Sedirnentolopy (1980) 27, 291-303
The shape of rock particles, a criticaI review
P. J. B A R R E T T
Antarctic Research Centre and Department of Geology, Victoria Universit.v of Wellington, Private Bag,
Wellington, New Zealand
ABSTRACT
An attempt was made to distinguish aspects of the shape of rock particles, and to discover by analysis
and empirical considerations the most appropriate parameters for describing these aspects. The shape
of a rock particle can be expressed in terms of three independent properties: form (overall shape),
roundness (large-scale smoothness) and surface texture. These form a three-tiered hierarchy of
observational scale, and of response to geological processes. Form can be represented by only two
independent measures from the three orthogonal axes normally measured. Of the four pairs of in-
dependent measures commonly used for bivariate plots, the twosphericitylshape factor pairs appear to
be more efficient discriminators than simple axial ratios. Of the two, the most desirable pair is the
maximum projection sphericity and oblate-prolate index for both measures show an arithmetic
normal distribution for the range investigated. A measure of form that is independent of the three
orthogonal axes, and measures derived from them, is the angularity measure of Lees. Roundness has
measures of three types, those estimating average roundness of corners, those based on the sharpest
corner, and a measure of convexity in the particle outline. Although each type measures a different
aspect, they are not independent of each other. Only roundness from corners is considered in detail.
As neither average nor sharpest corner measures are inherently more objective or more quantitative,
purpose should determine which is more appropriate. Of the visual comparison charts for average
roundness, Krumbein’s appears best. The Modified Wentworth roundness is the most satisfactory
for estimating roundness from the sharpest corner. The Cailleux Roundness index should not be used
because it includes aspects of roundness and form. Shape is a difficult parameter to use for solving
sedimentological problems. Even the best of the commonly used procedures are limited by observa-
tional subjectivity and a low discriminating power. Unambiguous interpretation of particle shape
in terms of source material and processes will always be made difficult by the large number of natural
variables and their interactions. For ancient sediments satisfactory results can be expected only from
carefully planned studies or rather unusual geological situations.
INTRODUCTION
There have been two main approaches t o investiga-
tions of shape of rock particles. The experimental
approach, using tumbling devices or abrasion mills,
allows observed changes t o be related to starting
materials, processes and time. In the empirical
approach, pebbles are measured in sedimentary
environments where the processes modifying pebble
shape are believed to be known. The problems of
0037-0746/80/0600-0291 $2.00
0 1980 International Association of Sedimentologists
measurement have also been examined, notably by
Griffiths and his co-workers (summarized in
Griffiths, 1967). As a result, there are many para-
meters for describing the shape of a pebble (Table
1) but none that is universally accepted. Confusion
appears to exist over what the various parameters of
shape actually measure and how they are related.
This paper aims to clarify the relationships between
various aspects of shape and to find the most
effective parameters t o estimate them.
292 P. J. Barrett
Table 1. Parameters and features used to describe aspects of shape of rock particles
Property Parameters or features
Form
Roundness
Elongation &flatness (Wentworth, 1922a; Zingg, 1935; Luttig, in Sames, 1966; Cailleux, 1947)
Angularity (Lees, 1964)
Sphericity (Wadell, 1932; Krumbein, 1941 ; Aschenbrenner, 1956; Sneed & Folk, 1958)
Form ratio (Sneed & Folk, 1958)
Factor ‘F’ (Aschenbrenner, 1956; shape factor of Williams, 1965)
Use of unranked shape classes (Holmes, 1960)
Roundness of sharpest corner (Wentworth, 1919, 1922b; Cailleux, 1947; Kuenen, 1956;
Average roundness for corners (Wadell, 1932; Russell & Taylor, 1937; Krumbein, 1941;
Average roundness from convexity of outline (Szadecsky-Kardoss, see Krumbein & Pettijohn,
Dobltins & Folk, 1970)
Pettijohn, 1949; Powers, 1953)
1938)
Surface texture *Markings due to contact with other rocks (pebble features catalogued by Conybeare & Crook,
*Surface texture resulting from internal texture, important for small pebbles of crystalline rock
1968; quartz grain features catalogued by Krinsley & Doornkamp, 1973)
*Numerical parameters have not yet been proposed.
