RESEARCH REPORT

Twisting space: are rigid and non-rigid mental transformationsseparate spatial skills?

Kinnari Atit • Thomas F. Shipley • Basil Tikoff

Received: 28 November 2012 / Accepted: 5 February 2013 / Published online: 20 February 2013

� Marta Olivetti Belardinelli and Springer-Verlag Berlin Heidelberg 2013

Abstract Cognitive science has primarily studied the

mental simulation of spatial transformations with tests that

focus on rigid transformations (e.g., mental rotation).

However, the events of our world are not limited to rigid

body movements. Objects can undergo complex non-rigid

discontinuous and continuous changes, such as bending and

breaking. We developed a new task to assess mental visu-

alization of non-rigid transformations. The Non-rigid

Bending test required participants to visualize a continuous

non-rigid transformation applied to an array of objects by

asking simple spatial questions about the position of two

forms on a bent transparent sheet of plastic. Participants

were to judge the relative position of the forms when the

sheet was unbent. To study the cognitive skills needed to

visualize rigid and non-rigid events, we employed four tests

of mental transformations—the Non-rigid Bending test (a

test of continuous non-rigid mental transformation), the

Paper Folding test and the Mental Brittle Transformation

test (two tests of non-rigid mental transformation with local

rigid transformations), and the Vandenberg and Kuse

(Percept Motor Skills 47:599–604, 1978) Mental Rotation

test (a test of rigid mental transformation). Performance on

the Mental Brittle Transformation test and the Paper Fold-

ing test independently predicted performance on the Non-

rigid Bending test and performance on the Mental Rotation

test; however, mental rotation performance was not a

unique predictor of mental bending performance. Results

are consistent with separable skills for rigid and non-rigid

mental simulation and illustrate the value of an ecological

approach to the analysis of the structure of spatial thinking.

Keywords Mental transformations � Rigid

transformations � Non-rigid transformations �Mental visualizations

Introduction

Spatial thinking ranges from concrete action planning to

abstract visualization of multi-dimensional space. While it

is possible that humans’ ability to visualize passing a plate

at dinner may to a first approximation be related to a sci-

entist’s ability to visualize the movements of continental

plates, it seems unlikely that a single cognitive process

subserves all concrete and abstract reasoning about the

range of spatial problems that confront a mobile symbolic

thinker. Spatial problem solving involves visualizing and

manipulating a broad class of spatial information—the

relations between locations, configurations, shapes, and

objects and how they can change over time (Newcombe

and Shipley in press). The ability to mentally represent and

transform spatial relations makes up one’s ‘‘spatial ability’’

(Carroll 1993) or ‘‘spatial skills.’’ Researchers in cognitive

This article is part of the special issue on ‘‘Spatial Learning and

Reasoning Processes’’, guest-edited by Thomas F. Shipley, Dedre

Gentner and Nora S. Newcombe. Handling editor of this manuscript:

Dedre Gentner.

K. Atit (&) � T. F. Shipley

Department of Psychology, Temple University,

1701 North 13th Street, Philadelphia, PA 19122, USA

e-mail: kinnari.atit@temple.edu

B. Tikoff

Department of Geology, University of Wisconsin-Madison,

Madison, WI, USA

B. Tikoff

Department of Geoscience, University of Wisconsin-Madison,

1215 W Dayton Street, Madison, WI 53706, USA

123

Cogn Process (2013) 14:163–173

DOI 10.1007/s10339-013-0550-8

science have endeavored to characterize the skills that

comprise spatial thinking and define the categories of

spatial skills; although there is general agreement about the

importance of spatial thinking and that there is not just one

skill (e.g., Guildford and Lacey 1947; McGee 1979;

Thurstone and Thurstone 1941), there is surprisingly little

agreement about the details of what makes up this skill set

(Caplan et al. 1985).

One suggested distinction in spatial skills that has been

agreed upon by many researchers is between two classes of

spatial transformations: (1) object-based transformations

and (2) egocentric perspective transformations—as

required during navigation (e.g., Hegarty and Waller 2004;

