Color naming reflects optimal partitions of

color space
Terry Regiera,b, Paul Kayb,c,d, and Naveen Khetarpala
aDepartment of Psychology, University of Chicago, 5848 South University Avenue, Chicago, IL 60637; cDepartment of Linguistics, University of California,

Berkeley, CA 94720; and dInternational Computer Science Institute, 1947 Center Street, Berkeley, CA 94704

Contributed by Paul Kay, November 21, 2006 (sent for review October 15, 2006)

The nature of color categories in the world’s languages is contested. these are the ‘‘bumps’’ on the surface. Jameson and D’Andrade (8)
One major view holds that color categories are organized around suggest that, given this irregularly shaped space, general principles
universal focal colors, whereas an opposing view holds instead that of categorization may account for universals of color naming (see
categories are defined at their boundaries by linguistic convention. also refs. 9–11). Although the general principle proposed was
Both of these standardly opposed views are challenged by existing originally characterized as ‘‘informativeness,’’ Jameson (12, 13) has
data. Here, we argue for a third view based on a proposal by Jameson suggested that the operative principle may be that of Garner (14):
and D’Andrade [Jameson KA, D’Andrade RG (1997) in Color Catego- that categories are constructed so as to maximize similarity within
ries in Thought and Language, eds Hardin CL, Maffi L (Cambridge Univ categories and minimize it across categories. Under this principle,
Press, Cambridge, U.K.), pp 295–319]: that color naming across lan- certain categorical partitions of the lumpy perceptual color space
guages reflects optimal or near-optimal divisions of an irregularly will be optimal in the sense that they optimize this measure, and
shaped perceptual color space. We formalize this idea, test it against others will be less so. The hypothesis is that optimal or near-optimal
color-naming data from a broad range of languages and show that it partitions correspond to observed universals in color naming. This
accounts for universal tendencies in color naming while also accom- proposal also may accommodate the finding that similar languages
modating some observed cross-language variation. sometimes have different boundary placements: such languages
may have distinct color-naming systems that differ minimally, if at
cognitive modeling 兩 color categories 兩 color terms 兩 semantic universals all, in optimality.
Jameson and D’Andrade’s (8) proposal is intuitively appeal-
ing. However, because the idea was originally advanced without
I t is often claimed that color categories in the world’s languages
are organized around six universal focal colors corresponding
to the best examples, or prototypes, of English black, white, red,
formalization and because its development has remained largely
informal (e.g., refs. 12 and 13), no formal test of this proposal
green, yellow, and blue or comparable terms in other languages against detailed empirical data has been attempted. Here, we
(1). On this view, the boundaries of color categories are pro- formalize this idea and test it against detailed color-naming data
jected from these universal foci and therefore tend to lie in from a wide range of languages. Our goal in doing so is to
similar positions in color space across languages. In contrast, the determine whether this proposal can avoid the challenges faced
opposing ‘‘relativist’’ view denies that foci are a universal basis by the two opposing views sketched above.
for color naming and instead maintains that color categories are Liljencrants and Lindblom (15) suggested that a similar idea may
defined at their boundaries by local linguistic convention, which explain universal tendencies in the structure of vowel systems in
is free to vary considerably across languages (2–5). vowel space. They proposed a formal model based on a measure of
Each of these two views is challenged by existing data. Universals overall perceptual contrast among vowel categories: The more
of color naming exist (6), and the best examples of color terms dispersed the categories in vowel space, the better the vowel system
across a wide range of languages cluster near the six proposed focal by their measure. (This dispersion corresponds to minimizing
colors (7). These findings are consistent with the focal-color similarity across categories; because they represented vowel cate-
gories as points, they did not also measure within-category simi-
account and inconsistent with the linguistic-convention account.
larity.) Liljencrants and Lindblom (15) explored the space of
However, despite this clustering near the six foci, the best examples
possible vowel systems and found that those with the greatest
of many color categories do fall elsewhere, a finding less easily
perceptual contrast corresponded fairly well to vowel systems found
accommodated by the focal-color account. Moreover, languages
in the world’s languages. If related ideas can account for universal
with the same number of categories, apparently organized around
tendencies in named color categories across languages, then that
the same or similar foci, sometimes differ in their placement of
would suggest a loose parallel between the forces that create
category boundaries (4), which suggests that category boundaries
categories of sound and those that create categories of meaning
are determined by more than the six proposed universal foci.
(16, 17).
A possible resolution of this tension is suggested by a proposal
advanced by Jameson and D’Andrade (8). Their proposal can be The World Color Survey
viewed as a natural generalization of the focal-color account, one
We took as our empirical base the color-naming data of the World
in which every color is focal (perceptually salient) to some extent,
Color Survey (WCS) (18, 19). These data were obtained by using
and some, such as the six listed above, are simply more focal than
a stimulus palette of 330 colors, as approximated in Fig. 1 Upper.
others:
The palette consists of 40 equally spaced Munsell hues, which are
One possible explanation [for universals in color naming] represented by the columns; achromatics, which are represented in
is . . . the irregular shape of the color space. . . . Hue
interacts with saturation and lightness to produce several
Author contributions: T.R. and P.K. designed research; T.R. and N.K. performed research;
large ‘‘bumps’’; one large bump is at focal yellow, and T.R. and N.K. analyzed data; and T.R. and P.K. wrote the paper.
another at focal red. . . . We assume that the names that get
The authors declare no conflict of interest.
assigned to the color space . . . are likely to be those names
Abbreviation: WCS, World Color Survey.
which are most informative about color (ref. 8, p. 312).
bTo whom correspondence may be addressed. E-mail: regier@uchicago.edu or
Thus, each point on the outer skin of color space is salient to some paulkay@berkeley.edu.
extent, but some colors on that skin are more salient than others; © 2007 by The National Academy of Sciences of the USA