THE MEANING O F ‘SHAPE’
Shape is the expression of external morphology, and
for some is synonymous with form (Shorter Oxford
English Dictionary, 1955; Gary, McAfee & Wolf,
1972). However, Sneed & Folk (1958) used the term
form for overall particle shape, to be obtained from
measurement of the three orthogonal axes, and
plotted on a form triangle. Used in this way ‘form’
clearly excludes other aspects of shape, such as
roundness. In contrast, Whalley (1972) saw farm as
the appropriate term for external morphology, but
regarded shape as only one of several properties
contributing to it.
Shape may also have different meanings for the
same person. For example, Griffiths (1967) has two
notions of shape, one being the expression of
external morphology (p. 1 lo), and the other ‘overall
shape’ being related to the original form of the
particle (p. 1 1 1 ) , and excluding roundness and
surface texture. Further on (p. 113 et seq.) he used
sphericity to estimate shape (meaning overall shape
presumably), though it is now clear that sphericity
contains only part of the information on overall
shape.
The two concepts of shape recognized by Griffiths
are maintained here, though terminology and usage
are clarified. Shape is taken to include every aspect
of external morphology, that is, overall shape,
roundness (=smoothness) and surface texture,
Form is used, following Sneed & Folk (1958), for
the gross or overall shape of a particle, and is
independent of roundness and surface texture.
T H E RELATIONSHIP BETWEEN
FORM, ROUNDNESS AND SURFACE
TEXTURE
Form, roundness and surface texture are essen-
tially independent properties of shape because one
can vary widely without necessarily affecting the
other two properties (Fig. 1). Wadell (1932, 1933)
long ago established the independence of sphericity
and roundness, but since then sphericity has come
to be recognized as only one aspect of form
(Aschenbrenner, 1956). Surface texture gives rise to
occasional practical difficulties in the measurement
of shape, but it is often not considered in discussions
of shape. Whalley (1972) stated ‘surface texture
can not be recognized in the projected outline of a
particle. . . ’, but this is not necessarily true for
crystalline rock particles, for example. Surface
texture bears the same relationship to roundness as
roundness does to form. These three properties
can be distinguished at least partly because of their
different scales with respect to particle size, and this
feature can also be used to order them (Fig. 2).
Form, the first order property, reflects variations in
the proportions of the particle; roundness, the
second order property, reflects variations at the
corners, that is, variations superimposed on form.
Shape of rock particles 293
v)
v)
F O R M
Fig. 1. A simplified representation of form, roundness
and surface texture by three linear dimensions to illus-
trate their independence. However, note that each of
these aspects of shape can itself be represented by more
than one dimension.
FORM
surface roughness of a pebble, though the well
rounded corners remain easily discernible. Striae,
chatter marks and other features may also be
acquired without changing the roundness. This
does not preclude the processes producing these
textures also changing the roundness over a long
period of time. Roundness of rock particles, which
normally increases through abrasion, can change
greatly without much effect on form. In contrast, a
change in form inevitably affects both roundness
and surface texture, because fresh surfaces are
exposed, and new corners appear, and a change in
roundness must affect surface texture, for each
change results in a new surface.
PARAMETERS FOR THE
ESTIMATION OF SHAPE
It is clear that no one parameter can be devised to
characterize the shape of a rock particle, and indeed
it is easy to see how several might be needed to
describe adequately each property that contributes
to shape. The precision or level of description (and
hence number of parameters) will depend on the
problem being studied. There are, however, at
least two properties that the parameters themselves
should have. (1) Each should represent an aspect
that has some physical meaning, so that they can be
related to the processes that determine particle
shape. (2) Each should represent a combination of
measurements from the same aspect of shape, that
is, from the same hierarchical level.
Various parameters that estimate particular
aspects of shape are discussed below, taking form
and roundness in turn. Surface texture will not be
considered further, as numerical parameters are yet
to be devised.
\ . .-_- ,‘ \
S U R F A C E T E X T U R E
Fig. 2. A particle outline (heavy solid line) with its com-
ponent elements of form (light solid lines, two approxi-
mations shown), roundness (dashed circles) and texture
(dotted circles) identified.
Surface texture, the third order effect, is super-
imposed on the corners, and is also a property of
particle surfaces between corners.
This hierarchical view of form, roundness and
texture is supported by the geological behaviour of
rock particles. Changes in surface texture need not
affect roundness. Weathering may enhance the
Form
Almost all parameters of particle form are based on
the longest, shortest and intermediate orthogonal
axes (Table 2). Shape parameters should be in-
dependent of size, and therefore normally take the
form of ratios of the axes. From three axes only two
independent ratios can be obtained, and this is the
limit for the number of independent parameters of
form. Zingg’s (1935) diagram, in which I /L is
plotted against S/I, is an early and clear expression
of this.
