A comparative investigation of integral- and separable-dimension stimulus-sorting behavior

Abstract

Studies focusing on unsupervised categorization and sorting behavior have traditionally identified and empirically isolated a few select strategies frequently employed by observers, one-dimensional (SD), family resemblance (FR), and exclusive-or (XOR). To date, these investigations have mostly involved creating sets of multidimensional stimuli that directly contrast utilization of these strategies coupled with task or stimulus property manipulations to see their effects on categorization and sorting behavior. Currently, we extend on this methodological approach by having observers sort integral-dimension stimuli for two recently developed constrained unsupervised categorization tasks employing three-dimensional Boolean category structures. These structures instantiate six different sorting strategies, including the aforementioned SD, FR, and XOR strategies. Additionally, we connect the prevalence of the strategies observed across both tasks to previous investigations employing separable stimuli with the same tasks and underlying Boolean category structures. In comparison, our results indicate significant reductions in rule-like sorting behavior (SD or XOR) across multiple structures with the integral stimuli. Associated with this decrease in SD and XOR behavior was a corresponding increase in FR and FR-related sorting behavior. Finally, we assess how well generalized representational information theory and the simplicity model can account for the pattern of results across both experimental paradigms and stimulus sets. In general, a formal model derived from “GRIT” outperforms the simplicity model, accounting for nearly the entire range of unsupervised sorting behavior (SD, FR, XOR) observed across the separable and integral tasks.

Appendices

Technical Appendices

A. Diagnosticity analysis for SD and XOR sorting across the construction and deconstruction tasks

A post-hoc analysis of how each of the color dimensions either facilitated or hindered SD classification across the different category instances in the construction task (Boolean structure: 3₂[3] – I) revealed that SD sorting was more pervasive when saturation (chroma) was the diagnostic dimension (86%) than when either hue (47%) or brightness (32%) were diagnostic, least favorable ${X}^{2}\left(1, N=326\right)=282.54, \, p<.001,h=0.87$. This result is particularly surprising given that previous research has shown brightness to be the color dimension most susceptible to unsupervised SD behavior (Ell et al., 2012). This difference could be the result of adding hue as a dimension of variation in the current study, as the previous sorting results by Ell and colleagues only utilized colors that varied over saturation and brightness (hue was held constant). Despite these differences at the instance level, however, SD sorting was still the dominant choice when each of the three dimensions were diagnostic. Consistent with the construction results, a post-hoc analysis of the 24 instances presented in the deconstruction task (Boolean structure: 3₂[5] – I) reveals that SD sorting was more pervasive when saturation was the diagnostic dimension (73%) than when either hue (38%) or brightness (27%) were diagnostic, least favorable ${X}^{2}\left(1, N=212\right)=137.4, p<.001,h=0.73$. The dimensional relationship for the construction task is shown in the leftmost plot (x-axis) of Fig. 7, whereas the dimensional relationship for the deconstruction task is shown in the rightmost plot (x-axis) of Fig. 7.

Unfortunately, one cannot perform the exact same post-hoc analysis of 24 instances for XOR sorting in the construction (Boolean structure: 3₂[3] – II) or deconstruction (Boolean structure: 3₂[5] – II) tasks as each dimension is diagnostic in relation to another dimension being diagnostic. Despite this limitation, we did correlate the orders produced by the dimensions that lead to more exclusive-or behavior. More specifically, we first ordered the instances based on an increasing proportion of choices that were exclusive-or. We then assigned the order number to that dimension if it was diagnostic; for example, the dimensional combination that produced the highest proportion of XOR sorts in the construction task (~ 62%) was chroma and hue. Thus, each of these dimensions receives a ranking value of 1. The combination that resulted in the fewest proportion of XOR sorts (~ 6%) included the hue and value dimensions. Each of these dimensions receives an additional ranking value of 24. Upon recording the ranking totals for each dimension across the 24 instances (16 values per dimension), where smaller ranking sums indicate more XOR behavior, we compared them to similar ranking totals (8 values per dimension) computed for the diagnostic dimensions of type 3₂[3] – I. As with the pattern of SD sorting, the dimensional relationship for the construction task is shown in the leftmost plot (y-axis) of Fig. 7, whereas the dimensional relationship for the deconstruction task is shown in the rightmost plot (y-axis) of Fig. 7.

