The Times-Table Mirror
Saturated in awe of the intricate patterning so prevalent in the world around me and faced with the luxury of exploring these realms, I found myself once again situated amongst the comfort of random human noise at another downtown coffee-shop, sipping tea with my calculator and notepad for hours on end.
Entwined in the fibres of ponderance and discovery, my mission on this occasion was to hunt down any musical treasures to be found in the times-tables.
First, I wrote out all 9 of the times-tables with the answers reduced to a single digit (a.k.a. "skein reduction"). The following example shows the skein reductions of the 4x table:
A repeating pattern of 4 8 3 7 2 6 1 5 9 was revealed by the 4x table. The rest of the times-tables gave the following repeating patterns:
In a zippy flash of eureka-gasm brainfartness I thought, "I will arrange each of these patterns into its own circular graph. I shall connect the dots!"
And so I connected the dots. I had imagined that some sort of nail-art type of imagery would appear, which it did. But some unexpected twists and turns emerged as well.
Firstly, I arranged the numbers 1 thru 9 clockwise around the outside of a circle, establishing a template for the graphs:
Next, I connected the dots around the circle in the skein order for each times-table. Using the 4x table as an example again (with the sequence 4 8 3 7 2 6 1 5 9 as shown on the previous page):
Cool looking pattern , eh?
But the real big surprise is the symetry in the sequence of shapes that are created through this procedure. Not including the 9x table, which seems to be reserved for its own special function as a sort of "mirror" or "grounding anchor" for the rest of the system, the series of 8 shapes evolves palindromically. They appear in the same order, backwards or forwards.
Also, polarity in regards to clockwise or counter-clockwise is reversed across the mirror. The 4x table shown above, for example, spins clockwise whereas its twin, the 5x table, is counter-clockwise.
The following diagram shows the pattern of patterns:
I originally had the diagram laid out lengthwise with all the shapes side-by-side, but I wanted it to fit on a single page without being too small to read, so I folded it in half, as shown. The folded arrangement has the added bonus of resembling a V-8 engine! The empty 9x graph at the bottom of the folded-looking diagram resembles the engine’s distributor, with the dot at 9 perhaps corresponding to the electrical supply wire. I wonder - what sort of pattern would emerge if the shapes were re-arranged to conform to the firing order of the sparkplugs of a real V-8?
Another idea for the deferral list, I suppose. Second edition... "Chaos In Boxes volume 2," perhaps?
[There’s this little yellow spider that keeps crawling on the computer screen; when the typing cursor moves towards the spider, it jumps out of the way every time, to avoid being attacked by the cursor. Silly little spider.]
The pattern gains new beauty and grace when arranged as a circle of circles:
Eventually I noticed that, by folding the circle in half along the dotted line, the numbers match up a certain specific way: the nonagons are numbered 1 and 8; the flowers are 2 and 7; the triangles are 3 and 6; the spikeys are 4 and 5. Each number pair totals 9, which is also the mirror number of the entire system - what an incredible coincidence!*
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