Finding SPOT

The SPOT System provides a notation for rapidly machine-sorting and comparing configurations of lines in visual patterns. But it began as a curiosity arising from an observation made during an idle pastime.

Back in the mid-70''s my curiosity was piqued by prime numbers, those natural numbers that seem somewhat aloof in that they can be evenly divided only by themselves or by 1. A traditional problem in mathematics is how to predict whether any given number (especially the really large ones) will prove to be prime. Given their crucial role in cryptographic techniques, such a method of finding really large primes could be quite valuable.

At first it seemed that it should be possible to "rope off" a segment of the natural numbers and to predict how many (and maybe even where) prime numbers would appear in that segment. All of the numbers in a given segment are either multiples (called "composites") of numbers less than the smallest number in the segment, or they are not. It seemed that if one could get a handle on how many smaller numbers are "operating through a segment", i.e., have multiples within the segment, then some rule should enable one to predict how many composites there are in the segment altogether. The number of primes would then be simply those numbers not included among its collection of composites.

To determine a segment I would pick two successive square numbers. For example, 11^2 and 12^2, in which case the segment would begin with 121 (11 squared) and continue through 144 (12 squared). In any such segment of natural numbers the composites are formed as multiples of numbers smaller than the lower end of my successive pair (in this example, by numbers less than 11). Hence, if one could calculate the number of composites in the segment, it would be simple to predict how many primes would occur in it.

I was able to get maddeningly close to a solution. Calculating composites for many different segments determined by successive squares repeatedly gave evidence of rhythmic implications of the smaller multipliers at work through any segment. Between mm (m times m, or, m^2) and nn, especially around the squares bracketing each end of the segment, and at the ''midpoint'' where the composite mn sits, prime numbers seemed to congregate, but never quite susceptible to any equation I could come up with (although I seemed to get closer and closer to finding a way of expressing the general case).

spot pathwayAs a visual aid to envisioning the pattern of composite numbers I tried several different ways of writing out a sequence of natural numbers. One way was to wrap the sequence in a spiral pathway about a center square, 0. Given the clock-wise wrapping that I happened to use I noticed immediately that even squarespot 100x100 primes numbers lined up on a diagonal up and to the right while odd squares aligned along a diagonal down and to the left. While I noticed the alignment I didn''t think too much about it at the time. It merely conveyed the regularities I had noted in where the primes seem to congregate at the ends and midpoint of each segment, seen here in their quasi-alignment in diagonal formations.

Eventually, pressed by other professional matters, I gave up the quest of predicting how many primes occur in a given number segment and went on to other things. Those "other things" included teaching myself to program in Fortran on a Digital Equipment DEC-10 with the notion, then deemed a "silly waste of time" by many, that I might be able to use a computer to render images on a teletype or plotter (then monopolized mostly by those in the scientific community).

Describing images by teletype via Fortran, to be output onto a plotter, proved an extremely tedious enterprise. Despite some progress, I finally concluded that either there had to be a better way than typing two Cartesian coordinates for every point on a contoured line (which comprises many vectors to describe changes of direction along the line). If I couldn''t find that better way, I would give up the project. It was just too tedious!

The next morning, as I awoke, I recalled the numbers on the spiral pathway and the regularities of its axial locations. It was a hunch that seemed worth pursuing.

quadratic gridIt seemed immdediately apparent that the axes and diagonals of a matrix determined by integers along a spiral pathway are simply graphic expressions of quadratic equations. From that realization came a stronger conviction that it should be rather straightforward to describe line vectors by just envisioning them on such a spiral matrix.

I wrote some procedures in BASIC (more straightforward for such explorations than Fortran) to translate back and forth between my spiral notation and Cartesian coordinates. I called it the Spiral Position and Orientation Translator and worked out a fairly efficient way of managing it on the computer. (Also, I must admit I named the system SPOT because it amused me to issue the command run SPOT, which is how a BASIC program was activated.)

A year or so later, the Apple II microcomputer became available and I purchased one to explore its six-color "hi-res" graphics capabilities (which seemed pretty advanced at the time!). By this time I had convinced myself that computers not only are misconstrued, but also mis-categorized under the rubric ''computer'', merely an accident of their initial application in aiming naval guns during World War II. They seemed more momentous as revolutionary communications technology than as rapid-calculating devices, an intuition which seems to have been confirmed in the years since. Using the SPOT System I put together a bit of educational software that was briefly marketed under the name Electroboard 2.0. It was a tutorial system whereby one could write up lessons or essays with accompanying graphics rendered via the SPOT System.

Then, thinking how to use such capability within a school setting, I began experimenting with hooking up computers and having them communicate by streaming the text and images back and forth in dialog with one another. When I shopped the package around to the various nascent software publishers of the era one editor actually asked me "Why would anyone want to link computers together to send images and text back and forth! But the advent of networking and then eventually the Internet and the World Wide Web soon overwhelmed my simplistic approach (and the stupefaction of that puzzled editor) and I went on to other things, such as using SPOT notation as basis of a notation for rapidly analyzing, sorting, and comparing vector configurations using ordinary text-processing and exploring applications within linguistics, media graphics and any other arena amenable to visualized vector patterns.