Abstract
Colour is commonly regarded as an absolute measure of object properties, but most work on visual communication signals is concerned with colour differences, typically scaled by just noticeable differences (JNDs). Object colour solids represent the colour gamut of reflective materials for an eye. The geometry of colour solids reveals general relationships between colours and object properties which can explain why certain colours are significant to animals and evolve as signals. We define a measure of colour vividness, such that points on the surface are maximally vivid and the ‘grey’ centre is minimally vivid. We show that a vivid colour for one animal is likely to vivid for others, and highly vivid colours are less easily mimicked than less vivid colours. Further, vivid colours such as black, white, red, blue and light, unsaturated shades are produced pure or orderly materials. This kind of material needs to created and maintained against entropic processes that would otherwise degrade or destroy them. Vivid coloration is therefore indicative of ecological affordance or biological function, so that it is valuable to have attentional biases towards these colours regardless of any specific significance.
1 Introduction
Animal and plant communication signals carry varied messages: some are attractive while others are defensive; courtship displays appeal to naïve viewers, whereas flowers and aposematic signals need to be memorable. Despite their various functions, and the diversity of colour vision in their natural receivers, signals generally include a limited palette colours including, black, white, saturated hues and light but unsaturated colours such as pink, whereas greys and browns are infrequent.
Any signal must attract attention and engender a response – it will be ineffective if it is overlooked or ignored. Why then should certain colours attract the attention of an eye with any given set of spectral photoreceptors, and why are the same types of spectra appropriate for animals with different types of colour vison? We show here how answers to these questions might be found if a colour is regarded not simply as a measure of the spectral composition of a light, but rather as a property of physical objects. This requires quantifying colour in new ways and dropping two of the most widespread procedures: the splitting of colour into independent chomatic and achromatic components, and, the scaling of colour distances using discriminability.
Most work biological signaling treats animal colour vision as means to discriminate between spectra, and is primarily concerned with the magnitudes of colour differences as measured by ‘just noticeable differences’ (JNDs) [Kelber et al., 2003, Kemp et al., 2015, Olsson et al., 2017]. Accordingly, one might predict that a colour signal will attract a receiver’s attention when it differs strongly from the background [Gittleman and Harvey, 1980], or the pattern itself has a high contrast [Rowe and Guilford, 1996, Aronsson and Gamberale-Stille, 2008]. In its everyday use ‘colour’ is not a relative or relational term like ‘contrast’, but is absolute, and part and parcel of object recognition. We do not say “the tennis ball is more yellow than the court,” but “the tennis ball is yellow”. Light reflected from a surface depends upon its chemical composition and (nanoscale) physical structure. Colour vision yields information about these properties. We speak of red faces, blue tea-mugs and so forth, and other animals may be similar. It follows that animals could find particular colours significant because they are characteristic of particular kinds of object.
The significance of a colour might be related to the specific coloration mechanism – for example if a pigment is costly to produce [Olson and Owens, 1998] – or to associations with particular beneficial or harmful objects [Endler and McLellan, 1988, Endler and Basolo, 1998, Palmer and Schloss, 2010]. In contrast with, and complementary to these approaches, we look here more broadly at the relationship between the composition of materials, their reflectances, and their colours. Perhaps surprisingly, we find that there are aspects of object colour that are consistent between different observers, which can be linked to underlying physical properties that are relevent to the psychology of an organism.
In particular, we analyse a property of colour we term ‘vividness’. Highly vivid colours for one organism are highly vivid for any other with the same or more number of photoreceptor types (rules 1 and 2). We then show that highly vivid colours are more informative, in that they correspond to fewer materials, and that those materials will be purer and more ordered than less vivid materials. From this we argue that psychological salience of these colours would be evolutionarily adaptive.
