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Mantis shrimp have 12 to 16 photoreceptors and humans have 2, and in rare occasions even 3. But if mantis shrimp can see 16, how many other colors are out there? How many photoreceptors is possible? How much of the spectrum is not ever going to be seen by any creature ever?

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The first source you posted pretty much answers your question. Here are some others, probably referring to the same study but talking about it differently:

The Mantis Shrimp Sees Like A Satellite (National Geographic; this article is referred to in your first source)

Study Offers Insights into Unique Color Vision of Mantis Shrimp

Mantis shrimp's super colour vision debunked (Nature)

Here is the paper that those three articles are based on:
A Different Form of Color Vision in Mantis Shrimp

And what seems to be the author's (slightly more recent) thesis on the subject:
Colour vision in mantis shrimps: understanding one of the most complex visual systems in the world

The big thing to understand is that there is no such thing as "the color spectrum". There is the electromagnetic spectrum, which is one-dimensional: you can describe an electromagnetic wave's position on the spectrum with one number, its wavelength. But if you look at a picture of the electromagnetic spectrum with the colors associated you can notice something immediately:

color spectrum

You don't see a uniform increasing of a single quantity from one number to the next, you see several distinct colors following each other.

That's because the point of color vision isn't to detect variations along the electromagnetic spectrum per se; the aim is to allow our brains to get more information about the world around, based on the fact that light comes in many different wavelengths. Different objects will reflect different combinations of those wavelengths, meaning being able to tell some of those wavelengths apart improves our ability to tell objects apart based on those differences.

The key here is that there isn't a perfect way of doing this; nobody has a system where you detect the exact wavelength of each photon and tell exactly how many of each wavelength you receive. What human eyes do is have some specific receptors be more receptive to some wavelengths than others, and then the brain compares how excited each receptor is and puts all that information together to generate a sense of "color", which is basically a way of telling some things apart when they reflect certain different wavelengths. It's related to wavelength, but isn't the same thing; for example our brains recognize wavelengths that excite the red and green receptors equally as "yellow". This includes wavelengths that fall between the red and green wavelengths, but also combinations of red and green wavelengths. That's how you get the color wheel, the ability to make some colors out of others, and the fact you can generate all colors that we see from three primary colors. An organism with an extra receptor between the red and green would be able to tell those different combinations of wavelengths apart so that "yellow" to us would look like several different possible colors to them. Conversely, trichromatic humans are able to tell colors apart, like red and green, that are one single color to dichromatic mammals.

So the answer to "how many colors on the spectrum are there" is "there is no universal color spectrum". Every organism with its own system of light receptors can theoretically have its own color spectrum; how many colors they see is a question of how many different combinations of wavelengths they can tell apart, and that's a function of how many receptors they have but there can also be other factors (like the sensitivity of given receptors, how much processing power they're dedicating to color vision, exploiting different aspects of physics that affect wavelength... In humans even language and culture can influence color discrimination to some extent).

And where does the mantis shrimp come in? Well this is interesting, because it's been generally thought as you put it that the mantis shrimp would be able to see many more colors than we do, due to its much more numerous receptors, but the recent studies mentioned in the first source you linked to suggest that the mantis shrimp is worse at discriminating colors than we are, i.e. it sees fewer colors. However it appears its color vision functions very differently, and that's where the 16 different receptors come in. It's possible that the different receptors allow the mantis shrimp not to detect more colors, but to process the colors it does detect much faster. To quote the Nature article:

If the shrimp eye compared adjacent spectra, like the human eye does, it would have allowed the animals to discriminate between wavelengths as close as 1–5 nanometres, the authors say. Instead, each type of photoreceptor seems to pick up a specific colour, identifying it in a way that is less sensitive than the human eye but does not require brain-power-heavy comparisons. That probably gives the predatory shrimp a speed advantage in distinguishing between different-coloured prey, says Roy Caldwell, a behavioural ecologist at the University of California, Berkeley.

And the National Geographic one:

Their working hypothesis is that the mantis shrimps analyse the outputs from all of their 12 receptors at once. Rather than making comparisons between those receptors, they pass the entire pattern of outputs onto the brain, without any processing. “One could imagine that they have a look-up table in their brain,” says Marshall. So rather than discriminating between colours like we do, their eyes are adapted for recognising colours.

