Colour

The science of colour (or for Americans color) shares with that of music the property of being an attempt to describe aspects of our physiological and psychological reactions to physical stimuli in situations where the physical stimuli have fairly simple mathematical relationships but the limitations and complexities of our sensory (and signal processing) apparatus cause us to try to fit them into a simpler framework in a way that paradoxically actually makes the analysis even more complicated. (The complication seems to me exacerbated by the fact that people have come up with somewhat conflicting models for simplifying different aspects of the situation but some people are led by popular accounts to treat these simplified models as absolute dogma – and so lots of effort gets expended in going from one to the other rather than seeing how they all arise from different ways of approximating the true story.)

In music, this is exemplified in the issues of harmony, chord, and discord (well-tempered vs equal-tempered scales for example), but here I want to concentrate on colour (where arguments about “primary” colours and which is the true “colour wheel” can waste a lot of time).

Observed Facts and The Physical Basis of Colour

White(ish) light from any source, when passed through a prism (or drops of water in the air), spreads out into a range or spectrum of coloured components.

People sometimes interpret the colours of this rainbow as distinct named bands (ROYGBIV, for example), but different people and cultures have come up with different numbers of bands and it appears that the range is truly continuous. Once light has passed through a prism, the output from a narrow part of the spectrum does not spread further, and this suggests that each point in the spectrum corresponds to a different basic kind of light – each with its own refraction angles.

The fact that similar spectra can be produced by passing the light through (or reflecting it off) a grid of parallel lines suggests that light is transmitted like a system of waves with the various kinds of light in the spectrum just corresponding to different wavelengths, and that the set of all physically possible kinds of light just corresponds to the set of all possible frequency distributions in the “fourier transform” of the signal.

Basic Psychology of Colour Perception

The pure frequencies of the spectrum are all seen by (most of) us as having different colours, but similar-looking coloured light from other sources sometimes can be split. For example, yellow light from the spectrum can be matched by a mixture of red and green, and violet light from the spectrum can be matched by a mixture of blue and a small amount of red. But there are also deeper purples and magentas that can be achieved by mixing red and blue which do not seem to match anything in the spectrum. By reviewing our own experience and comparing with that of experimental subjects we find that every kind of light gives a colour experience that can be at least approximately matched by some combination of three given sources so long as they include one that is predominantly at each end of the spectrum and one in the middle. The question of how to compare quality of matching (eg the number and choice of samples to compare and the brightness at which they are shown to the experimental subjects) is not very well defined and so it is not surprising that results differ, but there are a few widely used choices of Red, Green, and Blue phosphors that have over time come to be accepted as adequate.

Biological Facts and the Neurological Basis of Colour Perception

Examination of the human retina under a microscope shows the presence of various kinds of cells which further study shows to emit electrical signals in response to light. Some of these (the “rods”) are sensitive to all kinds of light and give about the same strength of signal to all of the frequencies in the spectrum of sunlight. Others (the “cones”) are more sensitive to some frequencies than others (and generally less sensitive overall).

Most people have three kinds of cones. One kind have their peak response towards the long wavelength end of the spectrum; some people used to call these the “Red-sensitive” or R cones but nowadays the fashion is to call them the L cones because even though they peak at longer wavelengths than the others they still fall off in the region of the spectrum that we see as bright red (so the strength of the L cone signal cannot be all that contributes to our sense of redness). The second group, which peak closer to the middle, are sometimes called “Green-sensitive” or G cones, but I will join those who refer to them as M cones instead. And for the third group, which peak nearer the short wavelength end and are sometimes called “Blue-sensitive” or B cones, I will follow the convention of calling them S cones and denote the strength of their signal by S.

It is not easy to measure the actual signals output from cone cells, but it is possible to extract the light sensitive chemicals and measure how their light absorbing properties depend on frequency and intensity of the incident light, and we may perhaps take this as a plausible proxy for the actual output signals.  Here’s one plot of how the energy absorption of each type of cone depends on the wavelength when the intensity is relatively low so as not to “saturate” the effect.

(Other sources may give somewhat different shapes because of using different methods for extracting the relevant parts of the cones and/or different illumination levels but the overall pattern seems to be similar in all cases.)

Note that the S signal (or at least the absorbance that we are taking as its proxy) is relatively weak, but this doesn’t necessarily mean that we see blue colours less vividly (though they do appear to fade more quickly in dim light) because our brain may respond more strongly to signals from the S cones  . For this reason the above graph also shows the normalized values L^,M^, and S^, where for each of X=L,M,S,  X^=X/X_max

Also, perhaps more significantly, we can see that the L signal is peaked quite close to the M and falls off a lot before the red end of the spectrum. So why do we nonetheless see true reds as quite vividly coloured?