1.1 Object Colour Solids and Vividness
Objects are seen by reflected light, and the gamut of all possible reflection spectra can be represented by their locations within in a Cartesian space known as an object colour solid whose axes correspond to photoreceptor excitations relative to the illumination spectrum (Figs 1, 2 Vorobyev [2003], Koenderink [2010]). Colour solids include the three main aspects of colour namely hue, saturation and brightness, whereas the more familiar chromaticity diagrams, such as Maxwell’s triangle discount brightness. An additional difference lies in the nature of the gamut boundaries. In a chromaticity diagram the boundary is defined by monochromatic spectra and (for trichromats) the purple line. As a monochromatic reflection contains negligible light highly saturated (or pure) colours are dark. By comparison object colour solid boundaries include both black (zero reflection) at the origin, and white corresponding to maximal reflection at all visible wavelengths, with intermediate the boundary surface being well approximated by spectral step-functions (Fig 2). Such spectra can be bright, and are more nearly physically realizable by natural pigmentation.
Following the logic that colour refers primarily to the physical properties of reflective materials (or objects) we propose a measure of colour within the object colour solid which we call vividness. Vividness resembles colorimetric parameters such as purity or saturation, but achromatic colours – black and white – and light unsaturated colours can be highly vivid. We show mathematically and empirically that the vividness of reflectance spectra is well correlated between different types of colour vision. Consideration of the relationship between vividness and the physical processes that generate object-colour demonstrates that orderly nanostructures or pure pigments are typically more vivid than their less orderly or pure materials. As order emerges against entropic tendencies, vivid or ‘bright’[Hamilton and Zuk, 1982] coloration is indicative of a functional role, and there-fore more “meaningful”, and hence such colours worthy of greater attention.
2 Modelling
We start with an account of why the achromatic colours black and white are maximally vivid, leading to a general model which includes chromatic colours.
2.1 Black and White
Models of colour vision and colour appearance usually treat chromaticity, which combines hue and saturation, as qualitatively distinct from brightness or luminance. This distinction is grounded in physiology and psychophysics [Livingstone and Hubel, 1988, Osorio and Vorobyev, 2005], yet black and white are ‘colours’ in ordinary English usage, and they are common in biological signals. In physiological terms an animal sees black when its spectral photoreceptors all have a low excitation, and white when they all have a high excitation. Hence, spectra that look black have low intensity at all visible wavelengths and spectra that look white have high intensity. As there is only one perfectly black spectrum, and a perfectly white surface must reflect maximally at all wavelengths it follows that spectra responsible for black and white are the same for all observers that share the same range of visible wavelengths, with minimal or maximal excitation across that range. These spectra can be designated as ‘extreme’, because they are at the limits of receptor excitation achievable by a surface.
Unlike black and white, intermediate reflectance spectra can produce different excitations according to the particular set of photoreceptors in a given eye. For the simplest case of two eyes each with a single type of photoreceptor, but tuned to different wavelengths, a stimulus of intermediate intensity can give different receptor responses in the two monochromatic observers. Consequently the appearance of greys for different types of monochromat is less predictable than are black and white. Similarly, for any given monochromatic eye only one spectrum can give black or white, but many different spectra can produce indistinguishable intermediate responses, a phenomenon which is known as colour metamerism[Logvinenko, 2009].
Black and white, produced by extreme spectra, are maximally vivid colours, while have greys have lower vividness.
The best known geometric representations of colour are chromaticity diagrams in which desaturated colours lie near the centre and the most saturated colours at the extremities, with the angle around the centre specifying hue.
One such diagram is Maxwell’s triangle [Maxwell, 1860]. Using a method based on colour mixing, Maxwell specified the colours of monochromatic lights as mixtures of red, green and blue primaries. When the brightness is ignored these results can be drawn in a triangle, with pure primaries at each corner, and mixtures within. The line of all the monochromatic lights, the “monochromatic locus” or “spectral line” forms a rounded Λ shape along two of the edges, the ends joined by a “purple line”. The fractional distance from the centre to the edge is then a measure of spectral purity, which is related to the colour’s perceived saturation [Wyszecki and Stiles, 2000].