1436 –1441 兩 PNAS 兩 January 23, 2007 兩 vol. 104 兩 no. 4 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0610341104
Fig. 1. Stimuli and example of a mode map. (Upper) WCS stimulus palette. (Lower) Mode map for Lele, a language with four major categories, and several
chips for which the modal response was some other category; each category is denoted by a color.

the left-most column; and eight levels of value (lightness; 10 for the plotted in CIELAB, they form a rather distorted sphere, with the
achromatics), represented by the rows. Each hue–value pair is at white point (A0 in Fig. 1) at the north pole, the black point (J0 in
maximum available chroma (saturation). Speakers of each of 110 Fig. 1) at the south pole, the intermediate grays running along the
languages of nonindustrialized societies named each of the color L* axis between these two poles, and all colored chips forming a
chips in this array.e There are clear universals of color naming in the bumpy, vaguely spherical surface around that axis, as shown in Fig.
WCS data (6, 7, 20–23) but also substantial cross-language varia- 2. These points are an approximation to the outer surface of the
tion. We hope to explain why color categories across languages have color solid, the space of realizable colors. There is a large protrusion

PSYCHOLOGY
the shapes and locations in color space they have. outward from the sphere’s surface around yellow and other smaller
In the present study, for each WCS language, we recorded the irregularities elsewhere. The hypothesis is that these irregularities in
modal color term for each chip in the array, i.e., the color term that the space, interacting with general principles of categorization,
was assigned to that chip by the largest number of speakers of that cause natural clusters to form that correspond to observed color-
language. We refer to the resulting labeling of the entire array as the naming universals.
‘‘mode map’’ for that language. For example, Fig. 1 Lower shows the
mode map for Lele, a language spoken in Chad; here, as in mode Partitions of Color Space. Imagine that each chip in the stimulus
maps throughout this paper, each color denotes a color term, or array of Fig. 1 has been labeled with some category; we wish
named color category.f This process produced a mode map for each to characterize how good a categorical partition of color space
of the 110 WCS languages. this arrangement represents. To that end, we defined an
objective function that measures the extent to which such an
Formal Specification assignment of category labels to chips maximizes similarity
Color Space. Because our tests depend on perceptual similarities within categories and minimizes similarity across categories.
between colors, which could be conveniently expressed by a dis- We refer to this quantity as ‘‘well-formedness’’: optimal par-
tance metric, and because the Munsell system does not have a titions of color space are those that maximize this well-
psychologically meaningful distance metric, we started by repre- formedness measure. We take the similarity of two colors x and
senting each of the colors in the stimulus palette in the CIEL*a*b* y to be a monotonically decreasing (specifically Gaussian)
(or CIELAB) color space, which does have such a metric. For function of the distance between the two colors in CIELAB
relatively short distances at least, the distance between two colors space:
in CIELAB space corresponds roughly to their psychological
dissimilarity (24).g When the colors in the stimulus array above are sim(x, y) ⫽ exp(⫺c ⫻ [dist(x, y)]2),