The concept of sphericity, as Wadell (1932, 1933)
294 P. J . Barrett
Table 2. Parameters for estimating aspects of form from three axes L =long axis, I -intermediate axis,
S =short axis, P =I/L, Q =S/I
Author Formula Name or description Range
Indices of flatness
Wentworth, 1922 L+ I
Cailleux. 1945 2 s
Zingg, 1935 I S
Luttig o n Sames, 1966)
-
L'T
I . 100
Flatness index
Ordinate and abscissa for a plot to
characterize shape
Elongation
Sneed & Folk, 1958
Indices of sphericity
Wadell. 1932
Krumbein, 1941
Sneed & Folk, 1958
Aschenbrenner, 1956
Other shape factors
Dobkins & Folk,
L
s . 100 Flatness
L
" h -
L
L - I
L - s
-
34 Vol of particle
~~ ~
Vol of circumscribing sphere
3 4 1 . S
L2
-
3 z/s2 -
L . I
12.8-
1 + P(1+ Q ) + 6 d I + P2(1 + Q2)
1970
Flatness
Flatness I' to the long axis
Intercept sphericity
Maximum projection sphericity
Working sphericity
Oblate-prolate index (OP index)
Aschenbrenner, 1956
Williams, 1965
sir.
L.S
I2
I - L . S if I2 > L . S
-
-
1 2
I 2 --+ i f L 2 < L . S
Shape factor F
Williams shape factor
I -CC
0- I
0- 100
0-100
0-1
0 - 1
0- 1
0- 1
0-1
0- 1
0- co
0- a,
0.- I
0-(- 1)
L . S
developed it, represents a different aspect of shape.
Wadell argued well for the sphere as a reference
form, and considered that deviations were best
represented by ratios of particle volume to the
volume of the circumscribing sphere (Table I ) .
Although Wadell is best remembered for his
demonstration that sphericity and roundness are
separate aspects of shape, his sphericity is sensitive
to roundness as well as form. Rounding the edges of
a cube changes its Wadell sphericity but not its
form. Therefore Wadell's sphericity is not a para-
meter of form alone, but includes a pinch of round-
ness, making it a difficult parameter to deal with.
conceptually at least.
Shape of rock particles 295
The differences between the procedures of Zingg
and Wadell for describing particle shape were
substantially reduced by Krumbein (1941a), who
derived an equation for estimating Wadell’s spheri-
city from measurement of the three orthogonal axes
of a particle. The principal assumption is that the
rock particle approximates an ellipsoid, Krumbein’s
intercept sphericity being a function of the volume
ratio of the ellipsoid defined by the three axes to the
circumscribing sphere. Whilst he regarded the
intercept sphericity as an approximation to true
sphericity Krumbein (1941a, p. 65) had in fact
created a conceptually purer parameter than
Wadell’s sphericity, for intercept sphericity measures
form alone. This was the time for the term equantcy,
proposed recently by Teller (1976) for intercept
sphericity, to be introduced.
Krumbein (1941a) recognized that lines of equal
intercept sphericity plot as hyperbolic curves on
Zingg’s diagram (Fig. 3), but it was left to Aschen-
brenner (1956) to recognize the need for a parameter
to describe variations in form for particles of equal
sphericity. His shape factor F (Table 2) had a range
from 0 to infinity, but Williams (1965) has provided
a transformation to give the shape factor a range
from $ 1 to -1 (Fig, 3).
Aschenbrenner’s (1 956) main purpose, however,
was to develop a measure of sphericity that used a
s/ I
I / L
*
ASCHENBRENNER
SCALE
WILL i: AMS
SCALE
Fig. 3. Zingg’s diagram, showing the relationship between
the axial ratios Z/L and SJZ, Aschenbrenner’s working
sphericity and Williams shape factor (from Drake, 1970).
reference form closer to real rock debris than an
ellipsoid. He wanted a plane-sided figure and chose
the tetrakaidekahedron which he thought repre-
sented a better aproximation to natural particle
shape. Also it was relatively easy to handle math-
ematically. He took true sphericity to be the ratio
of the surface area of the rock particle to the surface
area of the reference form, and derived a formula
that allowed sphericity to be calculated from the
three orthogonal axes, using the tetrakaidekahedron
as the reference form. However, he noted that it is
not possible to reach a sphericity of 1.0 unless the
reference form is an orthotetrakaidekahedron.