Kozhevnikov et al. 2006). This paper focuses on spatial

skills associated with object-based transformations that

could be used to reason about events—how objects change

(Shipley and Zacks 2008). Research on what defines spatial

ability regarding this specific class of transformations

remains unsettled because work that has attempted to

define spatial ability has focused on a relatively restricted

range of spatial information. Research has relied heavily on

a few measures (e.g., mental rotation), which sample only a

few of the many types of events human beings think about

(Gibson 1986; Shepard and Cooper 1986). This may in part

reflect the difficulty in constructing such measures, but it

may also reflect limitations in recognizing the breadth of

cognitive spatial skills that should be assessed. Using

extant measures to define categorical boundaries will only

work if the measures sample broadly from the spatial

thinking space. One notable program of work by Pani and

colleagues has sought to extend the territory of spatial

transformations by characterizing the processes involved in

sequences of mental rotation (e.g., Pani 1993; Pani and

Dupree 1994; Pani et al. 1995), as well as interactions

between rigid objects, such as reasoning about the pro-

jections of a shape onto a planer surface (Pani et al. 1996),

the orientation of the line defined by the intersection of two

surfaces (Pani et al. 1998), and the shapes resulting from

slicing the brain (neuroanatomy) (Chariker et al. 2011). As

there is no formal account of the universe of potential

spatial interactions among rigid objects, we are not sure

how extensively this impressive research program has

explored spatial thinking. One way to approach the prob-

lem of ensuring that research spans the breadth of spatial

thinking is to develop an account of the dimensions of

spatial cognition and ensure research covers this space.

Alternatively, one may analyze the types of spatial prob-

lems faced by humans and ask, does existing research

account for all (or at least many) of these problems?

A number of attempts have been made to characterize

the dimensions of spatial thinking by identifying com-

monalities in the cognitive mechanisms (e.g., Maccoby and

Jacklin 1974), identifying clusters in skill using meta-

analyses and factor analyses (e.g., Ekstrom et al. 1976;

Linn and Petersen 1985), using computational modeling

(e.g., Smith et al. 1982), and using neuroscience methods to

identify candidate biological substrates associated with

different types of spatial thinking (e.g., Mehta and New-

combe 1991; Milivojevic et al. 2003; Vogel et al. 2003).

Recently, Chatterjee (2008) proposed two broad dichoto-

mies of spatial thinking based on a review of psychologi-

cal, linguistic, and neuroscientific data, suggesting that

humans process intrinsic (objects’ shapes and part-based

representations) and extrinsic (relations among objects and

between objects and frames of reference) spatial repre-

sentations differently, and static (stationary) and dynamic

(moving) spatial representations differently. This low-

dimensional division of spatial thinking appears to

exhaustively categorize all spatial skills into the four broad

categories that involve the four types of spatial relations:

intrinsic static, intrinsic dynamic, extrinsic static, and

extrinsic dynamic.

In contrast to the breadth of the existing categorizations

of spatial skills, when we consider the types of events that

have been used to study spatial cognition, we find a rela-

tively narrow focus. Generally, the events employed to

study mental simulation of change by cognitive scientists

are rigid (transformations where distances between every

pair of points on an object is preserved) (e.g., Hegarty and

Waller 2004; Pani et al. 2005; Shepard and Metzler 1971).

Yet, humans can visualize events that involve complex

non-rigid changes. For example, we can readily imagine

the hood of a car before and after a collision. As a result of

the collision, the hood underwent a non-rigid transforma-

tion, where the distances among points on the hood chan-

ged as the metal crumpled.

In cognitive science, there is recognition that humans

can perceive non-rigid transformations (Gibson et al. 1978;

Gibson and Spelke 1983; Gibson and Walker 1984; Spelke

1982) and that the brain processes complex transformations

(e.g., Shepard 2001; Shepard and Cooper 1986; Shepard

and Metzler 1971), but there is almost no research on how

humans visualize complex non-rigid transformations, like a

car crash. To be sure, there are some notable exceptions to

this observation: Pittenger and Shaw (1975) studied rec-

ognition of shear and strain in aging, Hegarty et al. (2003)

studied reasoning about complex mechanical events

involving pulleys, and recently, Resnick and Shipley (in

press) studied expert geologists’ mental simulation of

brittle deformations (e.g., breaking) that involve multiple

translations of fragments. Aside from this handful of cases,

what we know about the mental simulation of non-rigid

events comes primarily from research on mental folding.

An extensive analysis of the research on mental rotation

and mental folding, which involves a sequence of piece-

wise rigid changes, concludes that there is likely partial

164 Cogn Process (2013) 14:163–173

123

overlap in the skills—the two skills share some cognitive

processes, but each also requires some unique mental

processes (Harris et al. in press).

