Hierarchical and Circumplex Structures 245contrast, like forceful-peaceful and quarrelsome-submissive.
Second, 5-D factors have different angles with the desirabil-
ity vertical axis: II, agreeableness, for example, is much
further north than is I, extraversion. When these angular
distances are forced to be equal, as in the model, content con-
trasts become contaminated by a desirability contrast. In the
example, peaceful is more desirable than forceful; therefore,
to the extent that they are at all judged opposite, that is partly
an artifact of a desirability difference.
It is fair to conclude that the double cone does not model
the underlying principle of chiasmic structure in an optimal
way. One could refine the model, but there is no need to do
so: Hofstee and Arends (1994) showed that the Abridged Big
Five circumplex (AB5C; see Hofstee, De Raad, & Goldberg,
1992) model to be discussed later can account for chiasmic
structure, and generates credible chiasms:
Daring Cautious
Reckless TimidIn two experiments, participants judged content contrasts
taken from AB5C chiasms to be superior over double cone
contrasts. This is not to say that chiasmic structure exists:
Hofstee and Arends reiterate a point already taken by
Peabody (1967) himself, namely, that desirability and content
cannot be separated. So the best one can do is create a chias-
mic illusion, as in the previous example. The algorithm goes
as follows: Take a particular circumplex; draw a diameter
separating desirable from undesirable traits; select two traits
on different sides of the diameter but close to it and to each
other, for example, cautious (slightly desirable) and timid
(slightly undesirable); together with their opposites, they cre-
ate the chiasmic illusion. It arises because in this case the
alleged content contrast is formed by two terms with an an-
gular distance that is only slightly less than 180 deg, instead
of 109.5 deg as according to the double cone model.
Do Chiasms Have a Future?
The double cone model was shown to be generalizable; it
may be possible to design a refined version by widening the
angle between content opposites, amounting to oblique
rotation. The more basic questions that remain, are What is
the taxonomic status of the underlying principle of chiasmic
structure? and What does it do to our conception of per-
sonality?
Whatever the refined model would be, it would focus on
traits that are close to the equator of a hypersphere whose
vertical axis is desirability: The model would focus on fairly
neutral traits. They form a small minority, so the focus
would be on a counterrepresentative subset of personality
variables. On the one hand, there is something venerable (to
use Saucier’s, 1994, term) to such a value-free approach;
personality psychologists, like everybody else, would prefer
practicing a discipline that is not submerged in extrascien-
tific values. On the other, desirability is not fruitfully consid-
ered as a mere response set or other artifact that is to be
separated from content: Hofstee and Arends (1994) empha-
sized that even in the classical example of chiasmic structure
cited earlier, stinginess is not merely undesirable thrift, but
an asocial version of it, whereas generousness differs from
extravagance in being prosocial; therefore, the evaluation
contrast is in fact one of content, as in the AB5C model. So
the most realistic conclusion is that chiasmic structure and
related models cannot be central to the concept of personal-
ity, even though they may have their place in specific
contexts (see Saucier, 1994; Saucier, Ostendorf, & Peabody,
2001).
Central features of the double cone model, however, ap-
pear to be valuable by themselves. One is the “circular pat-
tern” (Peabody & Goldberg, 1989, p. 556), as opposed to
simple structure, that is embodied in the model. Another is
orienting the trait space toward desirability as its reference
axis. These points are taken up later when developing an
integrative family of structure models.Generalized CircumplexesIn circumplex models, traits are assigned to segments of a
circle and are thus represented by their projection on the
bisectrix of that segment. Circumplexes picture tissues or
networks of traits: Contrary to hierarchies, circumplexes
have no superordinate and subordinate concepts. Eysenck
and Rachman (1965), for example, represented Hippocrates’
melancholic, choleric, sanguinic, and phlegmatic types as
mixtures of the positive and negative poles of neuroticism
and extraversion; presumably, however, Hippocrates would
have preferred a rotation by which an extravert is a mixture of
the choleric and sanguinic types, neuroticism is what melan-
cholics and cholerics have in common, and so on. Circles
enjoy full freedom of rotation.
Circles generalize to spheres, and spheres generalize to
hyperspheres—particularly, in this context, to the 5-D hyper-
sphere. An early example of a 3-D structure is Heymans’s
(1929) temperament cube. Not until the end of the twentieth
century, however, did 5-D researchers (Hofstee et al., 1992;
Saucier, 1992) construct circumplexes of more than two
dimensions.