Fig. 7

Diagnostic Color Dimension Utilization. Ranking totals for how often each dimension was utilized in SD (x-axis) or XOR (y-axis) sorting strategies in the Construction (left pane) and Deconstruction (right pane) tasks. The axis limits are scaled to contain both the theoretically minimum and maximum values for this method of totaling rankings. As detailed above, smaller totals are associated with dimensions that were utilized more often

Finally, Table 5 below displays response times for each of the six structure types across the Construction and Deconstruction tasks for both the integral and separable stimuli. As shown in the table, the only notable differences in response times depending on stimulus type occurred when the dimensions were integral, as opposed to separable, for the deconstruction task.

Table 5 Average (SD) Response Times and Independent-samples T test Results for the Tasks Implementing Integral Stimuli (Current) and Separable Stimuli (Doan & Vigo, 2016)

B. Representational information theory (Adopted from Vigo, 2015 and Vigo et al., 2020)

We begin by introducing one of the core models of Generalized Representational Information Theory (GRIT; 2013a, 2015). The model is referred to as the GISTM and is the basis for Generalized Representational Information Theory. Note that Vigo (2011) introduced Representational Information Theory (RIT) which is a precursor to GRIT. The main difference between RIT and GRIT is that the cognitive mechanism underlying GRIT does not limit the theory to categories defined only over binary dimensions. In our discussion below, we only discuss category structures defined over binary dimensions because in the experiment of this article we investigate only such category structures; thus, we will revert to the use of RIT because the results are computationally equivalent to results obtained using GRIT and the computations are easier in RIT. For comprehensive details, we recommend Chapter 9 of the book entitled Mathematical Principles of Human Conceptual Behavior (Vigo, 2015).

In what follows, a well-defined category is a set containing objects that are defined in terms of a preset finite number of dimensions and their values. In Representational Information Theory (Vigo, 2011, 2013a, 2015), the amount and quality of subjective information ${\hslash }_{s}$ conveyed by a subset R = $\left\{ {a_{1} , \ldots ,a_{n} } \right\}$ of a well-defined category F is defined as the percentage change (expressed as a proportion) in the structural complexity of F whenever R is subtracted from F. This is expressed by the following equation if we let G = F – R (i.e., G is obtained by subtracting from the original category F the subset or “concept cue” R of F).⁹

$$\hbar_{s} (\text R|\text F) = \frac{\psi (\text G) - \psi (\text F)}{{\psi (\text F)}} = \frac{{|\text G| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text G)}} - |\text F| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text F)}} }}{{|\text F| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text F)}} }}$$

(B1)

In the equation above, $\text R\subseteq \text F$ or $\text R\in \wp (\text F)$, G = F–R (i.e., F = G $\cup$ R, $\left|\text G\right|$ and $\left|\text F\right|$ are the number of elements in $\text G$ and $\text F$ respectively, and $\Phi$ is the degree of categorical invariance of a set of well-defined objects (see Vigo (2013b, 2015) for a detailed discussion of structural complexity and categorical invariance). Equation B2 below is a more detailed and general version of Eq. B1 (Vigo, 2015, p.173).

$$\begin{aligned} \hbar_{s} (\left\{ {a_{1} , \ldots ,a_{n} } \right\}|\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\}) = & \frac{{\psi (\text G) - \psi (\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\})}}{{\psi (\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\})}} \\ = & \frac{{|\text G| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text G)}} - |\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\}| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\})}} }}{{|\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\}| \cdot e^{{ - \frac{{D_{0} }}{D}\Phi^{2} (\text G \cup \left\{ {a_{1} , \ldots ,a_{n} } \right\})}} }} \\ \end{aligned}$$

(B2)

$$\psi \left(\text F\right)=\frac{\left|\text F\right|{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text F)}}{\lambda }$$

(B3)