One possible explanation (partly but not yet completely confirmed by actual analysis of optic nerve and brain neural connections – and also supported by psychological experiments in which people report how vividly coloured different light sources appear) is that the signals we identify as colours are calculated in terms of the percentage differences between L, M, and S signals rather than as those numbers themselves.

For example if we define R=2(L^-M^)/(L^+M^) , R^=R/R_max, B=2(S^-R^)/(S^+R^), and B^=B/B_max, (which can all be produced by running the LMS signals through a fairly simple sequence of neurons)

then the positive values of the R^ signal peak much further to the red end of the spectrum (and the negatives G^=-R^ and Y^=-B^

[insert graph of R^ and B^ signals, along with G^=-R^ and Y^=-B^]

This means that for any light source our sense of the colour and brightness of that source can be represented either by three positive numbers L,M,&S (actually four if we count the rods response) or by two signed numbers, RG and BY, for the colour and one positive number for the brightness.

Working in terms of the cone signals directly, we could represent all possible cone signals from each patch of the retina by points in (the first octant of) a 3d space (with one axis corresponding to the signal strength from each type of cone). Since scaling up all the strengths just corresponds to an increase of overall brightness we can identify the colour hues with the directions or with points on the first octant of a sphere of some fixed radius – which viewed from the middle direction looks like a triangle.

[insert rotatable 3d image of spherical surface in 1st octant of LMS space

and its triangular projection]

The corners of this triangle represent excitation of only one kind of cone – which might in principle be achievable by some chemical stimulus, but not by any kind of actual light since light of every pure frequency causes at least some stimulation of all three cone types (and in particular the L and M responses are generally quite close to one another)

This means that the set of response triples that can actually occur as a result of  stimulation by light is just a small part of the triangle. It includes, of course all of the response triples that can be stimulated by pure frequencies (which correspond to a curved line in the response space – curved because most spectral frequencies that stimulate both L and S also stimulate M so the corresponding points are not on the straight line joining the two endpoints) as well as any kind of weighted combination of those points. For any light source with spectrum of energy per unit of wavelength given by s(

which gives us what is called the “convex hull” of that curve of pure frequencies.

Except for the pure spectral colours, every physical colour spectrum has many others that give the same LMS responses and so look the same to us.

On the other hand, if we represent each LMS triple by its corresponding RG and BY values then we get a “map” of the spherical surface which  spreads things out much more clearly

[insert image of the RG vs BY horseshoe]

Every point in the original LMS triangle now corresponds to a point in the R^G^xB^Y^ coordinate plane but the coordinates are now much more closely related to our actual sense of “primary” colours and the part that corresponds to actual colour responses is more symmetrical (closer to being a circle)

Complementary and Contrasting Colours

Two coloured light sources that combine to make something that looks white are said to be complementary (where the word “complementary” with an ‘e’ refers to parts making up a whole, as opposed to “complimentary” with an ‘i’ which refers to making the other look good – or at least saying that it does).

While many people think of complementary colours as the most natural way of defining contrast, there may also be some reason for thinking that our sense of contrast is driven more by the results of the RG YB transformation and so to define contrast in terms of an RYGB colour wheel

Colours of Objects and Pigments

So far I have only been discussing the colours of light. But the colours of objects and pigments are a lot more complicated. This is because, although at the physical level the absorption/reflectance spectrum of an object or pigment is no more complicated than the spectrum of a light source, the colour we actually see from it depends on both its absorption/reflectance spectrum and the spectrum of the light with which it is illuminated. So, by seeing it under different lighting conditions we can with just our three kinds of colour cones learn far more about the detailed absorption/reflectance spectrum than just three weighted integrals.

Also, the mixing of pigments does not always just correspond to adding their absorption spectra. Depending on how the coloured particles are distributed and on whether the medium in which they are embedded is light dark or clear it may be either the absorptions or the reflectances (or some combination) that get added.

Most printer inks are designed to add their absorbancies in proportion when mixed and to be seen under a standard daylight spectrum and for that purpose is serves well to have fundamental pigments that are complementary to the RGB components used for analysis of the reflected light – namely Cyan, Magenta, and Yellow. But going beyond that basic statement takes us into the domain of the club of serious colour specialists – of which I am not a member.

Source: (1000) Why is orange colour in the rainbow? It’s the only one that doesn’t have its corresponding contrast colour in it. – Quora

Purely Probabilistic Positions?