From the perspective of wanting to study spatial cog-

nition as it applies to natural events, folding does not seem

representative of all the different types of transformations

available in the environment and only captures one aspect

of non-rigid changes, the fact that the change is not uni-

form within an object. No set of experiments has attempted

to study the simulation of more naturalistic non-rigid

events with continuously varying change. Here, we focus

on the simple event of bending. To study mental reasoning

about this event, we developed the Non-rigid Bending test,

which included items that required a non-uniform contin-

uous non-rigid transformation. We employ the test to try to

understand the relationship between mental folding and

mental rotation by studying reasoning about rigid and non-

rigid events. If reasoning about bending relates to measures

of reasoning about folding or breaking and not to measures

of rigid transformations (e.g., mental rotation), we may

conclude that rigid mental transformations are indeed

independent of those involved in reasoning about non-rigid

transformations in general. Alternatively, the skills needed

to reason about a sequence of piece-wise changes, such as

folding and breaking, might be distinct from those needed

to reason about continuous changes applied over an entire

array, such as bending or rotating. To further understand

how humans reason about non-rigid events, we looked to

see if skill in reasoning about one type of event predicted

skill on others. We tested participants reasoning about:

bending, breaking, folding, and rotating.

Methods

Participants

One hundred and seventeen college undergraduate psy-

chology students were recruited through Temple Univer-

sity’s Undergraduate Psychology Research Pool.

Participants provided informed consent and received credit

toward a research participation requirement for their

involvement in the study. Data from one participant were

discarded, because he did not complete the study. Data

from 116 participants were used in the analysis (38 men, 78

women, Mage = 20.60 years, age range 18–53 years).

Materials

Non-rigid Bending test

The Non-rigid Bending test was designed as an objective

test of a participant’s skill in mentally simulating bending.

For the test, 120 19.3 by 19.3 cm transparent plastic sheets

were printed with a black circle and star, shown in the two

examples provided in Fig. 1. The star was positioned on

the top half of the sheet, and the circle was positioned

10 cm from the bottom. A red line defined the bottom edge

of the sheet. The surface of the plastic sheet was defined in

one of two ways, either with random array of polygons or

with a 6 by 6 grid of lines. Half of the plastic sheets (60)

had a textured background (Texture Condition), with

polygons (1 by 2 mm—created using the ‘‘confetti’’ pattern

in Adobe Illustrator CS3) ‘‘sprayed’’ on the sheet with a

density of 10 polygons per square cm. The other half of the

sheets (Gridlines Condition) had 6 vertical and 6 horizontal

lines drawn 2.7 cm apart.1 An example of both types of

sheets is shown in Fig. 1. Each sheet was bent in three

ways to vary task difficulty (Appelle 1972; Rock 1973):

simple bend (the top was bent down to the level of the

bottom with no offset), moderate oblique bend (achieved

by bending a top corner down to the level of the bottom of

the sheet and offset by 10 cm from a simple bend), and

complex oblique bend (achieved by bending a top corner

down to the level of the bottom of the sheet and offset by

20 cm from a simple bend). Here, we refer to sheets bent

orthogonally as easy, sheets that had a moderate oblique

bend as medium, and sheets that had a complex oblique

bend as hard.

Three bends for each image yielded 180 stimuli for each

of the Texture and Gridlines conditions. Of these, 25

‘‘easy’’, 30 ‘‘medium’’, and 30 ‘‘hard’’ were selected at

random. For examples of the stimuli used in the Non-rigid

Bending test, see Fig. 2.

Participants were presented with the 85 stimuli twice

over the course of the task for a total of 170 trials. Trials

were presented in two blocks with a minute rest in between.

The stimuli were presented in a random order. For each

trial, the participants’ task was to determine what the sheet

would look like if it were unbent. Specifically, they were to

indicate whether the star would be to the left or right of the

circle if the sheet were flat (un-bent). Participants were

given 6 s to respond to each trial. The presentation of

stimuli and collection of data, accuracy and reaction

time (RT), were controlled by E-Prime (version 2.0).

1 Two separate conditions were designed to pick up potential

strategies that involved using a frame of reference within the bent

sheet to make the left–right judgments. For example, rather than

mentally unfolding the plastic sheet, a participant might try to

mentally construct an imaginary line parallel to the sides of the sheet

that extended from one form and then judge whether the other form

was to the left or right of this line. Such a strategy would have been

suggested by finding that the grid stimuli were easier than the texture

stimuli. No subjects spontaneously reported using this strategy, while

a number mentioned mentally unfolding the plastic sheet, and as

reported below, there was no evidence of a difference between the

two stimuli types.

Cogn Process (2013) 14:163–173 165

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Participants were instructed to press the ‘‘p’’ if the star

would be to the right of the circle, and the ‘‘q’’ if the star

would be to the left of the circle. Each trial began with a

fixation cross presented for 1.5 s.

Paper Folding test

The Paper Folding test was administered using the proce-

dure described in Ekstrom et al. (1976) Kit of Cognitive

Reference Tests. In this test, participants were presented

with figures representing a piece of paper being folded one

to three times and then having a hole punched in it. The

task was to visualize what the paper would look like when

unfolded and to select the correct answer from one of five

answer choices. Following Ekstrom et al.’s (1976)

instructions, a participant’s score on the task was corrected

for guessing by subtracting one-fifth of the number of

incorrect answers from the number of correct answers.