As may be seen in Eqs. B1, B2, and B3, the degree of complexity $\psi$ of a well-defined category of objects is defined as the product between the number of objects in the set $\left|\text F\right|$ and the negative exponent of its degree of categorical invariance $\Phi$ (Vigo, 2013b, 2015), where $\frac{{D}_{0}}{D}$ is a structure discriminability index for structures in which $D$ is the number of dimensions over which the category is defined and ${D}_{0}$ = 2 (the minimum non-trivial number of dimensions that define a category). The degree of categorical invariance $\Phi$ of $F$ is defined with the following generalized metric (Vigo, 2013b, 2015) where $s\ge 1$ is an integer, and specifically, s = 2 (Euclidean Metric), for categories defined over separable dimensions and s = 1 (Manhattan metric) for categories defined over integral dimensions.

$$\Phi \left(\text F\right)={\left[\sum_{i=1}^{D}{\left[0-\frac{\left|\text F\cap {T}_{i}(\text F)\right|}{\left| \text F\right|}\right]}^{s}\right]}^{1/s}={\left[\sum_{i=1}^{D}{\left[\frac{\left|\text F\cap {T}_{i}(\text F)\right|}{\left|\text F\right|}\right]}^{s}\right]}^{1/s}$$

(B4)

To explain the above equation, we shall set s = 2 (the separable dimensions case). Let’s encode the features of the objects in the category F using the digits "1" and "0" so that each of its objects may be represented by a vector of zeros and ones. Now, suppose that F consists of the following three objects: a small black triangle, a small black circle, and a large white circle. For example, the vector (1, 1, 1) stands for the first object when x = 1 = triangular, y = 1 = small, and z = 1 = black. Thus, the entire categorical stimulus can be represented by $\text F = \left\{ {(1,{1,1)},{ (0,1,1)},{ (0,}0,0)} \right\}$. If we perturb (i.e., transform) this categorical stimulus F with respect to the shape dimension (dimension 1) by assigning the opposite shape value to each of the object-stimuli in the set, we will get the perturbed categorical stimulus $T_{1} (\text F) = \left\{ {(0,{1,1)},{ (1,1,1)},{ (1,}0,0)} \right\}$ which indicates a transformation along the first dimension. More generally, if F is a categorical stimulus defined over D binary dimensions (where $D \ge 1$), then, for any dimension i ($1 \le i \le D$), the transformation $T_{i}$ on F is defined as follows: $T_{i} \left( {\text{F}} \right) = \left\{ {\left( {x_{1} , \ldots ,x_{i}^{\prime } , \ldots ,x_{D} } \right)\left| {\left( {x_{1} , \ldots ,x_{i} , \ldots ,x_{D} } \right)} \right. \in {\text{F}}} \right\}$ ,where $x_{i}^{\prime } = 1$ if $x_{i} = 0$ and $x_{i}^{\prime } = 0$ if $x_{i} = 1$.

Now, if we compare the original categorical stimulus to the perturbed set, we see that they have two object-stimuli in common. The object-stimuli that “survive” the transformation (from the standpoint of membership in the categorical stimulus) are called “invariants”. Thus, two out of three objects are invariants or remain the same. Intuitively, the ratio of invariants to number of objects in the categorical stimulus may be construed as a measure of the partial structural homogeneity or structural coherence of the categorical stimulus with respect to the dimension of shape and can be written more formally as $\left| {\text F \cap T_{i} (\text F)} \right|/\left| \text F \right|$. In this expression, $\left| \text F \right|$ stands for the number of object-stimuli in the categorical stimulus F and $\left| {\text F \cap T_{i} (\text F)} \right|$ for the number of object-stimuli that F and its perturbed counterpart $T_{i} (\text F)$ share.