What we interpret as the locations of elementary particles can certainly be compared with the predictions of regular mechanics. And they will often be quite close, so the classical predictions are actually useful. But the pattern of (usually small) variations from those predictions, while not “purely” probabilistic, does seem to have a component which cannot be explained in terms of some more precise classical properties that we just have not been able to properly measure. So our idea of a regular particle may just be something that does not really exist and what we interpret as its position may indeed by something that has an essentially probabilistic component.

Source: (1000) Alan Cooper’s answer to Is electron’s location purely probabilistic so its mechanism can’t be compared with regular mechanics, or is it just too small and too fast that the only way we can detect its location and interaction is through probability? – Quora

Maxwell’s Equations for Photons

In the quantum theory of electromagnetic fields Maxwell’s equations play two roles.

One is to describe the behaviour of the actual field observables which measure the combined effects of all possible numbers of photons, and the other is that they are satisfied by something that is as close as possible to being the “wave function” for a single photon.

I say “as close as possible” and put “wave function” in scare quotes because it does not satisfy all the properties of a non-relativistic wave function. In relativistic theories, the concept of strict localization does not exist. It can only be approximated for massive particles in frames where they have low momentum, and cannot be done at all for photons. But nonetheless, (as discussed in this survey article (.pdf) by Iwo Bialynicki-Birula) with appropriate normalization, a function satisfying the complex form of Maxwell’s equations can be used to generate probabilities for detection of a single photon in various experimental contexts.

See also the answer by ‘Chiral Anomaly’ to this question at physics stack-exchange.

Source: (1000) Alan Cooper’s answer to If particle nature of light is involved then what are Maxwell equations? – Quora

Sound from the Big Bang

A Quora question asks about the possibility of sound from the “Big Bang”:

I’ve been thinking… if we can look back and see the Big Bang and calculate when the universe was created and can see this because of the light traveling through space at the maximum speed that exists. The moment the Big Bang happened the sound had to be extremely loud and the first and only sound ever created which would still be traveling through space like everything else I would think right? So eventually it would make it to our universe and possibly be the catastrophic event that would destroy earth don’t you think?

It may be even more tempting to ridicule the idea of sound in the vacuum of space than the conception of the Big Bang as a localized event that happened someplace far away. But while the latter is a naive misconception, the former may not be so far fetched – at least if we “stretch” our definition of “sound”.

The theory of the “Big Bang” is not that it happened at some particular place, but rather that it happened *everywhere* 14 billion or so years ago; and just as we are now receiving “light” from when it happened 14billion or so light years away, so also whatever stars and beings evolved from what we see at that distance are just now receiving light that started with the part of the bang that happened right here. But although the “bang” was very hot and bright with lots of high energy short wavelengths, what we “see” of it now is very dim and of long wavelength (actually invisible) microwave radiation because the universe has expanded so much between then and now.

The same attenuation and extension of wavelength would apply also to sound waves if they could actually propagate through the vacuum of space, except that the source of what we would “hear” now would be much closer. In fact, back when the universe was denser there may have been variations of density which behaved like sound waves, but the expansion of the universe would have stretched them out by now so that their remnants would not be like audible sound waves (which in any case cannot exist in the vacuum that now fills most of space). Instead they would correspond to density variations on a much bigger astronomical scale. This may indeed be the source of some of the large scale non-uniformity we see in the density and distribution of gas and galaxies on a cosmic scale, but checking to what extent that is the case is a more serious exercise than just speculating that it may be so.

Source: (1000) I’ve been thinking… if we can look back and see the Big Bang and calculate when the universe was created and can see this because of the light traveling through space at the maximum speed that exists. The moment the Big Bang happened the sound had to be extremely loud and the first and only sound ever created which would still be traveling through space like everything else I would think right? So eventually it would make it to our universe and possibly be the catastrophic event that would destroy earth don’t you think? – Relativity IS Easy – Quora

Relativistic Mass

A Quora question asks: What is the equation that states that an object’s observed mass increases with its velocity?

It depends on what you mean by “an object’s observed mass”.

Nowadays the term “mass” is used exclusively for what used to be called the “rest mass” and is a property of the object alone that is independent of the relative velocity of the observer. So no physicist working today would say that “an object’s observed mass increases with its velocity”.

But there was a time in the past when some physicists used the term “mass” (usually, but not always, qualified with the adjective “inertial” or “relativistic”) to identify the multiplier needed to make a relativistically correct equation having the same form as Newton’s third law $#F=ma#$ (albeit only for the special case where the force and acceleration are parallel to the direction of relative motion between object and observer). So the equation you may be thinking of is $#m_{rel}=\frac{m_0}{\sqrt{1-v^2/c^2}}#$, but it is not a statement about what we now mean by “an object’s mass”. (And even adding the adjective “observed” or “apparent” doesn’t change that, as our observation of the “rest” mass is pretty much just as direct as that of the old “inertial” version.)