where dist(x, y) is the CIELAB distance between colors x and

eThe WCS data, including genetic affiliation and other particulars about the languages, are
y, and c is a scaling factor (set to 0.001 for all simulations
available at http://www.icsi.berkeley.edu/wcs/data.html.
reported here). This similarity function, which we adopt from
fIn this mode map, as in some others reported here, there are a few isolated chips for which the
the psychological literature on categorization (e.g., ref. 25),
modal color term was one that was not widely used; these chips are therefore colored
differently from most others in the array (e.g. here, the light-blue and brown chips in columns
has a maximum value of 1 when chips x and y are the same [i.e.,
4 –10). dist(x, y) ⫽ 0] and a value that falls off approaching 0 as the
gCIEL*a*b*
distance between chips x and y becomes arbitrarily large. This
is a 3D space. The L* dimension corresponds to lightness, whereas the a* and
b* dimensions define a plane orthogonal to L* such that the angle of a vector in that plane, similarity function thus captures the qualitative observation
rooted at the L* axis, corresponds to hue and the radius of such a vector corresponds to that beyond a certain distance colors appear ‘‘completely
saturation. Despite this reference to polar coordinates in linking positions in the space to different,’’ so that increasing the distance has no further effect
psychological quantities, the CIELAB distance metric is standard Euclidean distance. We
converted our Munsell coordinates to CIELAB by using Wallkill Color Munsell conversion on dissimilarity. The well-formedness function W is then
software, version 6.5.17, which assumes illuminant C, 2 degree standard observer. defined as follows:

Regier et al. PNAS 兩 January 23, 2007 兩 vol. 104 兩 no. 4 兩 1437
 different random initializations for each n; the resulting color-
naming scheme with the highest well-formedness value among
these 20 was taken to be optimal for that value of n.
Fig. 3 shows for n ⫽ 3, 4, 5, or 6, respectively, the optimal model
result obtained in the manner just described and data from lan-
guages in the WCS that are similar to the predicted optimal
pattern.h
Well-formedness optimization tends to place categories in
roughly the right places for these languages. For example, the model
correctly predicts that when a system has a separate yellow category
that category will tend to be lighter (nearer to white) than are
categories of other hues. It also captures the rather detailed fact that
in three-term systems, the composite red/yellow term does not
extend as far toward white as the separate yellow term does in the
four-, five-, and six-term systems. Thus, as predicted, there do exist
some languages with color-naming schemes fairly similar to the
theoretically optimal configurations produced by the model.
Fig. 2. The chips of the WCS stimulus array as plotted in CIELAB space. The
irregularity of the distribution can be seen, particularly in the outward pro-
However, the model does deviate from the observed pat-
trusion of the yellow region. terns, in some cases systematically. This deviation is especially
pronounced in the blue region. Starting with three-term

冘
languages and continuing through six-term languages, the
Sw ⫽ sim(x, y) category that includes green in the theoretically optimal
(x,y): configurations usually does not extend far enough ‘‘rightward’’
cat(x) ⫽ cat(y)

