The Alan Sondheim Mail Archive

June 4, 2007


I graph various forms of the equation y = tan(x^2); interesting phenomena
appear. Check out the .gif images at - the names
are equation00.gif, equation01.gif, etc. The graphs extend along the
x-axis with what appear to be constantly changing local symmetries. I have
experimented with different software/hardware, beginning with a TI-85
graphing calculator and a highly precision similar software program,
GraphCalc (obtained from Sourceforge). I've also used the graphing calcu-
lator and Mathematica in Mac OS9. Only in the last, Mathematica, did the
symmetries seem to disappear. I think the phenomena - the perception of
local symmetries - is the result of raster, i.e. the digitalization pro-
cess in the calculation of what are basically analogic functions. Raster
is tolerance-dependent; it's the digital 'jump' screened against the real.
The symmetries appear to be, masquerade as, independent 'things,' dif-
ferent from one another, lined up and sometimes intersecting in a chaotic
fashion. In other words, the appearance of things is constituted here by
the very absence of things; within the digital raster, every point, pixel,
is independent, disconnected, from every other.

Ah well, it's late and I'm not expressing myself well. I'll try again:
Given y = tan(x^2), the resulting graph on a digital computer seems to be
raster-dependent; the image appears to possess local and intersecting
symmetrical segments which seem chaotic. These segments can be considered
'things' in the sense of perceptually-defined contour-mapping. (In other
words, they appear to be things, local processes, local phenomena, whether
or not they are in 'actuality,' within the real.) Using a bad metaphor,
such 'things' are clearly gestalt images of disconnected pixels - i.e. a
line in the graph which appears connected, isn't. When sections of the
graph are enlarged, their morphology may radically transform. So what I'm
interested in is the digital representation of this particular group of
analogic functions, and the mathematics behind it. Is the representation
really chaotic? Are the symmetries really geometrically different from one
another, and if so, what's the mathematics behind this? And so forth. Any
help you might give me s greatly appreciated. In the meantime, the images
are beautiful. Check out the gifs and jpegs at -
look at the 'equation' files.


   I live not in myself, but I become
   Portion of that around me; and to me
   High mountains are a feeling, but the hum
   Of human cities torture: I can see
   Nothing to loathe in nature, save to be
   A link reluctant in a fleshy chain,
   Chased among creatures, when the soul can flee,
   And with the sky, the peak, the heaving plain
Of ocean, or the stars, mingle, and not in vain.

[...] [...]

   But let me quit man's works, again to read
   His Maker's, spread around me, and suspend
   This page, which from my reveries I feed,
   Until it seems prolonging without end;
   The clouds above me to the white Alps tend,
   And I must pierce them, and survey whate'er
   May be permitted, as my steps I bend
   To their most great and growing region, where
The earth to her embrace compels the powers of air.

(Byron, Childe Harold)

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