Hi, Asher,
Does the color map have to be in 3D space? Is it possible one could use other transforms that would allow more complete representations.
It is generally accepted that, for
homo sapiens (certainly not for many other species), the sensation of color is three-dimensional (in the mathematical sense): three values are sufficient to specify a color, and thus we can "plot" color in a three-axis space. And no space with more than three dimensions can do a better job of "specifying" a color - can give a "more-complete representation".
Bu that is not to say that other transforms (mathematically-redundant) cannot help us to better grasp the significance of color, or the working of color-handling chains.
A parallel in another realm is this. If we have a vessel that is a closed cylindrical cylinder, we can completely describe it (physically) in terms of two parameters, perhaps length and diameter. But, in a catalog of cylindrical air tanks, it is certainly useful to state perhaps length, diameter, and volume (and in a setting where this would be of any consequence, perhaps surface area as well).
In engineering, this is spoken of as "over-specifying" the vessel: we dare not, in specifying the construction of one, give length, diameter, and volume, not just because it is not needed, but because our values may conflict.
But when our purpose is to
describe, rather than
specify, this can in fact be very useful.
Another parallel is this. We can completely define the location of a point on the earth's surface in terms of its latitude and longitude. But it may be very useful, in describing the location of a radio tower, to not only describes its location in terms of latitude and longitude, but also in terms of coordinates in a mapping system, and also in terms of its location as a certain distance off the the side of the centerline of a particular highway at a certain milepost.
Now, can we think of a six-dimensional space in which the axes are:
• Latitude and longitude
• Easting and Northing on a grid
• Milepost and offset distance on some stated highway.
Yes, we can. But not all points in such a space are valid - only those whose latitude and longitude coordinates are consistent with their Easting and Northing coordinates and with their milepost and offset coordinates. That's actually only a tiny fraction of all the points in the space.
Since a six-dimensional space has no physical realization, I cannot describe this "sparse" space as comprising, for example, only a bunch of little threads. But it is conceptually much like that.
To get back to something we can better visualize, let's return to our universe of air tanks, for which we can have a length, a diameter, and (not independently) a volume. If we imagine a three dimensional space on which we will plot all available (or even all possible) air tanks, we find that they all must lie on a surface - the surface the satisfies the mathematical relationship between diameter, length, and volume. So it is really only a "two dimensional plot on a non-flat surface". But it could still be very useful to the connoisseur of air tanks.
Now as to what kind of "over-specified" color model might be valuable to us, we would first have to be able to articulate its objectives. In what way could we better visualize something about a color, and what, than in any three-dimensional description? What is it that we need to improve our appreciation of?
It is an interesting challenge.
Best regards,
Doug