冘
into blue/purple (see, e.g., Fig. 3, hue columns 30 –32). Thus,
Da ⫽ (1 ⫺ sim(x, y)) with regard to blue, this model makes the wrong prediction.
(x,y): Moreover, there are many languages in the WCS with color-
cat(x) ⫽ cat(y)
naming systems that are not very similar to the hypothetically
W ⫽ S w ⫹ D a. optimal model configurations. To give a sense of this disimilarity,
Fig. 4 displays the mode maps for four WCS languages with
Here, Sw is an overall measure of similarity within categories. extensions that diverge, sometimes sharply, from the predicted
Similarity is summed across unique pairs of chips (x, y) that are optimal configurations.
labeled with the same category [cat(x) ⫽ cat(y)]. Da is an In summary, aside from the problem with the blue region,
analogous overall measure of dissimilarity across categories. there exist languages for which well-formedness optimization
Here, dissimilarity [1 ⫺ sim(x, y)] is summed across unique pairs does a fairly good job of placing categories roughly where they
of chips (x, y) that are labeled with different categories [cat(x) ⫽ actually fall, as predicted. On the other hand, there are a number
cat(y)]. The well-formedness W of a particular assignment of of languages that do not much resemble these theoretically
category labels to chips is the sum of Sw and Da. The higher this optimal patterns. Nonetheless, we suggest that across all lan-
quantity, the more well-formed the configuration. guages, color-naming systems will be shaped to a detectable
If color naming across languages is shaped in part by the extent by well-formedness in a universal perceptual color space.
constraints embodied in this function, we would expect the We test this prediction below.
color-naming schemes of the world’s languages to correspond to
relatively high well-formedness values. Concretely, we predicted Well-Formedness of Attested and Unattested
the following: Color-Naming Schemes
If color naming across languages is shaped in part by the universal
1. Artificially generated color-naming schemes that lie at global structure of perceptual color space, we would expect to find traces
well-formedness maxima should resemble the natural color- of that structure in the color-naming system of any language, not
naming schemes found in some of the world’s languages. just those that are similar to the optimal patterns. To probe this
2. Given the pattern of color naming in any language, systematic idea, we started with Berinmo, a Papua New Guinea language that
distortions away from that pattern should tend to result in has been claimed to counterexemplify universal tendencies of color
lower well-formedness values than in the observed pattern. naming (ref. 2, but see refs. 20 and 21) and that therefore could be
These two predictions differ in strength: The first predicts that considered a conservative test of our proposal.
at least some languages fit a certain pattern, whereas the second We considered the Berinmo color-naming data and 19 hypo-
predicts that all languages fit another pattern. We tested both thetical variants of it, which were obtained by rotating the actual
predictions against the color-naming data of the WCS. data by 2, 4, 6, etc. (and ⫺2, ⫺4, ⫺6, etc.), hue columns in the
stimulus array (around the ‘‘equator’’ of the color solid) as illus-
Optimal Color-Naming Schemes trated in Fig. 5.i
We used simulations to construct theoretically optimal color- This procedure yielded a set of systematic variants of the Berinmo
naming schemes by maximizing well-formedness (W). Specifically, color-naming system in which the same configuration of categories
we obtained the theoretically optimal color-naming scheme with n relative to each other is maintained while the absolute position of
categories for each of n ⫽ 3, 4, 5, and 6. In each case, we began by
randomly assigning each chip in the stimulus array to one of the n
hAlthough all simulations are based on distances in 3D CIELAB space, we display the results
categories. We then adjusted category extensions through steepest
as overlays of the actual 2D stimulus palette, which is based on the Munsell system. We
ascent in well-formedness. Each of the 330 chips in the array was display our results this way because the palette is widely used as a reference frame in the
selected in random order and assigned the category that produced literature on color naming and cognition.
the greatest overall increase in well-formedness (cf. ref. 26). This iTherotation is in Munsell coordinates, although our well-formedness calculations are in
process was repeated until no further increase was possible. This CIELAB. We chose to rotate in this manner because it is simple to convey the idea with these
optimization process as a whole was conducted 20 times with displays and because doing so does not affect the logic of our argument.

1438 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0610341104 Regier et al.

 PSYCHOLOGY

Fig. 3. Model results for n ⫽ 3, 4, 5, and 6, each compared with color-naming schemes of selected languages from the WCS.

this configuration of categories along the hue axis is varied. color-naming system of Berinmo to have higher well-formedness
Critically, only one of these variants (the unrotated variant) we than any of the comparable rotated variants. Why? Because, by
know to be actually attested. If color naming is shaped in part by the hypothesis, the boundaries of the naturally occurring Berinmo
universal structure of perceptual color space according to the system lie where they do in large part because of the structure of
well-formedness model, we would expect the attested (unrotated) perceptual color space, whereas this is not true of the artificially

Regier et al. PNAS 兩 January 23, 2007 兩 vol. 104 兩 no. 4 兩 1439
 Fig. 4. WCS color-naming systems that are dissimilar from the predicted optimal configurations.