Fig. 8

Logical manifold transformations across the dimensions of shape, color, and size for a 3₂[3] type structure instance

The first pane of Fig. 8 above illustrates this qualitative transformative process. Doing this for each of the dimensions generates the partial invariance scores of the categorical stimulus F which are then arranged as a vector referred to as the logical manifold (structural manifold in the general theory) of the Boolean categorical stimulus F. The described procedure defines the $\Lambda$ operator which operates on Boolean categorical stimuli to generate their respective logical manifolds as shown by Eq. B5 below.

$$\Lambda ({\text{F}}) = \left( {\frac{{\left| {{\text{F}} \cap T_{1} ({\text{F}})} \right|}}{{\left| {\text{F}} \right|}},\frac{{\left| {{\text{F}} \cap T_{2} ({\text{F}})} \right|}}{{\left| {\text{F}} \right|}}, \ldots ,\frac{{\left| {{\text{F}} \cap T_{D} ({\text{F}})} \right|}}{{\left| {\text{F}} \right|}}} \right)$$

(B5)

Each component of the resulting logical manifold vector represents a degree of partial invariance for the categorical stimulus. But we also wish to characterize the global or total degree of invariance of the categorical stimulus and that is what Eq. B3 (the invariance law or GISTM) above accomplishes. We do this by taking the Euclidean distance of each logical manifold from the zero logical manifold whose components are all zeros (i.e., 0 = (0,…,0)). The zero logical manifold represents the condition of complete structural or relational disorder (i.e., incoherence) where object stimuli in the category are not interrelated from the standpoint of categorical invariance. Distances $\Phi$ from this extreme condition represent degrees of coherence/homogeneity as determined by the $\Lambda$ operator. Accordingly, degrees of similarity to this extreme condition as a function of distance represent degrees of disorder and contribute directly to the complexity of the category. Thus, Eq. B3, the law of invariance (or GISTM), follows directly from Shepard’s similarity measure (where similarity is characterized as the negative exponent of distance) and it states that the complexity $\psi$ of a well-defined category of objects F is directly proportional to its cardinality $\left|\text F\right|$ and inversely proportional to the exponent of its overall degree of categorical invariance $\Phi$ (or $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{\Phi }$, in the more general theory).

Finally, the variable measure $\lambda$, referred to as structural equilibrium in GIST, acts as a moderator for the degree of perceived complexity by compensating for the fact that the greater the proportion of fully diagnostic dimensions (that is, dimensions with zero invariants per the logical manifold operator), the easier it is for a classification rule to be identified for the category F in question, and this greater ease facilitates classification performance despite the increased complexity contributed by these dimensions not having invariants. This makes sense when considering that once ideotypes are extracted by the logical manifold operator, the next step, that of classification rule formation (or some other representation), involves a different procedure: namely, that of determining which dimensions partake in the formation of the rule. Thus, if z represents the number of dimensions with zero invariants (i.e., whose logical manifold kernel is equal to zero) and D is the number of dimensions associated with the category F, then $\lambda$ is defined as follows: $\lambda =\frac{z}{D}$+1 (one is added to avoid division by zero).

C. How to compute representational information with structural equilibrium (Adapted from Vigo, 2015 and Vigo et al., 2020)

Next, we give an example of how to compute the information values for each of the objects in a stimulus set (i.e., categorical stimulus) using the GISTM with structural equilibrium ($\lambda$). The reader will find more details and examples in Vigo (2013b, 2015) and Vigo et al. (2020). We begin by computing the degree of categorical invariance of the stimulus set. First, we can represent this stimulus set by the following concept function:

$$\text F= {x}^{\mathrm{^{\prime}}}yz+{x}^{\mathrm{^{\prime}}}{y}^{\mathrm{^{\prime}}}{z}^{\mathrm{^{\prime}}}+x{y}^{\mathrm{^{\prime}}}z+x{y}^{\mathrm{^{\prime}}}{z}^{\mathrm{^{\prime}}}$$

(C1)

Or, alternatively, we can encode the dimensional values of the stimulus set using the digits “1” and “0” such that:

x = white, y = 1 = round flask, and z = 1 = large

$x^{\prime}$ = black, y = 1 = triangular flask, and $z^{\prime}$ = 0 = small

The category above results in the following, $\text F=\{011, 000, 101, 100\}$. This set can be transformed along the color dimension by assigning the opposite color value to all its objects to obtain the perturbed set $\{111, 100, 001, 000\}$. Comparing the original set to the perturbed set, we observe that they have two objects in common with respect to the dimension of color. That is, two out of four objects remain the same (this ratio of 2/4 for the color dimension is represented as the first kernel in Eq. C2). This ratio is a measure of the partial invariance of the category with respect to the dimension of color. Conducting a similar analysis with respect to the dimensions of shape and size, and arranging the values as an ordered set or vector, we get the logical or structural manifold of the concept function $\text F$ above (in other words, we apply the lambda operator defined in appendix B on the concept function F as shown in the following equation).