For colour vision of non-human species similar diagrams are typically based on photoreceptor spectral sensitivities[Renoult et al., 2017]. For an eye with n spectral types of photoreceptor (contributing independently to colour vision) the chromaticity diagram is n − 1 dimensional. Chromatic spaces are good for describing the colours of lights, but less suitable for reflectance spectra, because a reflectance can only approach the boundary by reducing the amount of reflected light, making the colour darker – maximal spectral purity is black! Hence chromaticity diagrams typically exaggerate differences between dark colours.
Object colour solids are so-named because they appropriate for representing ‘object’ or reflectance spectra. They are useful in colour reproduction and the formulation of dyes and pigments, because the available gamut can be compared to the colour range visible to the human eye. The axes of colour solids [Wyszecki and Stiles, 2000, Schrödinger, 1920, Vorobyev, 2003, Koenderink, 2010] correspond to photoreceptor excitations (or a similar set of primaries) normalised to the illumination intensity. As photoreceptor spectral sensitivities overlap, colour solids do not fill the space defined by the axes, but are roughly ellipsoidal with two pointed corners (Fig 1). Monochromatic spectra lie an infinitesimal distance from the origin which is black, and maximal reflectance (white) is at the opposite vertex. Humans have three types of cone photoreceptor, and hence a 3-dimensional object colour space, but the same geometrical principles apply to any type of colour vision: for most mammals, which are dichromats, the space is 2-dimensional, whereas the spaces of birds are probably 4-dimensional [Kelber et al., 2003, Vorobyev, 2003]. As there are both empirical and mathematical relationships between these spaces we refer to all of them as colour solids.
2.2 Colour Solids and Vividness
We now more formally define the colour solid and vividness. For a given set of photoreceptor spectral sensitivities (si(λ), i ∈ 1 … n) and an illuminant (l(λ)), reflectance spectra can be organised into an geometric object known as the object colour solid [Schrödinger, 1920, Wyszecki and Stiles, 2000], shown in figure 1. This object is formed from the colours (a vector of photoreceptor quantum yields, (q1 … qn)) associated with all theoretically possible reflectance spectra: that is all distributions of reflectance values between zero and one, taken over visible wavelengths (Λ): To obtain the colour object solid, photoreceptor responses (qi) are normalised to the quantum yield of a perfectly reflecting surface1 . This means that black has x coordinates of (0, 0, …) and white has coordinates (1, 1, …). Throughout we express the x coordinates in terms of a relative quantum yield function : The object colour solid is convex, and lies within a unit n-cube (Fig 1). It is pointed at the diagonally opposite black and white corners, but the other corners and edges are smoothed due to the spectral overlap of the photoreceptors. Except for the boundaries, every point in the solid maps to more than one reflectance spectrum, corresponding to colour metamerism[Logvinenko, 2009]. The boundaries represent reflectances with the highest spectral purity for a given luminance. Unlike chromaticity diagrams, the colour solid therefore accounts for the trade-off of saturation against luminance, so light colours can lie on the boundary. This accords with the intuition that we do not see light colours as necessarily less pure than dark.
