rotate this way or that, to any angle. And of course this same principle can be extended to three dimensions to define a vector at the origin that can be steered to any direction. This suggests the operational principle for how electrochemical oscillations in the brain might "sweep out" the spatial pattern of our experience in perception and visual consciousness.
For those of us who have believed for some time that the brain paints out its picture of experience, this insight from Clifford Algebra resolves a number of thorny issues. The raster scan of an old-fashioned television screen sweeps image rows from top to bottom, giving the image a texture of horizontal lines that reveal the scanning priority, rows first then columns. If this principle were extended into three dimensions, a third sweep, depth plane by depth plane would fill in the third dimension with a clear priority of row, column, then depth plane, x, y, and z in that sequence. If human perception were to employ that kind of raster scan principle, we would see it as an artifact in our vision, as clear as we see it on a television tube.
Instead we see an apparantly seamless experience devoid of any kind of textured artifacts of the scanning sequence. Geometric Algebra suggests how this is achieved - by setting the scan priority relative to the object itself, rather than to a global coordinate frame. The scan sweep of the vector occurs along the vector itself, painted out by synchronized oscillations of the (x,y,z) component waveforms.