Compare all of these hues... Please Register or Log in to view the hidden image! To a real pic of the spectrum of sunlight through a prism: Please Register or Log in to view the hidden image! Notice there is no magenta in the real spectrum, as explained in my previous post.
Those pics above of an optical phenomenon--diffraction of ordinary sunlight--mean that color perception is in the same space as the "appearance" of colors in diffracted light. An optical diffraction grating is a device with a periodic, yet linear(!), structure. The linearity is present in the spiral groove in DVDs and CDs, which are commonly available diffraction gratings. This suggests that color generation, via a process called diffraction, has an underlying vector space. An ordinary flourescent light emits a spectrum with "fixed" color peaks in it and you can prove it with a diffraction grating--there is a definite notion there of distances between colors, and of an affine space, or at least of how far you need to be from the grating to see the different colors. Of course, people like QuarkHead and iceaura know all that. I don't see how diffracted light goes anywhere near the same combination of colors being more than one perceived color--that sounds like a discrimination problem, as in the kind optical illusions have, so it isn't in this spectral (i.e. "pure") color space anyway. At least, that's what Maxwell thought--he and Boltzmann have an energy distribution Maxwell-Boltzmann statistics, named after them. Erwin Schrodinger thought so too, he rigorously defines the cone over the simplex (graph) of fixed color wavelengths. Wavelengths are easier to work with than frequencies, sometimes, but any physicist should know how to transform a wavelength into a frequency.
No such papers have been posted here. Every related paper posted here presents locations and distances defined over wavelengths, not colors. Everyone here has always agreed that a vector space, a distance function, and all necessary properties for a Hilbert space, can be defined over the visible light portion of the EMR spectrum and the combinations of the wavelengths derived from it. I see wavelengths amd combinations thereof, represented in the colors assigned to them (for various perfectly good reasons) by the maker of the diagram. That assignment is not reliable in the larger world of this thread, the world in which a "color space" rather than a wavelength space is the topic, which will be found (see links for examples) to contain no such one to one correspondence of colors and wavelength combinations. It's a perception reality, not a "discrimination problem". And yes, lots of human color perceptions are missing from the space of "spectral" or "pure" etc "colors". That's because they are wavelengths, not colors, in the first place. Short version: Wavelengths are not colors.
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So what did Maxwell, or Schrodinger, represent locations or distances over in their papers? The ones they published? What about Grassman? I'm beginning to think you aren't capable of understanding why wavelengths and colors are related. Colors however, exist in the so-called time domain, wavelengths are time-independent. That's something you learn in 1st year electronics.
The Hilbert space of color wavelengths is the Hilbert space of color perception. Perception is well-understood and well-mapped to these standards with thousands of test subjects. Your continued insistence that colors aren't wavelengths is just like saying the 24 bit string that specifies an sRGB color isn't a color. That's exactly right, the string is a string and the color is a color, but the string "is" a color when your display controller sends it to a pixel or two. That is, when it transforms the string into a color (in real time!).
Sorry, are you saying you can't see that x is the color red, and y is the color green? They even added a helpful pair of colored arrows! This x/y = red/green, says nothing to you that makes sense? It doesn't suggest that the "maker" of the diagram actually has plotted some kind of mathematical function?
Those are places where the red end of one spectrum is overlapping the blue/violet end of a different spectrum. If you look closely at any individual spectrum, you should be able to make out that there is no magenta in it. But you probably knew this, and that is probably why you said sorry in advance. Please Register or Log in to view the hidden image! No worries either way.