Aschenbrenner arbitrarily and perhaps regrettably,
set the formula for his ‘working sphericity’ half-
way between the two (Table 2). Although he could
derive a formula using the orthotetrakaidekahedron
capable of yielding a sphericity of 1.0, the reference
form would itself have a ‘true sphericity’ of only
90.1. He appears not to have recognized that the
difference in sphericity values results from a differ-
ence in roundness of the reference forms.
Sneed & Folk (1958) suggested that the sphericity
of a particle should express its behaviour in a fluid.
Noting that particles tend to orientate themselves
with maximum projection area normal to the flow,
they proposed a maximum projection sphericity
derived from the ratio of a sphere equal to the
volume of the particle to a sphere with the same
maximum projection area. Sneed and Folk did not
compare their measure with other measures of
sphericity, but simply presented the results of a
major study on river pebbles using the new measure.
The widespread acceptance of their measure may
reflect as much the usefulness of the results as the
power of their argument for the measure. The use of
behaviouristic measures can lead to problems in
interpretation. A measure may be inappropriate
when the behaviour assumed in deriving it may be
unimportant or different in the particular situation
in which one wants to use the measure. Should a
measure appropriate for water-deposited pebbles
be used for pebbles deposited from ice? Perhaps the
answer can be avoided by noting that the formula
for Sneed and Folk‘s measure is very close to that of
intercept sphericity of Krurnbein (1941a), which it
was designed to replace. The only difference is that
maximum projection sphericity uses the short axis
as a reference, whereas intercept sphericity uses the
long axis (Table 2). Thus the two formulae appear
to be equally valid measures of sphericity from a
conceptual point of view.
296 P. J. Barrett
Sneed & Folk also proposed the use of a tri- The measures proposed by Folk and his students
angular diagram for plotting pebbles’ form, the allow the same pebble data to be plotted in two
three poles representing platy, elongated and com- different ways (Fig. 4): (1) sphericity against OP
pact (equant) pebbles (Fig. 4). Unlike most such index on orthogonal axes; (2) S/L against ( L - I ) /
diagrams where the location of a point is determined (L- S) on triangular graph paper (the form dia-
by the proportions of the three end members, the gram).
location here is determined by the value of the apex In the equivalent diagram using the procedures
end member-compactness, measured by S/L, and of Zingg, Aschenbrenner and Williams (Fig. 3), the
a proportion (L-Z) / (L-S) measured parallel to same pebble data can be plotted as: (1 ) Z/L against
the base, which divides pebbles into three classes, S/Z; (2) Aschenbrenner working sphericity against
platy, bladed and elongated. The diagram empha- Williams shape factor.
sizes the fundamental character of these shapes, Each of the four plots derives from the same
and the way in which they converge on a single type, basic data, the lengths of the three principal axes.
compact. Therefore a trend in one diagram cannot be legiti-
The relationship between the form triangle and mately confirmed by a similar trend in another.
maximum projection sphericity is similar to that The only other common form index that uses the
between Zingg’s diagram and intercept sphericity. same three axial measurements is the flatness index
For each maximum projection sphericity value of Wentworth (1922a) (Table 2). The index was
there is a unique curve on the form triangle. The adopted by Cailleux (1945) and now his name is
need for a complementary shape property was not commonly associated with it.
immediately recognized, but in 1970 Dobkins & As each pair of measures expresses the same
Folk offered the OP index (oblate-prolate index, information they are now compared, using two
Table 2), which was based on the ratio L - Z , criteria, namely: (1) their effectiveness in discriminat-
L - s ing between different shapes, measured by the ratio
though it also took into account degree of com- of range to average standard deviation; and (2)
pactness. The OP index ranges from - 03 to + 03, the degree to which each measure follows a normal
unlike most shape measures, which range from 0 or distribution, extreme deviations making a measure
-1 to + l . difficult to use for statistical tests.
-
TER
BLADED
Fig. 4. Folk’s form diagram, showing the relationship between the defining parameters S / L and (L-Z) / (L-S) , and
maximum projection sphericity and oblate-prolate index (from Dobkins & Folk, 1970).
Shuppe of rock particles 297
The data used for the evaluation are from pebbles
in the range 8-64 mm, collected and organized into
sets of 20-30 pebbles. Each set represents a particular
rock type and sedimentary environment. The
pebbles came from two areas, Hooker G
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