$$\Lambda \left(\text F\right)=\left(\frac{2}{4},\frac{0}{4},\frac{2}{4}\right)$$

(C2)

Relative degrees of global invariance can then be measured by taking the Euclidean (for separable dimensions) or Manhattan (for integral dimensions) distance of each structural or logical manifold from the zero logical manifold whose components are all zeros (i.e., (0,..., 0)). Thus, the overall degree of invariance $\Phi$ of the stimulus set when the metric is Euclidean is given by:

$$\Phi \left(\text F\right)=\sqrt{{\left(\frac{2}{4}\right)}^{2}+{\left(\frac{0}{4}\right)}^{2}+{\left(\frac{2}{4}\right)}^{2}}\approx 0.71$$

(C3)

Using the degree of categorical invariance, Vigo (2013b, 2015) computes the perceived degree of structural complexity $\psi$ of any well-defined category of objects (with structural equilibrium as a moderator) as being directly proportional to its cardinality (i.e., size) and indirectly proportional to the exponent of its degree of invariance as shown below:

$$\psi \left(\{011, 000, 101, 100\}\right)=\frac{p{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text F)}}{\lambda }=\frac{4{{e}^{-}}^{.67(.5)}}{1.33}\approx 2.15$$

(C4)

Now that we have obtained the degree of perceived structural complexity of the stimulus set using the structural complexity measure, we are now ready to calculate representational information as shown below. In general, a set of objects is informative about a category whenever the removal of its elements from the category increases or decreases the structural complexity of the category. That is, the amount of representational information (RI) conveyed by a representation R of a well-defined category $\text F$ is the rate of change of the structural complexity of $\text F$. Simply stated, the representational information carried by an object or objects from a well-defined category $\text F$ is the percentage increase or decrease (i.e., rate of change or growth rate) in structural complexity that the category experiences whenever the object or objects are removed. Note that this is usually expressed as a proportion that may be rendered as a percentage by multiplying by 100.

In Representational Information Theory (Vigo, 2011) and Generalized Representational Information Theory (Vigo, 2013a, 2015), the amount and quality of subjective information ${\hslash }_{s}$ conveyed by a subset R of a well-defined category $\mathrm{F}$ is defined as the percentage change in the structural complexity of $\mathrm{F}$ whenever R is subtracted from $\mathrm{F}$. This is expressed by the following equation:

$${\hslash }_{s}\left(\text R|\text F\right)= \frac{\psi \left(\text G\right)- \psi \left(\text F\right)}{\psi \left(\text F\right)}= \frac{\frac{\left|\text G\right|{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text G)}}{\lambda }-\frac{\left|\text F\right|{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text F)}}{\lambda }}{\frac{\left|\text F\right|{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text F)}}{\lambda }}$$

(C5)

In the equation above, $\text R\subseteq \text F$ or $\text R\in \wp (\text F)$, $\text G=\text {F--R,}$ $\left|\text G\right|$ and $\left|\text F\right|$ are the number of elements in $\text G$ and $\text F$ respectively, and $\Phi$ is the degree of categorical invariance of a set of well-defined objects (see Vigo for a detailed discussion of structural complexity and categorical invariance). As may be seen in the equation above, the degree of complexity $\psi$ of a well-defined category of objects is defined as the product between the number of objects in the set $\left|\text F\right|$ and the negative exponent of its degree of categorical invariance $\Phi$ (Vigo, 2013b, 2015), where $\frac{{D}_{0}}{D}$ is a discriminability index for structures in which $D$ is the number of dimensions over which the category is defined and ${D}_{0}$= 2 (the minimum non-trivial number of dimensions that define a category relationally).