Despite many strident claims in other answers that it was “incorrect”, the alternative choice of using the word “mass” for $#m_{rel}#$ was in fact perfectly valid if applied correctly. It just wasn’t very useful because the resulting number $#m_{rel}#$ is not a property just of the object itself but depends also on the observer and is different (with a slightly more complicated formula) for accelerations and forces in directions other than that of the relative motion.

Source: (1000) Alan Cooper’s answer to What is the equation that states that an object’s observed mass increases with its velocity? – Quora

Explaining Relativity Without Equations

Can you explain time dilation and space contraction in relativity without using complex mathematical equations?

Yes. Any decent introductory text on relativity does this – but probably just in one or two sentences before going on to derive the actual formula (for which the apparent level of complexity of the resulting equations may depend on the reader’s experience).
The basic idea is that if two identical side-by-side trains are passing by one another and a light signal is sent when the ends from which it is sent are together, then if the trains are in relative motion the signal will reach the far end of one before the other. So if observers on both trains measure the same speed of light then their units of length and/or time must be different. Working out exactly how the coordinates used by each observer are related to those of the other does involve the use of mathematical formulas and equations, but they are well within the scope of high school algebra so whether or not you call them “complex” is a matter of perspective.

Source: (1000) Alan Cooper’s answer to Can you explain time dilation and space contraction in relativity without using complex mathematical equations? – Quora

Time Contraction

Special Relativity tells us that two inertial observers in relative motion each perceive the other to be ageing more slowly – ie each infers that the tick intervals of the moving clock appear to be dilated. But can time contract as well as dilate?

Yes, but with the proviso that the dilation or contraction is just a description of how the progress of one clock appears relative to another and that two observers will not necessarily agree on which events in their lives are simultaneous – and so can only compare average (rather than instantaneous) clock rates using the total time intervals on their clocks between events where they are together.

Two observers who separate and reunite will agree that the total time experienced by the one that felt more forces of acceleration (or of resistance to gravity if spending time near a massive object) will be less than that experienced by the other. This means that from the point of view of the one who was more accelerated (or spent more time at the bottom of a potential well) the clock of the other appears on average to have been speeded up (ie tick intervals appear contracted), while the one who remains unaccelerated interprets this as meaning that that the other’s clock tic intervals were, on average, dilated.

Source: (1000) Alan Cooper’s answer to Can time contract, as opposed to dilation? – Quora

Yet Another Run at the Twin “Paradox”

What’s with the “if” in this question? And what do you mean by the word “co-moving” other than perhaps stationary with respect to one another?
The standard version of the Twin “Paradox” starts with two twins, who are obviously “co-moving” in that sense at birth, and a distant star which is also “co-moving” (ie stationary with respect to the twins). Then at some point one of the twins travels to the star and back (usually with unspecified periods of acceleration and mostly constant speed in both directions).
Any correct application of Special Relativity predicts that when they re-unite the traveller is younger. The age difference can be calculated in terms of any frame of reference and (for any specified acceleration history – including that of instantaneous speed jumps) the answer is always the same so there is no real paradox.
The alleged “paradox” arises only in the mind of someone who notices that the traveller perceives the homie to have been ageing more slowly during the constant-speed legs of the trip and then just ignores the fact that the traveller also perceives a sudden rapid ageing of the homie during the turn-around. (During that turn-around the traveller feels the force of acceleration and so is aware of being in a non-inertial frame, whereas the homie feels no such forces. So the situation is definitely NOT symmetrical.)

Source: (1001) Alan Cooper’s answer to How do you solve the Twin Paradox if everyone (including the waypoint) is co-moving at the get-go? – Quora

Wigner’s Friend

If we define both the observer and the “observed” as both being part of say an even bigger system, would the wave function still collapse in this system?

This conundrum is known as the Wigner’s Friend problem, though it is also often asked with reference to Schrodinger’s cat.

In my opinion its best resolution is in the understanding that the wave function or quantum state is not a property of the system itself but of its relationship to an observer, and I think this view is a better reading of what Hugh Everett was describing in his “Relative State” interpretation of quantum mechanics [which was re-presented later (mostly by others) as a “Many Worlds” interpretation where observations (and other interactions) continually cause the creation of new “branches” (in a way that Everett himself apparently once described as “bullshit” in a marginal note on someone else’s elaboration of the MWI)].

Source: (1001) Alan Cooper’s answer to Is the collapse of the wave function in Quantum Physics based on a system frame of reference? If we define both the observer and the ‘observed’ as both being part of say an even bigger system, would the wave function still collapse in this system? – Quora