Wisdom, justice and mercy. Generosity, fairness and self-interest. Two dual descriptions of one spectrum. Decency is about balance among them. [cf: red, green, blue; cyan, magenta, yellow; two pictures of `colour' (leaving out black and the dual it would need for the RGB scheme to be as full as CMYK); white is all that stuff near `the middle'. Equally, cf: the low-entropy states - relative to a description of something in terms of suitable co-ordinates - are the ones `well away from' the co-ordinate axes and the associated planes and whatever of higher dimension, where appropriate.] Decency's path need not lead simply `up the diagonal', though: it responds to what's around it and adapts to become able to navigate a path which keeps `near' the diagonal. A true dynamics of how folk develop needs, at least, the

why isn't rgb enough ? not dual to cmyk [r,g,b] ~ [c,m,y,k] iff [r,g,b] = k.(c.C+m.M+y.Y) might not be the right model but suffices to show a distinction which would matter: the C, M and Y involved have C parallel to [0,1,1], M parallel to [1,0,1] and Y parallel to [1,1,0] we can re-phrase the above, with a little care, to get [c,m,y] parallel to r.R+g.G+b.B with R parallel to [-1,1,1] (because M+Y-C = [2,0,0]), G parallel to [1,-1,1] and B parallel to [1,1,-1]. The reality of colour perception (ignoring here the case of four-colour vision, and beyond, as reasonably straightforward exercises in generalisation for the enthusiastic) says that we have receptors of four kinds, known as red, green, blue and `overall', though the latter is `implemented' differently (rods rather than cones), and is optimised for different things. The signal from a receptor has the form of an `inner product' between the receptor's absorption spectrum and the spectrum of the light incident on that receptor (assessed with some time-granularity, which I'll not pretend to cover here). The strenghts of the red, green and blue signals for a given illumination are correlated because the absorption spectra are `nowhere negative' and overlap (that is, there are frequencies at which more than one of the spectra are positive), while the spectrum of `incident light' is also positive. Take a dye: stain transparent material with it and look at how the spectrum of light coming out of it depends on that of the light that went in. It turns out to have the form of a (usually diagonal) linear function: its detail is as follows: one may describe the light incident on the transparent material, or that emerging from it, in terms of a scalar function of frequency - the energy density per unit frequency of the light - which is known as the `spectrum' of the light. Its form is (F| spectrum :E) with F the (positive) linear space of frequencies (which is one-dimensional) and E the positive linear space of intensities. Strictly, this description loses some phase information, but of a kind that I'd expect entropy to destroy very fast. Let Spectra = {(F|:E)} be the space of mappings (F|:E): though its members mostly aren't linear, Spectra is still a linear space. It is important that E is positive (or, at least, has a lower bound) and linear: it is largely irrelevant that F is also. the spectrum of the emergent light, e, may be obtained from that of the incident light, i, by a linear mapping (Spectra|:Spectra), at least for a broad range of spectra, for a wide variety of dyes. That linear mapping is, furthermore, expressible as a measure on F, of form ({F|:Scalars}| T :Scalars) implying ({(F|:L)}| :L) for arbitrary linear L as, for any basis (dim| :dual({F|:L})), ({F|:L}: x-> (F|:E)}), as i-> (F| f-> integral(F| g-> i(g).t(g,f) :E) :E) - so e(f) = integral(F| g-> i(g).t(g,f) :E), which is in E, and t takes two frequencies. Furthermore, there's a constraint (conservation of energy), on the form of t(g,f) - implicitly, that e is never negative and integral(F| f-> f×e(f) :F⊗E) is less than integral(F| f-> f×i(f) :F⊗E). Note that F⊗E is, again, one-dimensional and positive (and here we do depend on F being (linear and) likewise), so we can make sense of `less than'. This much is even true for rubies and players of kindred tricks. function of frequency of light, called the absorption spectrum of the stained object, for which the emergent light differs from the incoming light by exception to diagonal: ruby. a certain kind of green is transmuted to a red. If you take a mixture of dyes and consider how they reflect light, the spectrum of the reflected light depends on that of incident light and the `average' absorption One can only represent, using RGB, a cube of space. Written by Eddy.$Id: decency.html,v 1.2 2001-10-17 17:11:31 eddy Exp $