“Oddly enough, the closest device to stomatopods would be a satellite,” says Marshall. “Remote sensing algorithms have look-up tables of colour to fill in the image that the satellite forms.”

For your other questions:

How many photoreceptors is possible?

Theoretically the limit would only be how many you can pack in an eye and how many chemical pigments exist, but in practice there is a trade-off between color and spatial discrimination (after all any color receptor is a light receptor that's limiting itself to certain wavelengths only, so you lose in overall light perception). Another article in the Science journal the article on Mantis shrimp color vision appeared in suggests the optimal number is isn't over 4 (in organisms that use our system of color vision of course, not the mantis shrimp's):

In fact, theory predicts that two to four receptor types are optimal for discriminating the spectra of natural materials and maximizing the number of objects that could be distinguished by color.

And:

How much of the spectrum is not ever going to be seen by any creature ever?

Again, considering every set of photoreceptors leads to its own color spectrum (and every brain probably cooks it to its own sauce beyond that), that question applied to the set of all colors isn't really meaningful. The answer would probably be along the lines of "almost all of it" because the set of actual color vision systems in the natural world is an infinitesimal percentage of the set of theoretically possible color vision systems... but it's not like those colors are "out there to be seen" in anything but the most abstract sense; they exist in the same sense that a child that's the combination of a sperm and egg that haven't met exists.

If by "spectrum" we mean "electromagnetic spectrum", then there is likely an upper and lower bound to the wavelengths that can be detected, where wavelengths on the upper bound are too deadly for a cell to survive let alone detect (though bodies can and do sacrifice cells to exploit inhospitable environments, see the cells lining our stomachs, so even that might not be an absolute obstacle), and on the lower bound aren't energetic enough to cause pigment molecules to react to them. I don't know what those bounds are though; it's likely that more important constraints are how useful it is to detect those higher and lower wavelengths, that any organism would bother evolving the ability. (also, for the lower wavelengths, whether their detection is associated with forming an image. For example we can detect infrared radiation as heat but we don't think of that as "seeing". On the other hand "eyespots" that detect visible light without forming an image are often thought of as a primitive form of sight; there is a level of arbitrariness there but it's also true that lower wavelengths have different optical properties).

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  • $\begingroup$ "You don't see a uniform increasing of a single quantity from one number to the next, you see several distinct colors following each other." That is simply an artifact of the computer process used to make that image, likely in sRGB, which has far less of a color range than Adobe RGB or ProPhoto. Using a larger color space and having a monitor capable of displaying it would would produce a gradient with no human discernible steps to it. $\endgroup$ – AMR Jun 7 '17 at 23:27
  • $\begingroup$ @AMR it's not a question of discernible steps (in fact the previous image I had for this didn't have such visible ones but it was https), it's a question of yellow not being perceived as a shade of red or blue. Compare with one-dimensional entities that are perceived as one-dimensional (if not linear), like temperatures from warm to hot, sound from quiet to loud, luminosity from dark to light... $\endgroup$ – Oosaka Jun 8 '17 at 5:49
  • $\begingroup$ Then your answer is not explaining that point correctly. Electromagnetic waves are not one-dimensional. They have amplitude and frequency as well as wavelength. All of these contribute to the energy of that photon, which in turn excite electrons in opsins that can then create an action potential that causes the neuron to fire. The number of conjugated bonds determines what range of wavelengths the opsin will respond to. You are also making assumptions that I am not sure you can back up, and you should avoid doing that. You could add as a comment or clearly label that it is your opinion. $\endgroup$ – AMR Jun 8 '17 at 6:26
  • $\begingroup$ @Rozenn Keribin: But I do percieve the yellow grdually transitioning to orangish-yellow, then orange at one end, and greenish-yellow then green at the other. And I dare say any interior decorator worth his/her salt could put at least a couple of dozen names to various shades in the transition: peach, tangerine, chartreuse, &c. $\endgroup$ – jamesqf Jun 8 '17 at 6:38
  • $\begingroup$ @jamesqf, AMR: these points should be discussed in chat I think. chat.stackexchange.com/rooms/60137/… $\endgroup$ – Oosaka Jun 8 '17 at 17:43

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