I see wavelength combinations presented as corresponding one - one to specific colors by whoever made the diagram. For that, I need a screen or other presentation medium carefully designed to present that engineer's color assignment - to be compatible with that engineering setup. No other wavelength presentation setup - such as the view of wavelengths from my window - will do: the colors will be different, and stand in different relationships to each other. Color perception space cannot be described as a Hilbert space, because it is not a vector space and it has no distance function. Meanwhile, there is no such thing as a physical, universal, non-arbitrary "color wavelength" - that is: there is no wavelength that corresponds to one and only one perceived color, and there is no perceived color that corresponds to one and only one wavelength. That is one major reason why creating a Hilbert space of wavelengths does not simultaneously create one of colors. (Another: there is no ordering of colors as there is of wavelengths - no ordering of wavelengths matches an ordering of perceived color ). Wavelengths are not colors. Repeat: wavelengths are not colors. Assigning colors to them, no matter how "natural", well motivated, or useful, does not change that basic physical fact. It is an engineer's or physicist's or other such person's choice and decision and inevitable simplification and distortion, subject to circumstantial and alterable criteria.
arfa brane keeps insisting there exists such a thing as "colour space" which is a vector space. For this to be true, certain conditions need to hold - for every colour \(C\) there exists another colour \(C^{-1}\) such that \(C\cdot C^{-1} = C_0\) where \(C_0 \cdot C = C\). Also you need a multiplier from a scalar field such that, for any colour \(C\) that for \(\alpha \in \mathbb{F}\) then \( \alpha C_1 = C_2\) in the same same space. arfa brane made the claim - the onus is on him to show these conditions hold
Not exactly. What I keep insisting is that the vector space of colors is well-defined--by Maxwell and Helmholtz, Young etc, oh yeah, and by Erwin Schrodinger. I can only assume that you have never used colors to represent a vector space; this is something engineers have been obliged to do, because engineering requires standards. Please, if you feel like ever doing it, google the words "color vector space", and maybe even look at the introduction of a few of the linked papers, or maybe the posts to stackexchange or whatever. But I'm not convinced you've researched the subject very much because you seem to know nothing about the history of color models. Do you know (I bet you don't), that Riemann said color space is one of only two kinds of inhomogenous manifold? Is that a gotcha? I think so. If you and iceaura insist the models are wrong, then the onus is on you to demonstrate how the development of models of color since say, Newton, has been misguided and simply can't work--colors are not wavelengths, whatever significance that is supposed to have. What significance does it have? Why is there a problem with fixing a wavelength and calling it a primary color (or fixing anything)? Where does this fail to be meaningful (given, as I may have mentioned, humans have been designing electronic color displays for almost a century). They've been painting with colored paints for quite a lot longer. But that can't be possible, apparently, because nobody can mix two colors and reliably produce a third color, how can anyone hope to paint a realistically colored and shaded painting of anything at all? So the blue LED curve is misleading too? Humans don't universally see the color blue, when a blue LED emits some light . . . This attempt by engineers to reliably produce a color (by fixing its wavelength), is also some kind of optical illusion, perhaps?
One other query: can someone here explain how engineers design color displays which have electronic components, and avoid vector spaces. After that easy one, explain how to design and build a range of amplifiers using electronic components, but not vectors.
Good Heavens, yet another construction! arfa brane, you start by telling us that "colour space" is an Hilbert space, then no, it's an affine space, then no, it's a Euclidean space and now it's a projective space. You even mention inhomogeneous manifolds, as though you are familiar with them. In no case are you able to give an argument (let alone a proof), you merely quote, with seeming approval, random internet sources. As these spaces, although related, differ in very important respects, it is tempting to suggest you don't even understand the sources you are quoting, never mind the mathematics involved.
So you're prepared to dismiss the possibility that colors are a Hilbert space based on what one poster, me, has posted? You aren't prepared to read anything about color models? What the hell is that? But you seem unable or unwilling to agree with the argument that you can't design an electronic display easily, unless you define a vector space. What do you suppose is the reason different people (actual scientists) have models in these "related, but different in important respects" vector spaces? Or inhomogenous manifolds (as if color is inhomogenous, pfft). Why would Riemann say color is a manifold? What was his problem?