The exact calculation of the colour solid is rather involved. Numerical solutions can be obtained by dynamic programming [Wyszecki and Stiles, 2000] or our own method (see SI 2 for details of both), but for the current purpose it is better to start with Schrödinger’s approximation [Schrödinger, 1920, Vorobyev, 2003], which uses spectra formed by step changes in intensity between zero and one. Figure 2 illustrates Schrödinger’s spectra. The number of step changes depends on the number of spectral classes of photoreceptor: for an eye with n spectral classes the maximum number of step changes needed to approximate the colour solid boundary is n − 1. For dichromats, they are single steps, the two series (step up and step down) ranging from black through red, orange, then yellow to white, and from black through blue then cyan to white. For trichromats there are two steps in the visual range, while for a tetrachromat the Schrödinger spectra include those that have a reflectance of 0 up to some wavelength λ1 then 1 until another, λ2, then 0 until λ3 then 1, and in addition their inversions, i.e. those that have values of 1 then 0 then 1 then 0. Whilst the Schrödinger spectra are only approximations to the extreme spectra that lie on the boundary of the solid, the extreme spectra, like Schrödinger spectra, will only ever have either maximal (1) and minimal (0) intensity at every wavelength2
Schrödinger’s approximation of boundaries of the object colour solid holds best when photore-ceptor spectral sensitivities have a single peak. Some receptor spectral sensitivities are bimodal (e.g. Fig 3), in which case Schrödinger’s spectra typically lie slightly inside the boundaries of the corresponding colour solids. Such deviations are in practice small, because photoreceptor spectral sensitivities are typically smooth with a dominant peak. As long as this approximation holds a conclusion analogous to that for black and white applies; namely that for eyes with the same number of photoreceptor classes, the Schrödinger spectra corresponding to boundary colours are the same. Differences between receptor sensitivities mean that the shape of the object colour solid varies between species, and the locations of individual spectra within the boundary are not directly comparable (Fig 3), but they will be on the boundary nonetheless. It follows that Schrödinger spectra for one eye are Schrödinger spectra for an eye with a larger number of photoreceptor classes: the set of Schrödinger spectra for an n-chromat contains the Schrödinger spectra for an m-chromat, when m ≤ n.
The tendency for the same spectra to lie at the boundary of the colour solid extends to colours not on the boundary, although, as was the case for black, white and grey, the variablility in the position is greater for more central colours (Fig 3). Whilst the theoretical limitations on the position, illustrated in Fig 3, are broad, the spectra that achieve these limits are difficult to realise practically. This is evident in Fig. 4.
In practice, spectra of natural objects are a small subset of physically possible spectra [Maloney, 1986, Osorio and Bossomaier, 1992, Vorobyev et al., 1997], and natural spectra tend to be smoother than Schrödinger spectra [Maloney, 1986] (compare Fig 1 and Fig 2). Also, spectra with multiple transitions within the visible range are unusual, so that higher order Schrödinger spectra are seldom approached in nature; exceptions include some structural colours, such as that found on the nape of the feral pigeon, which can appear greyish to trichromats, but is likely to be vivid for birds [Osorio and Ham, 2002].
2.3 Definition of Vividness
We define vividness as a number ranging from zero at the centre of the colour solid to one at the boundary, which is linear with respect to the colour solid coordinates (xi). For an observer/illumination combination that is described by n relative quantum yield functions (f1 … fn) the vividness of a reflectance r is: where b(x) is the position of the boundary in direction of the vector x − ½ from the centre (½).
The numerator is the Euclidean distance from the centre of the solid. The division by ‖b(x) − ½‖ maintains an invariance between species. It has the effect of adjusting the numerator to reflect more physical, rather than perceptual, properties. The latter being more difficult to assess for non-human species (and even for humans).
2.4 Properties of Vividness
The properties of colour solids we have discussed so far can be formally expressed with two rules, the first is motivated by our argument above and it is true insofar as it is approximate, and contingent on having “well behaved” relative quantum yield functions, the second is a geometric fact:
Rule 1: Approximate Equality
For two observers with equal numbers of spectral receptor classes (n), the vividness of a stimulus is approximately the same for both observers. This rule is motivated by the foregoing discussions, and is corroborated by figures 3 and 4. For more vivid colours the range of discrepancy decreases, and the approximation is better – a consequence of there being “fewer” metamers. This phenomenon can be seen in figure 3.
As this rule concerns the relative quantum yield functions f, which are obtained from both spectral sensitivities and the illuminant, rule 1 is both a statement about changes in photoreceptor sensitivities and in illumination.
Rule 2: Monotonicity with Dimensionality
For a given illumination, an increase in the number of receptor types will result in an increase in vividness, so that colours are more vivid for species that have a large number of photoreceptor classes (e.g. birds) than for those with fewer (e.g. mammals). This effect can be observed in figure 4, and a proof is given in SI 1.