Scores could range from -4 to 20 points. The test has two

parts with 10 questions each. Participants were given 3 min

to complete each half.

Mental Brittle Transformation test

The Mental Brittle Transformation test is a test of skill in

reasoning about breaking developed by Resnick and

Shipley (in press). A shortened version of the original task

was used in this study. The stimuli were created by

inserting a repeating character between each letter of a

familiar word, and then, the whole item was cut into pieces

which were moved in one of three ways: faulted, where the

cuts and displacement of parts were consistent with the slip

motion of a geological fault; random, where the fragments

were displaced in random directions; and exploded, where

the fragments were displaced radially away from a central

point. Resnick and Shipley (in press) found that using the

three types of changes offered a good range of difficulty in

identifying the original word. Participants were given

1 min to visualize what each word looked like before it was

broken. Participants were given the opportunity to view

each stimulus twice. Three faulted words, three exploded

words, and three randomly displaced words were pre-

sented. The items were presented in a random order. To

score the test, one point was given for each correctly

identified word. Scores could range from 0 to 9.

Mental Rotation test

Participants completed the Peters et al. (1995) paper and

pencil version of the Vandenberg and Kuse (1978) Mental

Rotation Test. For this test, the participant was presented

with five line drawings of 3D forms similar to those used

by Shepard and Metzler (1971). The target form is on the

left and four answer choices on the right. For each of the 24

problems, participants were asked to identify the two

choices that were identical but rotated versions of the target

form. Using Vandenberg and Kuse’s (1978) suggested

method of scoring, participants only received credit for an

item if both figures were identified correctly. Scores could

range from 0 to 24 points. The test has two parts with 12

problems each. Participants had 3 min to complete each

half.

Procedure

This study had a between-subject design. Participants were

alternately assigned to one of the two background condi-

tions of the non-rigid bending task: Gridlines (n = 57) and

Texture (n = 59). One person was tested at a time in a

closed and quiet room. After arriving for the study, the

A B

Fig. 1 Illustrations of the plastic sheets used to generate the stimuli in the Non-rigid Bending test: a textured background condition, b gridlines

background condition

166 Cogn Process (2013) 14:163–173

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participant first completed the consent process. After giv-

ing consent, he or she completed the Non-rigid Bending

test, and then the three other tests. The Mental Brittle

Transformation test, the Mental Rotation test, and the

Paper Folding test were given in a random order. Partici-

pants averaged 50 min to complete the study (range

45–60 min).

Results

Initial analysis of the Non-rigid Bending test

To investigate the effect of the amount of bending on

performance on the Non-rigid Bending test, a one-way

ANOVA was conducted on all stimuli.2 Levene’s test for

homogeneity of variances was found to be violated for the

present analysis, F(2, 345) = 18.29, p \ .001, and there-

fore, a Kruskal–Wallis H Test was used instead. The

analysis revealed a significant effect of level of bending,

H(2) = 119.85, p \ .001. Bonferroni-corrected Wilcoxon

Signed Rank post hoc comparisons revealed that as the

level of bending increased in complexity, accuracy

decreased. Participants’ accuracy on the Non-rigid Bending

test was significantly lower on the ‘‘medium’’ trials

(M = .85, SD = .13) than on the ‘‘easy’’ trials (M = .92,

SD = .13), p \ .01, and lower on the ‘‘hard’’ trials

(M = .72, SD = .18) than the ‘‘medium’’ trials (M = .85,

SD = .13), p \ .001. This suggests that the Non-rigid

Bending test is a valid measure of non-rigid mental

transformation.

Reaction time on the Non-rigid Bending test

To investigate the effect of the level of bending on RT for

the Non-rigid Bending test, a second one-way ANOVA

was conducted. The analysis again revealed a significant

effect of level of bending, F(2, 345) = 17.76, p \ .001.

Tukey’s post hoc comparisons revealed that as the level of

bending increased, RT increased. Participants’ RT on the

Non-rigid Bending test was significantly greater on the

‘‘medium’’ level trials (M = 1.66 s, SD = .60 s) than on

the ‘‘easy’’ level trials (M = 1.38 s, SD = .57 s), p \ .01,

and greater on the ‘‘hard’’ trials (M = 1.85 s, SD = .66 s)

Fig. 2 Examples of stimuli from the Gridlines background condition

in the Non-rigid Bending test: a is an example of an ‘‘easy’’ trial—an

image of a sheet with an orthogonal bend, b is an example of a

‘‘medium’’ trial—an image of a sheet with a moderate oblique bend,

c is an example of a ‘‘hard’’ trial—an image of a sheet with a complex

oblique bend

2 To investigate the relationship between the background pattern and

the amount of bending on performance on the Non-rigid Bending test,

a mixed-design ANOVA with a within-subject factor of level of

bending (easy, medium, hard) and a between-subject factor of

condition (Gridlines, Texture) was conducted. Mauchly’s test indi-

cated that the assumption of sphericity had been violated

(v2(2) = 56.37, p \ .001), and therefore, degrees of freedom were

corrected using Greenhouse-Geisser estimates of sphericity (e = .72).