derived variants, in which the boundaries were deliberately shifted ing easily visualizeable cross-language comparison. For this
away from their natural positions. Moreover, we would expect reason, we transformed all well-formedness values to the range
well-formedness to drop off as a function of the amount of rotation (0–1) to make them comparable across languages: for each
away from the naturally occurring Berinmo color-naming scheme. language L, the minimum well-formedness value that any rota-
These expectations were confirmed, as shown in Fig. 6, although tion of L received was mapped to 0; the maximum value that any
⫹2 columns rotation was a close competitor for maximum well- rotation of L received was mapped to 1; and the values for all
formedness. The fact that well-formedness is maximized for the other rotations of L were linearly transformed to lie between
unrotated version of Berinmo indicates that the attested Berinmo these two extremes.] Fig. 7a shows the transformed well-
color-naming system is more consistent with the universal structure formedness value averaged across all WCS languages under each
of perceptual color space, coupled with general principles of rotation.
category formation, than are any of the hypothetical rotated In general, well-formedness is highest when languages are unro-
variants. Thus, it appears that the Berinmo color-naming system is tated, and greater rotation away from the naturally occurring
located where it is along the hue dimension not because ‘‘color system results in correspondingly less optimal values, as predicted.
categories are formed from boundary demarcation based predom- The fact that the mean transformed well-formedness value for
inantly on language’’ (2), but rather because the structure of unrotated languages is near one, with little variation, indicates that
perceptual color space makes its actual location the optimal loca- most languages have maximum well-formedness when unrotated.
tion. At the same time, the near maximum at ⫹2 columns rotation
To probe this issue directly, we also determined for each language
is also informative: It shows that small variations in boundary
in the WCS which rotation of that language yielded the highest
placements sometimes lead to only very modest differences in
well-formedness value. We then tallied how many languages had
well-formedness, which may explain why similar languages can
differ somewhat in their boundary placements (4). their well-formedness maximum at each rotation. The results are
shown in Fig. 7b. Most languages (82 of 110) have their well-
A More General Test formedness maximum at 0 columns rotation. There are some
To test this idea more generally, we conducted the same rotation- languages that have their maximum elsewhere, but most are fairly
based analysis on each of the 110 languages of the WCS. We near 0 columns rotation.
predicted that well-formedness would be higher for the actually These results show that the point that applied to Berinmo applies
observed data for a given language than for hypothetical versions more generally across languages: The color-naming systems of the
of that language, which were derived from the original by world’s languages tend to be positioned in hue just where the
rotation, as in the preceeding section. [The well-formedness structure of perceptual color space predicts they should be.
values of different languages often differ substantially, hamper-

Fig. 5. Berinmo color categories unrotated (Top) and rotated four (Middle) Fig. 6. Well-formedness for Berinmo when rotated 0, 2, 4, 6, etc., hue
and eight (Bottom) hue columns. Each colored region corresponds to a named columns. The configuration that yields greatest well-formedness is the unro-
color category. tated (attested) version.

1440 兩 www.pnas.org兾cgi兾doi兾10.1073兾pnas.0610341104 Regier et al.

 partitioning an irregularly shaped color solid. This view has in
common with the focal-color account that there are universal
perceptual constraints governing the position and shape, not just
the connectedness, of color categories across languages. But in
contrast with the focal-color account as usually articulated, there is
a potentially unlimited repertoire of foci. Every color is ‘‘focal’’
(perceptually salient) to some degree, although some more than
others, and categories are formed by general principles of catego-
rization operating over the resulting uneven landscape. By the same
token, there are both similarities to and differences from the
linguistic-convention account. By casting color naming in terms of
a well-formedness measure based on similarity within and across
categories, we support the claim, and help to explain the fact, that
color categories tend to occupy connected regions of color space.
Moreover, our findings leave open the possibility that linguistic
convention may play some role in determining color category
boundaries. We have seen that not all languages are ‘‘optimally’’
well-formed, and linguistic convention may be one force that can
pull a particular language away from a perceptually optimal parti-
tioning of color space. However, our results pose a direct challenge
to the proposal that language is free to carve up color space in any
conceivable manner as long as the resulting categories are con-
nected. For the joint effect of the irregularity of color space and
general principles of category formation appears to influence the
placement of color categories across the world’s languages. Thus,
our results provide an explanation for both the connectedness of
color categories and the particular positions in color space they tend
to occupy while also allowing for the possibility of a certain amount
of language-specific adjustment of these universal tendencies.
Fig. 7. Rotation analysis of WCS data. (a) Well-formedness averaged across all 110
There is another possible reason why many languages do not
WCS languages as a function of rotation. For each rotation, the dot shows the match the theoretically optimal configurations. All of the optimal
average transformed well-formedness value across languages and the bar shows the configurations were obtained by starting from a random assignment