These rules3 allow one to describe the relationship between any two observers.
As vividness increases the constraints on the variety of spectra that can realize the colour become increasingly restrictive, until, at the boundary of the solid there is a unique spectrum. As there are “more”4 spectra that correspond to each of them, less vivid colours are more prone to metamerism. This rule can be compared to the metamer mismatching transformation that occurs under variable illumination [Tokunaga and Logvinenko, 2010].
Colour purity [Wyszecki and Stiles, 2000] resembles vividness for the chromatic plane, and has fairly similar properties. This is because the chromaticity space is a scaled cross section of the colour solid at fixed luminance as it goes to zero (i.e. a slice of the solid very near black). Vividness also resembles other measures, such as chroma [Endler, 1990]. In addition to the difficulties we have already highlighted with chromaticity, purity is not defined for monochromats, is not mathematically well behaved for dichromats, and can be quite complex for tetrachromats and above (see supplementary material SI 3).
2.5 Mixing
Vividness has a fundamental relationship to physical order, as exemplified by the case of conservative mixing. That is to say, where two colours, xa and xb, are mixed in a ratio of za : zb and the resulting colour is: where k = za/(za + zb) and the various values of k correspond to various positions on the line segment between xa and xb. This holds regardless of the spectra that produce xa and xb.
The distance of points on this line segment are closer to (or the same distance from) the centre of the solid than the more distant of xa and xb. If the solid were a perfect sphere (i.e. if the distance to the boundary were fixed) we could conclude directly that vividness of the mixture was smaller than that of the colours being mixed. As the solid is not a sphere, we must make a further observation: where the surface is (strictly) convex, a line segment connecting two points on the surface of the solid passes entirely within its volume (for a proof see supplementary material SI 1). The colour solid’s boundary colours have the distinguishing feature that they cannot be made by a mixing other object colours.
The rule applies to all observers, in spite of the fact that numerical values of V for a particular object colour may well be different.
Rule 3: Convexity of Mixing
for all k ∈ [0, 1]. Mixing two colours results in a colour that is less vivid than the most vivid of the two (and perhaps less than both). Rule 3 can be written in a more general form, for a mixture of multiple colours, as: This expression describes a phenomenon familiar to anyone who has mixed paints, once a duller colour is mixed into a more vivid one there is no way to recover the original vividness except by adding an even more vivid paint.
This is not a property unique to vividness, but we must bear in mind the significance of this rule for our argument: Without this rule we would only have mathematical results about the geometry of colour solids, but with this rule, we can talk about the physical properties of vividly coloured materials, and thereby talk about their ecological function and how they are perceived (in a Gibsonian sense [?], at least).
3 Discussion
Object colour solids are useful representations of the colours of reflective surfaces, as opposed to lights. The mathematical properties of these spaces along with empirical evidence (Fig 4) leads to three main findings relevant to the evolution of colour in communication signals.
Firstly, we define a measure of colour, vividness, which corresponds to the distance of a colour from the (grey) centre of the soild, and show mathematically that a spectrum which is highly vivid (i.e. near the boundary of the colour solid) for one type of colour vision will be highly vivid for any type that has the same number or more spectral types of photoreceptor. We find empirically that the vividness of natural reflectance spectra is correlated for eyes with different sets of photoreceptors (Figs 4 and 3).
Secondly, colours on the boundary of the solid are attributable to a single reflectance spectrum, and the number of spectra that map to a given point in the colour solid increases towards the centre, so that many spectra can look ‘mid-grey’. Highly vivid colours are therefore more likely to be associated with a specific physical cause (i.e. material), and they will be less easily reproduced by alternative means than less vivid colours. Thirdly, mixing colours inevitably reduces the vividness of the more vivid colour, so that pure materials tend to be more vivid than mixtures. This is why vacuum-cleaner dust is greyer than home furnishings. For structural colours increasing regularity of nano-structure increases vividness, and mixing of pigments can only render them less vivid.
