246 Structures of Personality Traits
Heymans’s Cube
Heymans (1929) constructed a network model with three
dimensions—emotionality, primary versus secondary function
(comparable to extraversion-introversion), and activity—
forming the axes of a cube. Types are located at each of
the eight vertices of the cube, among which are the four
Hippocratic types; for example, the sanguinic type is at the
vertex where low emotionality, primary function, and high
activity meet.
Heymans tended to conceive the temperament space as
unipolar: The type characterized by the absence of emotion-
ality, activity, and secondary function is named amorphous.
One amendment therefore is to move the origin of the trait
space to the center of the cube. Next, it is difficult to conceive
of activity and primary function as orthogonal; different
dimensions (and types) would be chosen in a contemporary
three-dimensional model. Finally, one would prefer rounding
the cube to a sphere. On the one hand, it is thus gratifying to
note that time has not stood still, and that Heymans’s cube is
now obsolete by reasonable standards. On the other, it is
equally gratifying to recognize Heymans’s model as a fore-
runner of the generalized circumplexes that did not appear
until the end of the twentieth century.
Saucier’s Rhombicuboctahedron
Saucier (1992) presented an integration of interpersonal and
mood circumplexes and the Big Five Factors I, II, and IV. He
drew attention to the fact that simple structure does not mate-
rialize in these domains; many variables are interstitial in that
they are closer to the bisectrix of the angle between two fac-
tors than to the factors themselves. When simple structure is
nonetheless imposed, interstitial variables are likely to be
assigned to different factors by different investigators, even
though the positions of variables and factors are closely
comparable. Saucier constructed 6 bipolar scales as bench-
marks for the interstitial positions, in addition to the 3 bipolar
factor markers: a IIIversus IIIscale (friendly vs.
unfriendly), a IIIversus IIIscale (dominant vs. sub-
missive), and so on. He depicted the resulting trait struc-
ture as a rhombicuboctahedron, a prism showing the 18
(i.e., 2 [36]) unipolar benchmarks as facets.
Saucier’s model may be alternatively conceived as an
abridged three-dimensional circumplex, depicted by three or-
thogonal circles based on two of the three factors at a time.
Each circle contains two bisectrices of the angles between the
factor axes; in the model, a variable is represented by its pro-
jection on the vector (out of 9 bipolar or 18 unipolar vectors)
to which it is closest. This representation has the advantage
that it is easily carried to the fifth dimension (discussed later).
Saucier showed that the IIIIV sphere was the most in-
terstitially structured of all 10 spheres that are contained in the
5-D hypersphere; that difference, however, is quite relative in
view of the many mixtures involving Factors III or V.
Like Wiggins’s (1980) two-dimensional interpersonal cir-
cumplex, Saucier’s model uses octants, which are 45 deg
wide, corresponding to a correlation of .707. Therefore, the
variables assigned to such a segment may still form a fairly
heterogeneous set. Hofstee et al. (1992) distinguished traits
that had their primary loading on one factor and their sec-
ondary loading on another (e.g., III; sociable, social) and
traits with a reverse pattern (III; merry, cheerful). This
strategy amounts to slicing up a circle into 12 clock segments
of 30 deg, corresponding to a correlation of .866. A reason for
making these finer distinctions is that 30 deg is about the
angular distance at which vectors are still given the same sub-
stantive interpretation (Haven & Ten Berge, 1977). If this
amendment is worked into Saucier’s model, it becomes
identical to a three-dimensional version of the abridged
circumplex.The Abridged Big Five Circumplex ModelThe AB5C model consists of the 10 circumplex planes that
are based on 2 of the 5 factors at a time. Thus, variables are
represented by their projections on the closest plane or, more
precisely, on the closest of the 6 bipolar clock vectors
(running from 12 o’clock to 6 o’clock, 1 to 7, and so on) in
that plane. The hypersphere contains large empty spaces be-
tween the model planes, so it may look as if the abridgement
is rather drastic. However, varimax rotation puts the variables
as close to the planes as possible; Hofstee et al. (1992)
showed that it does a better job at this than at maximizing
simple structure, which is putting the variables as close to
the single factors as possible. Thus, representing traits by
their two highest loadings seems acceptable; a model includ-
ing tertiary loadings is entirely conceivable, but it would be
much more complex and add very little.
More aptly than by a spatial configuration, the AB5C
tissue is depicted by a table using the 10 factor poles (I,I,
II,II, and so on) as both warp and weft, the column de-
noting the primary loading, and the row, the secondary load-
ing of the traits assigned to a cell. Of the 100 cells in that table,
the 10 combinations of the positive and negative poles of the
same factor are void; the remaining 90 contain the unipolar
facets generated by the model. The gain over the simple-
structure model is enormous. That model accommodates only