Using this we can calculate the information conveyed by the first element, that is the subset $\text R=\{011\}$, of the set $\mathrm{F}=\{011, 000, 101, 100\}$. The set $\text G$ is now defined as:

$$\text G=\text F-\text R=\left\{011, 000, 101, 100\right\}-\left\{011\right\}=\{000, 101, 100\}$$

(C6)

The logical manifold of the set $\text G$ is obtained as described earlier. We transform the set along each of its three dimensions and compare the perturbed set with the original to observe the number of objects that are common between the two sets. The logical manifold is:

$$\Lambda \left(\text G\right)=\left(\frac{2}{3},\frac{0}{3},\frac{2}{3}\right)$$

(C7)

Using the logical manifold we now obtain the degree of categorical invariance as

$$\Phi \left(\text G\right)=\sqrt{{\left(\frac{2}{3}\right)}^{2}+{\left(\frac{0}{3}\right)}^{2}+{\left(\frac{2}{3}\right)}^{2}}\approx 0.94$$

(C8)

and subsequently obtain the degree of perceived structural complexity as.

$$\psi \left(\text G\right)=\frac{p{{e}^{-}}^{{\frac{{D}_{0}}{D}\Phi }^{2}(\text G)}}{\lambda }=\frac{3{{e}^{-}}^{.67(.89)}}{1.33}\approx 1.24$$

(C9)

Substituting the values of $\psi \left(\text F\right)$ and $\psi \left(\text G\right)$ above gives us the information conveyed by the object 011 in $\text R=\{011\}$ about its set of origin $\text F=\{011, 000, 101, 100\}$: that is,

$${\hslash }_{s}\left(\text R|\text F\right)= \frac{\psi \left(\text G\right)- \psi \left(\text F\right)}{\psi \left(\text F\right)}= \frac{1.24- 2.15}{2.15}\approx -0.42$$

(C10)

Repeating the above analysis, we can obtain the amount of information conveyed by all possible single element representations of $\text F$. The following table shows these approximate information values.

$\text R$	$\text F$	$\text G={\text F-\text R}$	$\psi \left(\text G\right)$	${\mathrm{\hslash }}_{s}\left(\text R|\text F\right)$
{011}	{011, 000, 101, 100}	{000, 101, 100}	1.24	− 0.42
{000}	{011, 000, 101, 100}	{011, 101, 100}	1.34	− 0.38
{101}	{011, 000, 101, 100}	{011, 000, 100}	1.34	− 0.38
{100}	{011, 000, 101, 100}	{011, 000, 101}	1.50	− 0.30

The cognitive basis for this removal is selective attention. GRIT posits that observers have the ability of psychologically “subtracting” an object from a category (as is the case with a single element concept cue) temporarily by selectively attending to all the members of the category except the object in question (while at the same time suppressing the object in memory). The results of the coherence and complexity judgments of the category with and without the object are then computed and stored in working memory where the computation of the most informative object takes place per GRIT’s prescription described above.

D. Note on metric selection

At first glance, using the Euclidean metric to account for separable-dimension results and the Manhattan metric to account for integral-dimension results may seem counterintuitive, given that models of categorization behavior implementing principles of similarity assessment and prior categorization results are often interpreted using the opposite relation (Nosofsky, 1992; Shepard, 1987). However, it is important to note that these models rely on computing relative distances between exemplars in a multidimensional psychological space. GIST, on the other hand, relies on computing relative distances between the relational invariance structure inherent to sets of exemplars (i.e., ideotypes), not between individual exemplars, in such psychological spaces. Under this global level of analysis, the entities of interest are ideotypes, and these, as structures, are more likely to look alike: a property that is better captured by the Manhattan metric. Vigo et al. (2022) delve deeper into the theoretical assumptions underlying metric selection for the GISTM when separable as opposed to integral stimulus dimensions are used in categorization tasks. And, although both these alternatives predict equally the learning difficulty orderings associated with the integral dimensions of color for the 3₂[4] family of structures, only the first provides a satisfactory explanation for the ordering in terms of a cognitive mechanism. Regardless, in the interest of mathematical parsimony, we use the second approach that involves using the Manhattan metric.