Due to entropy order does not arise by chance in nature, so vivid colours are indicative of some functional role. This need not be as a signal; the vivid colours of leaves and blood are due to high concentrations of light harvesting and oxygen transport pigments. This is not to say that and a dull coloured tissue cannot have a specific function, the implication only works one way – vividness requires some kind of order, but order does not necessarily result in vivid colour. Nonethe-less, if an object has a functional role it is a priori worthy of attention, which is a requirement for any signal.
Historically, vivid coloration has been associated with the phenonemon of life, and this in turn has been related to thermodynamics. In Tropical Nature [Wallace, 1878] Wallace argues that colourfulness is a consequence of “vital energy”, and hence a natural attribute of living organisms. While Schrödinger proposed in his book What is Life? [Schrödinger, 1944], that life is characterised by order away from chemical equilibrium, that is by being non-entropic. Here we have seen why vivid colours are indeed likely to be associated with a system – such as life – that counters the effects of entropy.
3.0.1 Aposematism and Mimicry
The strong contrasts and vibrant colours of aposematic displays illustrate some of our findings. Vivid colours are salient, so they have the potential to promote both innate and learnt responses, and they and will be seen consistently by a broad range viewers, so they need not be predator-specific. For instance in tropical forests predators of insects include monochromatic strepsirrhine primates, dichromatic mammals and snakes, trichromatic primates and amphibians, and tetra-chromatic birds and lizards [Kelber et al., 2003]. A consequence of rule 2 is that the achromatic extreme colours – black and white – will be effective for all receivers, those approximating the dichromatic Schrödinger spectra (which we would see as two series black, red, yellow, white and black, blue, cyan, white) would be effective for dichromats and above, the trichromatic series which adds purples and greens for trichromats and above, and so on. Similar principles apply in marine environments where predator colour vision ranges from monochromacy in cephalopods through di, tri and tetrachromacy in various fish to the multispectral system of stomatopods[Marshall et al., 2015].
A further benefit of vivid defensive signals arises because as vividness increases copying a colour entails more exact matching of the spectrum. A defended aposematic model may in principle outma-noeuvre a Batesian mimic by adopting more vivid colours outside the mimic’s physiological ‘gamut’ [Franks et al., 2009, Briscoe et al., 2010, Bybee et al., 2011] and conversely there might be a pressure for Müllerian mimics to share less vivid colours.
3.1 Modelling of Colour in Biological Signals
Investigation of colour signaling by animals and plants often starts by modelling photoreceptor responses to reflectance spectra. Receptor responses do not directly specify colour differences or colour appearance, which normally requires recourse to psychophysical models [Kelber et al., 2003, Kemp et al., 2015]. Models based on chromaticity assume that lightness (or luminance) is discounted, and although some add achromatic contrast as a separate parameter[Siddiqi et al., 2004, Olsson et al., 2017], all such models can lead to difficulties. For example they predict that dark colours are unrealistically distinct, and there may be an implicit assumption that the strong achromatic contrasts, which are present in many signals are either irrelevant or have a qualitatively different function from chromatic components.
Object colour solids represent the full gamut of colours visible to an eye, and importantly offer a natural means of representing colour as a property of reflective materials rather than spectral lights. It is straightforward to define the locations of reflectance spectra in the object colour solid, and hence to explore broader questions about the gamuts of colour signals directs at various types of receivers (cf [Osorio and Vorobyev, 2008]. Vividness is a simple and well defined measure of colour, within a colour solid which can be related to the physical properties coloured materials, and gives insight into how and why animals with diverse visual systems might evaluate colour. Of course it remains an empirical question whether vividness is a useful measure of colourfulness. One could test whether vividness should predict attention or salience better than colour saturation or purity, or to a scale based on colour distances measured in terms of just noticeable differences within the colour solid.