The analysis revealed no effect of condition, n.s. An analysis of RT

revealed a similar pattern: There was no effect of condition and no

significant interaction, n.s. As there was no effect of background

pattern condition on accuracy or RT on the Non-rigid Bending test,

data from the two conditions were collapsed for further analysis.

Cogn Process (2013) 14:163–173 167

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than the ‘‘medium’’ trials (M = 1.66 s, SD = .60 s),

t(115) = -8.17, p = .05. Thus, analogous to the findings

in mental rotation that RT increases with increases in the

level of simulated rotation required for the item (Shepard

and Metzler 1971), we find that RT on the Non-rigid

Bending test increases with the amount of simulated

bending required for an item.

Performance across all measures

Due to the relatively high performance on the ‘‘easy’’

(M = .92, SD = .13) and ‘‘medium’’ (M = .85, SD = .13)

trials on the measure, subsequent analyses focused on the

‘‘hard’’ trials (M = .72, SD = .18), as this condition had

the largest range of individual performance and overall

variance. To study the relationship among the four tasks,

Pearson’s correlations were conducted.

A summary of the correlations, means, and standard

deviations for all four of the tasks is shown in Table 1. All

correlations were positive. Using Cohen’s (1988) guide-

lines, accuracy on the Non-rigid Bending test and accuracy

on the Mental Brittle Transformation test were moderately

correlated, r(114) = .381, p \ .001. Accuracy on the Non-

rigid Bending test and accuracy on the Paper Folding test

were also moderately correlated, r(114) = .36, p \ .001,

and accuracy on the Non-rigid Bending test and accuracy

on the Mental Rotation test were weakly correlated,

r(114) = .28, p \ .01.

Based on this administration, the Paper Folding, Mental

Brittle Transformation, and Mental Rotation tests were all

reliable (Chronbach’s alpha statistics were .79, .87, and

.80, respectively). Both conditions of the Non-rigid Bend-

ing test, Gridlines and Texture, were also reliable

(Chronbach’s alpha statistics were .98 and .95,

respectively).3

To further investigate the relations between the spatial

skills used in the Non-rigid Bending test and the spatial

skills used in each of the other measures (Mental Rotation

test, Paper Folding test, and Mental Brittle Transformation

test), a multiple linear regression analysis was conducted to

evaluate how well performance on the other spatial tests

predicted performance on the Non-rigid Bending test. The

predictor variables included performance on the Mental

Rotation test, Paper Folding test, and Mental Brittle

Transformation test. The criterion was performance on the

Non-rigid Bending test. The linear combination of the three

spatial measures significantly predicted performance on the

Non-rigid Bending test, with the combined spatial

measures explaining 22.0 % of the variance, R2 = .22,

F(3,112) = 10.51, p \ .001. Of the three spatial measures

used in the study, only performance on the Mental Brittle

Transformation test (b = .29, p \ .01) and Paper Folding

test (b = .25, p \ .01) significantly predicted Non-rigid

Bending test performance, with performance on the Mental

Brittle Transformation test having the largest effect. Per-

formance on the Mental Rotation test did not predict Non-

rigid Bending test performance, n.s. A summary of the

results from this multiple linear regression is shown in

Table 2.

Because results from the previous multiple linear

regression indicated that performance on the Mental

Rotation test did not significantly contribute to predicting

performance on the Non-rigid Bending test, a stepwise

regression was used to further investigate the relationship

between the four measures in our study. In order to find the

best subset of measures that predict bending performance, a

forward selection stepwise regression was conducted with

performance on the three measures (Mental Brittle Trans-

formation test, Paper Folding test, and Mental Rotation

test) as the predictors, and performance on the Non-rigid

Bending test as the criterion. In this type of regression, the

variable most strongly correlated to the criterion is inserted

first (Thompson 1978). Consistent with the outcome of the

previous regression, only two of the three predictors sig-

nificantly contributed to the model. The combination of

Table 2 Multiple linear regression analyses predicting performance

on Non-rigid Bending test

Predictors b SE b b

MBT .55 .04 .29**

PF .01 .00 .25*

MRT .06 .10 .06

The predictors in this multiple linear regression were average per-

formance on the Mental Brittle Transformation test (MBT), Paper

Folding test (PF), and Mental Rotation test (MRT)