You have posted nothing in support of that bizarre claim. Everything you have posted is based on wavelengths. The best of them, the closest you get to relevance here, are approaches that sharply limit human color vision to laboratory arrangements designed - intentionally and explicitly and specifically - to provide the absolutely necessary one-one correspondence between color and wavelength (or wavelength category, or whatever) that a vector space or distance function for colors based on wavelengths requires. No such correspondence exists in real world settings, in either direction - as my links above demonstrate, for you to revisit and verify at your leisure. For example: Your feelings may be contributing to your confusion. Read your sources with less feeling and more attention to detail, maybe. Here is a quote from the preview of chapter 5 in your link, for example (bolding and punctuation in original). It concerns the definition of the "color manifold", which turns out to be a manifold of wavelength categories: Outside of the laboratory, of course, such carefully limited presentations of "lights" (wavelengths) are rare and unrepresentative, and no such categories of "lights" (categories of wavelengths) can be defined. That is certainly not a criticism of the research, which is valuable and informative (and a lot of work - setting things up so a given wavelength combination elicited a single given color assignment by the "normal" (?) human visual system, excluding all the other assignments routine in the real world, was clearly difficult) - it just means one can't define a vector space or distance function over the colors themselves. Wavelengths are not colors. There's a reason I keep repeating that.
But that's exactly what scientists, like Grassmann, Schrodinger, Maxwell and Helmholtz actually do, they define a vector space over colors. If I said there's such a thing as a color tensor, and it's very useful in image analysis, what does "you can't define a vector space over colors" mean then? If a vector space can't be defined over colors, which I have to assume means the colors in the display I'm looking at (which as I mentioned I can look at quite closely, in which case I see only three colors but at different magnitudes), then why have so many people done it? "You can't define a vector space over colors", is refuted by the existence of many papers published over a century or so, that do define a vector space over colors. Those would be all the papers you haven't seen because you're not really interested in educating yourself, are you? I don't see wavelengths, unlike you, I see colors when I look at a color display, or an LED emitting light in the visible range. Wavelengths of light aren't really things you can see, in my experience.
They don't. They define a vector space over wavelengths, categories of wavelengths, arrangements of wavelengths, etc. I have been quoting your own links to that effect, including your latest just above that post. Do you read your links? It would mean you have not defined a tensor over colors. You have instead defined a tensor over a wavelength space, to which you have assigned color labels. Or it might mean you have been holding out, and refusing to present links or other evidence to a tensor you know about that was defined without reference to wavelengths. That would be interesting - it looks impossible, to me, because unlike wavelengths colors are not ordered or additive etc. But if there is one, there it would be, right in front of me. Nothing you have linked here does that. Everything you have linked here defines its vector spaces over wavelengths, categories of wavelengths, etc. What papers are you talking about? ? Nobody has claimed colors don't exist or can't be seen, merely because they cannot be structured as a vector space or support a distance function.
Actually that isn't true. The vector space is defined over the outputs of LEDs, each emitting specific colors, for instance. Or it's defined over the space of fixed colors in ink cartridges, or over the space of fixed colors in tubes of oil-based paints. Once you fix at least two colors you have a vector space of colors, with a color basis. I can only assume you haven't tried googling "color space", and actually reading the papers that define color vectors, tensors etc. There is a Grassman function with a color basis. There are graded algebras, vector pencils and blades, all defined on color spaces. And so there's a way to visualise say, a Clifford algebra with color vectors. Of course, you can claim that the models are all just mathematical approximations. I don't see how the approximation of colors means my computer display can't output actual colors because colors aren't wavelengths. It just doesn't scan.
What is color. There are two parts to this question; one is the perception part, the other is the generation part. I would say the first part happened . . . first, then we thought about how to "make" different colors, we've been painting our bodies for tens of thousands of years apart from decorating ourselves with colorful objects, or shiny ones. So today what we do to generate colors reliably is build displays with hundreds of thousands of individual, identically manufactured, pixels each having three lights which each emit light at a different wavelength. The wavelength differences are important, without them there is no way to see . . . different colors (resp. color difference). That we also have standard color palletes is understandable; that we want sources of color which are stable (i.e whose wavelengths stay fixed or perceivably so) is also quite understandable. That vector algebra has been developed over "the space of" colors is also quite understandable, because you can't design electronic circuits without vector algebras. Without a formal mathematics, electronics would be hit and miss, which is isn't and hasn't been since at least Maxwell. Now disagree with me that this author is also talking about color vectors (resp. is not talking about 'wavelength' vectors):