4 Conclusion
In The Origin of Species Darwin writes. “…the belief that organic beings have been created beautiful for the delight of man, … has been pro-nounced as subversive of my whole theory, …” [Darwin, 1859]. Contemporary literature rarely addresses this question directly, but offers a range of accounts of why certain colours or patterns should be attractive to animals [Zahavi, 1975, Grafen, 1990, Guilford and Dawkins, 1991, Anderson, 1994, Johnstone, 1995, Endler and Basolo, 1998]. Some refer to the nature of the sensation, and postulate general aesthetic principles, whereas others highlight the specific value of a stimulus. For example it is debated whether animals use particular colours in courtship displays because they resemble objects of value such as food items [Allen, 1879, Endler and Basolo, 1998] or because the colours are indicative of the quality of a potential mate[Hill, 1991]. Here we find that these distinctions may be blurred, because consideration of the physical causes of the coloration suggests that certain colours should a priori be significant regardless of specific associations with objects of relevance to the animal, or indeed its particular type of colour vision. There is no need to invoke evaluative judgements [Prum, 2012] or co-evolutionary processes [Fisher, 1930] such as those associated with sexual selection. In addition the laws of thermodynamics mean that there is an immediate cost to producing pure materials and hence vividly coloured tissues, but this is not part of the main thrust of our argument, which is instead about a general psychological bias towards vivid colours.
Models of visual salience to humans typically include components that are akin to vividness [Niebur and Koch, 1996], and this kind of low level prediction can be compared and contrasted by higher level (evaluative) theories based on asking subjects about the aesthetic value of a colour. Palmer and Schloss’s [Palmer and Schloss, 2010] valence theory proposes that humans prefer colours associated with desirable objects, to those associated with decay, excrement and so forth. Given that decomposition and biological waste tend to be chemical mixtures, with low vividness, whereas pure materials tend to be more vivid it would be interesting to test whether vividness predicts colour preference as well as the valence. At a more practical level, we can offer some assurance to field biologists that it is reasonable to generalise from our own colour perception to that of other animals, despite their physiological differences [Bennett et al., 1994].
Data Availability
The code used to generate figures here have been made available, along with documentation and usage examples, as an open source project currently hosted at https://github.com/lucas-wilkins/lemonsauce. The numerical methods are also detailed in the supplementary information.
The Munsell chip data are available from the Spectral Color Research Group, University of Eastern Finland at http://www.uef.fi/web/spectral/-spectral-database [Parkkinen et al., 1989].
Acknowledgments
We would like to thank Mary Stoddart, Almut Kelber and Justin Marshall for their comments, and Justin Marshall also for the spectra used in figure 1.
Footnotes
↵1 The maximal reflectance could instead be set according to a standard such as BaSO4 pellet – this does not affect the arguments we make here (see appendix SI 1.3) and should, in fact, result in a more practical measure of vividness.
↵2 When discretised approximations like those in the appendix are used to calculate boundary spectra, a step transition between two discretisation points will appear at an intermediate intensity. This is a discretisation artifact.
↵3 For a fully formal approach we require another rule stating that V is not dependent on the order in which the relative quantum yield functions are specified.
↵4 Usually there is only one spectrum corresponding to a point on the boundary of the solid, but there might be more in cases where the surface is perfectly flat. Inside the boundary, or at points on the boundary without non-zero bounded curvature, there is an infinite number. Measuring the absolute sizes of the sets of spectra corresponding to a single colour is non-trivial because there is no ‘infinite-dimensional’ analogue to the Lebesgue measure. In this work we avoid this problem by always comparing between two different finite dimensional colour solids (⊆ [0, 1]n), rather than looking for a way of measuring spectrum-space ([0, 1]∞).
↵5 The equality constraints (Dx = e) are usually explicit in numerical procedures, but in mathematical treatments they are usually understood as a pair of inequality constraints: Dx ≤ e and −Dx ≤ −e.