* p \ .01

** p \ .001

Table 1 Summary of correlations, means, and standard deviations

for scores on the Non-rigid Bending test (NRB), Mental Rotation test

(MRT), Paper Folding test (PF), and Mental Brittle Transformation

test (MBT)

Measure NRB MRT PF MBT M SD

1. NRB – .28* .36** .38** .72 .18

2. MRT – – .46** .35** .34 .18

3. PF – .27* 9.17 4.22

4. MBT – – – – .28 .31

* p \ .01

** p \ .001

3 Due to the dichotomous nature of the Non-rigid Bending test, split-

half reliability was also calculated for both Gridlines and Texture

conditions (Spearman-Brown coefficients were .89 and .60,

respectively).

168 Cogn Process (2013) 14:163–173

123

performance on the Mental Brittle Transformation test

(b = .31, p \ .01) and the Paper Folding test (b = .28,

p \ .01) predicted 21.7 % of the variance in Non-rigid

Bending test performance, R2 = .22, F(2,113) = 15.63,

p \ .001, with performance on the Mental Brittle Trans-

formation test still having the largest contribution. Perfor-

mance on the Mental Rotation test did not contribute

significantly to the model, n.s. A summary of the results

from this stepwise regression is shown in Table 3 (top

section).

To further understand the relations between the skills

assessed by each of the measures, three additional forward

selection stepwise regressions were conducted. Each of the

stepwise regressions was used to examine how combina-

tions of the four measures in this study (Mental Rotation

test, Paper Folding test, Mental Brittle Transformation test,

and Non-rigid Bending test) predicted performance on each

of the individual tests. A summary of the results from all of

the stepwise regressions is shown in each section of

Table 3. The Mental Rotation and Non-rigid Bending tests

did not significantly predict performance on each other

after taking account of performance on the Paper Folding

and Mental Brittle Transformation tests. Similarly, per-

formance on the Paper Folding and Mental Brittle Trans-

formation tests was not related once performance on the

other two tests was taken into account.

Discussion

This study investigated the relations among rigid and non-

rigid mental transformations. We developed a new measure

of bending, a skill entailing a continuous non-rigid mental

transformation. Participants’ performance on the Non-rigid

Bending test appeared to be related to the amount of mental

bending required for each item. This finding is analogous to

findings in mental rotation research that performance is

related to the amount of simulated rotation (Shepard and

Metzler 1971). Performance on this new test was predicted

by performance on measures of other non-rigid transfor-

mations (folding and breaking), but not rigid transforma-

tions (rotating). Interestingly, performance on folding and

breaking was both related to rotating and bending, but

breaking and folding were not strongly correlated with

each other. Combining the findings of this study with dis-

tinctions in spatial thinking made in the extant literature

(e.g., Chatterjee 2008; Resnick and Shipley in press;

Shepard and Cooper 1982), we propose that the skills used

for rigid mental transformations are distinct from those

used for non-rigid mental transformations, and skills used

to manipulate intrinsic spatial properties are distinct from

those used for extrinsic spatial properties. These relation-

ships are illustrated in Fig. 3. This figure spatially

represents an interpretation of our results. Skills that share

underlying cognitive processes are shown as partially

overlapping. So, for example, mental bending shares pro-

cesses with both breaking and folding, but not the same

ones, since folding and breaking do not share any resource

that is common to all four visualization skills.

We found a relation among our three measures of non-

rigid mental transformations (breaking, folding, and

bending), and a lack of relation between bending and

rotating. Resnick and Shipley (in press) found a similar

disassociation in experts. Expert geologists and expert

chemists performed equally well on a measure of mental

rotation, but the geologists were superior to the chemists in

performance on a measure of mental breaking. Taken

together this suggests that the spatial skills used for non-

rigid mental transformations are distinct from the spatial

skills used for rigid mental transformations. Thus, we

interpret the vertical position of tests in Fig. 3 as indicating

the degree (or amount) of non-rigid mental simulation

required to solve the tasks. Rigid rotation requires little or

no skill, bending the most, and folding and breaking an

intermediate level because they require local piece-wise

Table 3 Stepwise regression analyses predicting performance on

each spatial measure

Criterion Predictors R2 DR2 b SE b b

NRB Model 1 .15 .15**

MBT .23 .05 .38**

Model 2 .22 .07*

MBT .18 .05 .31*

PF .01 .00 .28*

MRT Model 1 .22 .22**

PF .02 .00 .46**

Model 2 .27 .06*

PF .02 .00 .40**

MBT .15 .05 .25*

PF Model 1 .22 .22**

MRT 10.72 1.92 .46**

Model 2 .27 .06*

MRT 9.09 1.94 .40**

NRB 5.75 1.94 .25*

MBT Model 1 .15 .15**

NRB .64 .15 .38**

Model 2 .21 .07*

NRB .51 .15 .31*

MRT .45 .15 .27*

Measures used in these stepwise regressions include average perfor-

mance on the Non-rigid Bending test (NRB), the Mental Brittle

Transformation test (MBT), Paper Folding test (PF), and the Mental

Rotation test (MRT)

