Example 6.  Anti-Color, Real Color, and Super-Color in Tuple-Images

Here, we define the concepts of 'anti-color' and 'super-color' to be, respectively, lesser than and greater than the R, G, B values used in color imagery.  These concepts are embodied in the following computer-generated graphics. In the image shown below (6.1), one can think of the row of lines on the top and the right side of the image as comprising 64 different 'color guns'.  Some of these guns 'fire' anti-color, and some shoot real color.  Only the ‘real color’ guns are visible in this example.  Anti-color guns are outside real color and they are arbitrarily assigned to the color black (K) to help highlight nearby real colors.  Interesting intersections appear when rows and columns containing color and anti-color combine by simple addition of their R, G, B triples

The ‘invisible’ anti-color and visible color can intersect to make a real color dimmer or extinguish it entirely.  Real colors can add to make super-color which is also 'invisible' and encoded as black (K).  As computed, this image contains only a 64 x 64 matrix with 10 pixels of 'padding' on the top and right side to show the pure gun color as undisturbed lines.  For enhanced visibility, the image size has been increased by a factor of ~ five, yielding a 400 x 400 24-bit image.  The image shown above was computed by selecting only four numbers, grouping them three at a time, and feeding them into the guns as R, G, B triples. By using all four numbers at these three positions to exhaustion, there are 64 combinations: 4 x 4 x 4 = 64 that comprise the 64 guns.  In this specific case, the 64 triples were selected as 3-tuples from the set of four numbers: {-0.5, 0, 0.25, 0.5}.  Lower numbers in the R, G, B-valued guns, such as (-0.5, -0.5, -0.5) are at both the extreme top left and at the bottom right of this image.  There addition at an intersection yields (-1.0, -1.0, -1.0), at the lower left, which is an 'invisible' anti-color that is encoded as black (K).   The highest-valued intersection occurs at the upper right where the R, G, B color (1.0, 1.0, 1.0) is generated by (0.5, 0.5, 0.5) intersecting additively with itself, and this appears white (W) on a (0 - 1) R, G, B scale.

Please note that this image has the overall shape of a single (1) arrow head pointing towards the upper right that is repeated on a medium scale of 1/16th the image's area (11 small arrow heads), and again at 1/256th of the image area (11 x 11 = 121 smaller arrow heads).  This 'small-medium-large' repetition pattern was particularly obvious in the figure selected above.  This fractal-like characteristic of the graphic, where microscopic structures mimic macroscopic levels at several levels of magnification, is apparently due in part to the ordering of numbers within the list of tuples.  According to Mathematica's implementation of the function "Tuple":

Tuples [list, n] generates a list of all possible n-tuples of elements from list.

The elements of list are treated as distinct, so that Tuples [list, n] for a list of length k gives output of length .

The order of elements in Tuples [list, n] is based on the order of elements in list, so that Tuples[{, … , }, n] gives {{, , … , }, {        , , … , }, … , {, , … , }}.

Any further interpretation of the patterns generated by this method can be explored by the reader using the source code itself, published here in notebook format (requires the Mathematica program) and in PDF.  This version of the source code loops 10 times and explores all possible 3-tuples from a set of eight numbers, four of which employ a random number generator.  As such, these images are more complex than the ones published here that explore all 3-tuples from a set of only four numbers.  The former sets of images are larger and more difficult to display on the web.  The .PDF file is the best for viewing details of the images, and it is highly compressed for downloading.

In the attached anti-color folder, we have generated 100 unique images using 3-tuples of a four number set (as in the image, above); except theses numbers are somewhat randomized in range. In each cycle, one image is produced from the 3-tuples, constrained by picking from the following four sets of numbers:

{(-1.0 to -.75) and -0.25}

{(0.0 to 1.0) and 1.0 }

{(1.0 to 0.0) and 0.0}

{0.25 and (1.0 to .75)}

Numbers within parenthesis ( ) indicate a range that is both indexed to the loop number and used to set the limits of a random number generator.  The same method can be viewed in the source code attached above.  As written, this version of the anti-color code can loop up to 100 times before the indexing needs to be reformatted.

Other examples of this methodology can use anti-color and super-color that combine to make real color, or one color-type may overwhelm the other to remain in the invisible ranges of either anti-color or super-color.  Super-colors are defined as having any single R, G, B value above 1.0.  This has to be considered on a per channel basis, as a single R, G, B triple such as (-0.5, 0, 1.5) contains an anti-color, real color, and super-color, respectively.  Any (R, G, or B) channel that is either negative or is greater than unity casts a veto vote on any real color and therefore turns the entire R, G, B triple to black (K).  Other 'voting' alternatives can be considered, such as creating floors and ceilings at  0 - 1, respectively,  per channel.  This has the effect of creating more real-color pixels; however, the patterns are less interesting than those shown here.

On average, a 0.5 MB tif file is required to encode a single (64 triplet) image and to preserve its color integrity.  Because of bandwidth and unwanted compression on the web (e.g., tif to jpeg and gif) the files shown here are unintentionally compressed and some color information has been lost.  Within MS Windows, these images play in the 'Windows Picture and Fax Viewer'.  Images are not presented here as animated gifs, because that format, at best, uses a single super-palette (for all images) of only 256 colors which causes severe dithering and/or color loss.

Possible applications of this methodology include image encryption, where a key is needed to reveal the valid real color range of any particular pixel. It is possible that engineers working in image compression could utilize some further finding based on this initial work.  In addition, any graphical entity that shows symmetry and hierarchical organization is potentially important in structural biology - where symmetry and organization are fundamental to certain structures and function, e.g., viruses and bacterial spore coats.  Finally, pseudocolor work is an excellent pedagogical technique for getting students interest in mathematics, in this case the combinatorial aspect of tuples and the use of  tensors.

Source code is given in .nb and .PDF formats.