PSYCHOLOGY
standard error. (b) Number of WCS languages exhibiting a well-formedness
maximum at each rotation.
of category labels to chips. In contrast, the color-naming system of
a given language has a history: It has evolved not from a random
state, but rather from an earlier category system, usually one with
Discussion exactly one fewer categories (1, 19). Thus, some languages may not
approximate maxima in well-formedness as much as they do points
The literature on color naming has recently been dominated by
on an evolutionary path leading from one maximum in well-
the opposition of two major views: that color categories are
formedness to another.
organized around universal foci (1, 7, 23) and that categories are
determined at their boundaries by linguistic convention (2–4). Note. We have recently become aware of an independent formalization of
For the latter view, the only major universal constraint on color Jameson and D’Andrade’s proposal. N.L. Komarova, K.A. Jameson, and L.
naming is that a category must occupy a connected region in Narens (personal communication) explore Jameson and D’Andrade’s pro-
color space; aside from that constraint, the location of the posal as a basis for the evolution of stable systems of color categories. Our
category and its boundaries in color space are a ‘‘free parameter’’ goal of testing the idea against existing empirical data led us to a different
(5), subject to presumably arbitrary cultural determination. formalization.
The model and data presented here do not align directly with We thank Kimberly Jameson, Roy D’Andrade, and Susanne Gahl for
either of these two traditionally opposed positions. Instead, they comments on earlier drafts and Tony Belpaeme for discussion of some
support the proposal of Jameson and D’Andrade (8): that color of the issues pursued here. This work was supported by National Science
naming is determined in part by general principles of categorization Foundation Grants 0418283 (to T.R.) and 0418404 (to P.K.).

1. Berlin B, Kay P (1969) Basic Color Terms: Their Universality and Evolution (University 14. Garner WR (1974) The Processing of Information and Structure (Erlbaum, Potomac,
of California Press, Berkeley, CA). MD).
2. Roberson D, Davies I, Davidoff J (2000) J Exp Psychol Gen 129:369–398. 15. Liljencrants J, Lindblom B (1972) Language 48:839–862.
3. Roberson D, Davies IRL, Davidoff J (2002) in Theories, Technologies, Instrumen- 16. Hespos S, Spelke E (2004) Nature 430:453–456.
talities of Color: Anthropological and Historical Perspectives, eds Saunders B, van 17. Regier T (2005) Cognit Sci 29:819–865.
Brakel J (Univ Press of Am, Lanham, MD). 18. Cook RS, Kay P, Regier T (2005) in Handbook of Categorization in Cognitive Science,
4. Roberson D, Davidoff J, Davies IR, Shapiro LR (2005) Cognit Psychol 50:378–411. eds Cohen H, Lefebvre C (Elsevier, Amsterdam), pp 223–242.
5. Roberson D (2005) Cross-Cult Res 39:56–71. 19. Kay P, Maffi L (1999) Am Anthropol 101:743–760.
6. Kay P, Regier T (2003) Proc Natl Acad Sci USA 100:9085–9089. 20. Kay P, Regier T (2007) Cognition 102:289–298.
7. Regier T, Kay P, Cook R (2005) Proc Natl Acad Sci USA 102:8386–8391. 21. Kay P (2005) Cross-Cult Res 39:39–55.
8. Jameson KA, D’Andrade RG (1997) in Color Categories in Thought and Language, 22. Lindsey DT, Brown AM (2006) Proc Natl Acad Sci USA 103:16608–16613.
eds Hardin CL, Maffi L (Cambridge Univ Press, Cambridge, UK), pp 295–319. 23. Webster MA, Kay P (in press) in The Anthropology of Color, eds MacLaury RE,
9. Yendrikhovskij S (2001) Color Res App 26:S235–S238. Paramei GV, Dedrick D (Benjamins, Amsterdam).
10. Griffin LD (2006) J R Soc Interface 3(6):71–85. 24. Brainard DB (2003) in The Science of Color, ed Shevell SK (Elsevier, New York), pp
11. Dowman M (2007) Cognit Sci 31:1–34. 191–216.
12. Jameson KA (2005) J Cognit Cult 5:293–347. 25. Nosofsky R (1986) J Exp Psychol Gen 115:39–57.
13. Jameson KA (2005) Cross-Cult Res 39:159–204. 26. Hopfield JJ (1982) Proc Natl Acad Sci USA 79:2554–2558.

Regier et al. PNAS 兩 January 23, 2007 兩 vol. 104 兩 no. 4 兩 1441