* p \ .01

** p \ .001

Cogn Process (2013) 14:163–173 169

123

rigid transformations. Thus, we suggest that previous

findings of a partial overlap in performance on rotation and

folding tests (see Harris et al. in press) reflect a pair of

underlying cognitive processes necessary to simulate

folding as a partially non-rigid and partially rigid event.

This account suggests the same is true for mental simula-

tion of breaking. Note, in this interpretation, we are not

committing to a particular spacing on this dimension, and

the folding is positioned halfway between bending and

rotating in the diagram for simplicity, in fact the two

intermediate cases maybe closer to one end than the other.

The correlational data do not allow precise relative posi-

tioning at this point.

Although folding and breaking may both require rigid

and non-rigid reasoning processes, they share little vari-

ance. We hypothesize that this reflects a fundamental dif-

ference between the two types of transformations: breaking

results in many separate objects, whereas folding produces

a single object. This distinction matches Chatterjee’s

(2008) intrinsic–extrinsic dichotomy and is represented by

the horizontal axis of the diagram in Fig. 3. Although we

can talk about a broken object as one thing, we may need to

reason about breaking in terms of the extrinsic relations

among the fragments of the original object, and conversely,

paper folding may require going from the segmented

independent parts of an object to a single integrated whole.

By this account, the intermediate position of bending and

rotating on the intrinsic/extrinsic dimension implies that

these visualizations may require a combination of intrinsic

and extrinsic cognitive processes. As noted above, we are

not committed to the particular metric spacing of the

transformations, so bending and rotating likely, on balance,

lean more toward intrinsic processing. That they are not

purely intrinsic is consistent with there being multiple

strategies for approaching these test items. Rotating for

example may be solved with either a holistic strategy that

operates on the single object (e.g., Cooper, 1975; Shepard

and Metzler 1971) or a piece-wise strategy of rotating one

part at a time (e.g., Goksun et al. in press; Just and Car-

penter 1976; Khooshabeh et al. in press).

By considering the nature of natural events and devel-

oping tools that can measure skills in reasoning about a

broad array of mentally simulated events, we have identi-

fied a new type of spatial reasoning and achieved some new

insights regarding previous findings on mental rotation and

folding. The motivation for this study and the context for

the interpretations we have offered reflect an interest in

broadening the scope of spatial cognition research to con-

sider complex spatial problems and the information

humans use to solve them. We refer to this as an ecological

approach to spatial cognition. The ‘‘ecological approach’’

discussed here is analogous to Gibson’s ecological

approach to visual perception (Gibson 1986) in that there is

an emphasis on how the structure of information available

in the environment guides an organism’s actions. Similarly,

we suggest that the deformational information available in

the environment guides the mental transformations and

spatial reasoning skills. We diverge from Gibson in not

advocating a theory of direct perception or action. We want

to emphasize the potential value of an analysis of the

potential structured information available in the ambient

optic array that could inform the observer about past

events, independently from discussion of how the infor-

mation influences behavior. Taking an ecological approach

and studying skills required to visualize transformations

present in the environment should help assure that what is

learned in the laboratory can inform cognitive science

about the many spatial problems humans reason about in

their lives.

Evaluating how well the spatial problems that are used

to study spatial cognition represent the spatial problems

encountered by humans as individuals or a species is dif-

ficult. We advocate an ecological approach to this chal-

lenge, one that focuses on the structure of the information

available to solve spatial problems. Some previous research

has also taken this approach (e.g., Gilden and Proffitt 1989;

Kaiser et al. 1986; Proffitt and Gilden 1989). For example,

Chariker et al. (2011) begin to characterize the spatial

challenges faced by neuroanatomy students as they attempt

to learn to recognize 3D structures from 2D cross sections.

In the study reported here, we focused on a spatial problem

Fig. 3 A Venn Diagram illustrates the shared variance among the

four spatial skills measures. In the diagram, the overlap betweencircles indicates shared variance and suggests a shared set of spatial

skills. No overlap between circles indicates that there was little shared

variance, and possibly no significant overlap in skills. We hypothesize

that the vertical axis represents a continuum of non-rigid to rigid

transformations and that the horizontal axis represents the continuum

of extrinsic to intrinsic spatial skills

170 Cogn Process (2013) 14:163–173

123

faced by geologists when they see layers of rocks that are

bent or curved as a result of a geological transformation—

bending. For the initial foray into this research domain, we

focused on the structured information in geology for sev-

eral reasons: the discipline represents the array of

mechanical, thermal, and chemical spatial relations that

result from objects’ interactions; geologists self-report high

levels of spatial skills across a range of problems on both of

Chatterjee’s (2008) dichotomies (Hegarty et al. 2010); and

the cognitive skills employed by geologists have begun to

be characterized (Kastens and Ishikawa 2006).

To develop a program of research on understanding

events, we look to structural geology, which is the study of

the three-dimensional distribution of rocks and how this

spatial pattern records the history of the rocks. Consider the

events of the earth from the perspective of a structural

geologist. As rocks move relative to other rocks, the

motion may be associated with ductile changes (a contin-

uous change in shape—e.g., bending) or brittle changes

(discontinuous changes—e.g., breaking) in the rocks. In

some regions (e.g., tectonic plate margins), movements

involve a combination of ductile and brittle changes,

whereas in other regions, the motions of the region can be

described as a rigid motion. Thus, geologic events may be

divided into rigid and non-rigid changes. For rigid changes,

all the points in the changing object maintain their spatial

relations to each other. For non-rigid changes, or defor-

mation, which is a change in the shape or dimensions of a

rock resulting from stress or strain, the spatial relations

may change. All deformations, rigid and non-rigid, can be

described by a displacement field, which is a three-

dimensional array of vectors describing the local spatial

changes in a three-dimensional volume. Although defor-

mation may be quite complex, any deformation field can be

decomposed into four basic components: translation, rota-

tion, dilation, and distortion (Malvern 1969). The compo-

nent fields may be summed to recover the original complex

deformation. Thus, all complex non-rigid changes can be

characterized by the sum of four simple spatial changes.

Cognitive science has notably not investigated mental

simulation of all four components of complex non-rigid

changes. As noted, most research has focused on rotation,

and some work has investigated mental translation (e.g.,

Shepard 2001; Shepard and Cooper 1986; Shepard and

Metzler 1971). The non-rigid transformation employed in

this experiment could be constructed from rotations and

translations. The critical distinction between rigid and non-

rigid transformations that we see in our data may reflect

fundamental differences in mentally simulating a single

rotation and mentally simulating transformations that vary

in magnitude across the object or image. Given the lack of

research on mental simulation of dilation and distortion, we

suggest that there is critical need for a research program in

this area that addresses the questions of how the mind

simulates all four different types of changes, how it sim-

ulates spatially varying changes, and how it simulates

combinations of changes. The value of such a program

would be to expand our understanding of the varieties of

transformations people can mentally simulate, and identify

potential constraints. We are not committed to a decom-

positional account where we expect to explain all mental

transformations based on the sum of four basic mental

transformation primitives. To the contrary, we are merely

pointing out that the landscape of what we do not know

about mental transformations seems to be fairly extensive.

A key idea here is that the physical sciences, such as

geology, can provide the requisite description of spatial

information; cognitive scientists may profitably look to

these sciences for help in understanding the categories of

spatial problems that we face in the world. This approach

also identifies experts in the natural sciences as people who

will be particularly good at solving the types of spatial

problems that are central to that science (Lubinski and

Benbow 1992). Thus, an ecological approach to delineating

spatial problems can also provide insight into the types of

skills that should be developed and fostered in the class-

room. Since high spatial ability has been found to predict

success in science disciplines (Shea et al. 2001), identify-

ing and categorizing the spatial information in spatial

problems could guide new approaches to curriculum

development to help train students’ spatial thinking skills,

develop their general problem-solving skills, and enhance

learning in the physical sciences.

We found that measures of mental bending, breaking,

folding, and rotating demonstrates a complex, overlapping

pattern of variance. We interpret this pattern as reflecting

an underlying dimension of spatial reasoning processes that

are used to mentally simulate rigid and non-rigid trans-

formations. Future work supporting our specific interpre-

tations would also support an ecological approach and

illustrate the value of considering the types of spatial

information characterized by the natural sciences—those

sciences charged with describing the world we live in and

must reason about.

Acknowledgments Except the first author, all subsequent authors

are listed alphabetically. This research was supported by a grant to the

Spatial Intelligence and Learning Center, funded by the National

Science Foundation (SBE-0541957 and SBE-1041707), and by a

Fostering Interdisciplinary Research on Education grant, funded by

the National Science Foundation (grant number DRL-1138619).

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