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- measurement have also been examined, notably by
tions of shape of rock particles. T h e experimental Griffiths and his co-workers (summarized in
approach, using tumbling devices or abrasion mills, Griffiths, 1967). As a result, there a r e many para-
allows observed changes t o be related to starting meters for describing the shape of a pebble (Table
materials, processes a n d time. In the empirical 1) but none that is universally accepted. Confusion
approach, pebbles are measured in sedimentary appears t o exist over what the various parameters of
environments where the processes modifying pebble shape actually measure and how they are related.
shape are believed to be known. The problems of This paper aims to clarify the relationships between
0037-0746/80/0600-0291 $2.00 various aspects of shape and to find the most
0 1980 International Association of Sedimentologists 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 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 Roundness of sharpest corner (Wentworth, 1919, 1922b; Cailleux, 1947; Kuenen, 1956;
Dobltins & Folk, 1970)
Average roundness for corners (Wadell, 1932; Russell & Taylor, 1937; Krumbein, 1941;
Pettijohn, 1949; Powers, 1953)
Average roundness from convexity of outline (Szadecsky-Kardoss,see Krumbein & Pettijohn,
1938)
Surface texture *Markings due to contact with other rocks (pebble features catalogued by Conybeare & Crook,
1968; quartz grain features catalogued by Krinsley & Doornkamp, 1973)
*Surfacetexture resulting from internal texture, important for small pebbles of crystalline rock
*Numerical parameters have not yet been proposed.

THE MEANING O F ‘SHAPE’ the gross or overall shape of a particle, and is

independent of roundness and surface texture.
Shape is the expression of external morphology, and
for some is synonymous with form (Shorter Oxford
English Dictionary, 1955; Gary, McAfee & Wolf, T H E R E L A T I O N S H I P BETWEEN
1972). However, Sneed & Folk (1958) used the term FORM, R O U N D N E S S A N D SURFACE
form for overall particle shape, to be obtained from TEXTURE
measurement of the three orthogonal axes, and
plotted on a f o r m triangle. Used in this way ‘form’ Form, roundness and surface texture are essen-
clearly excludes other aspects of shape, such as tially independent properties of shape because one
roundness. In contrast, Whalley (1972) saw farm as can vary widely without necessarily affecting the
the appropriate term for external morphology, but other two properties (Fig. 1). Wadell (1932, 1933)
regarded shape as only one of several properties long ago established the independence of sphericity
contributing to it. and roundness, but since then sphericity has come
Shape may also have different meanings for the to be recognized as only one aspect of form
same person. For example, Griffiths (1967) has two (Aschenbrenner, 1956). Surface texture gives rise to
notions of shape, one being the expression of occasional practical difficulties in the measurement
external morphology (p. 1lo), and the other ‘overall of shape, but it is often not considered in discussions
shape’ being related to the original form of the of shape. Whalley (1972) stated ‘surface texture
particle (p. 1 1 1 ) , and excluding roundness and can not be recognized in the projected outline of a
surface texture. Further on (p. 113 et seq.) he used particle. . . ’, but this is not necessarily true for
sphericity to estimate shape (meaning overall shape crystalline rock particles, for example. Surface
presumably), though it is now clear that sphericity texture bears the same relationship to roundness as
contains only part of the information on overall roundness does to form. These three properties
shape. can be distinguished at least partly because of their
The two concepts of shape recognized by Griffiths different scales with respect to particle size, and this
are maintained here, though terminology and usage feature can also be used to order them (Fig. 2).
are clarified. Shape is taken to include every aspect Form, the first order property, reflects variations in
of external morphology, that is, overall shape, the proportions of the particle; roundness, the
roundness (=smoothness) and surface texture, second order property, reflects variations at the
Form is used, following Sneed & Folk (1958), for corners, that is, variations superimposed on form.
 Shape of rock particles 293

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
v)
v) 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

FORM
ESTIMATION OF S H A P E
Fig. 1. A simplified representation of form, roundness
and surface texture by three linear dimensions to illus- It is clear that no one parameter can be devised to
trate their independence. However, note that each of characterize the shape of a rock particle, and indeed
these aspects of shape can itself be represented by more it is easy to see how several might be needed to
than one dimension.
describe adequately each property that contributes
to shape. The precision or level of description (and
FORM 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.
\ . . - ,_
‘ - \
SURFACE TEXTURE
Fig. 2. A particle outline (heavy solid line) with its com- Form
ponent elements of form (light solid lines, two approxi- Almost all parameters of particle form are based on
mations shown), roundness (dashed circles) and texture the longest, shortest and intermediate orthogonal
(dotted circles) identified.
axes (Table 2). Shape parameters should be in-
dependent of size, and therefore normally take the
Surface texture, the third order effect, is super- form of ratios of the axes. From three axes only two
imposed on the corners, and is also a property of independent ratios can be obtained, and this is the
particle surfaces between corners. limit for the number of independent parameters of
This hierarchical view of form, roundness and form. Zingg’s (1935) diagram, in which I / L is
texture is supported by the geological behaviour of plotted against S/I, is an early and clear expression
rock particles. Changes in surface texture need not of this.
affect roundness. Weathering may enhance the 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 Flatness index I-CC
-
Cailleux. 1945 2s
Zingg, 1935 I S Ordinate and abscissa for a plot to 0-I
characterize shape
L'T
Luttig o n Sames, 1966) I . 100 Elongation 0- 100
L
s . 100 Flatness 0-100
L
"
Sneed & Folk, 1958 Flatness 0-1
-h
L
L- I Flatness I' to the long axis 0-1
-
L-s
Indices of sphericity
Wadell. 1932 34 Vol of particle 0- 1
~~ ~

Vol of circumscribing sphere

Krumbein, 1941 3 4 1-
.S Intercept sphericity 0- 1
L2
Sneed & Folk, 1958 3 z/s2 Maximum projection sphericity 0-1
-
L.I
Aschenbrenner, 1956 12.8- Working sphericity 0-1
1 + P(1+ Q ) + 6 d I + P2(1+ Q2)
Other shape factors
Dobkins & Folk, 1970 Oblate-prolate index (OP index) 0-co

sir.
Aschenbrenner, 1956 L.S Shape factor F 0- a,
-
I2

Williams, 1965 I - L . S if I2> L . S Williams shape factor 0.-I

-
12
I 2 --+ i f L 2 < L . S 0-(- 1)
L.S

developed it, represents a different aspect of shape. separate aspects of shape, his sphericity is sensitive
Wadell argued well for the sphere as a reference to roundness as well as form. Rounding the edges of
form, and considered that deviations were best a cube changes its Wadell sphericity but not its
represented by ratios of particle volume t o the form. Therefore Wadell's sphericity is not a para-
volume of the circumscribing sphere (Table I ) . meter of form alone, but includes a pinch of round-
Although Wadell is best remembered for his ness, making it a difficult parameter to deal with.
demonstration that sphericity and roundness are conceptually at least.
 Shape of rock particles 295

The differences between the procedures of Zingg reference form closer to real rock debris than an
and Wadell for describing particle shape were ellipsoid. He wanted a plane-sided figure and chose
substantially reduced by Krumbein (1941a), who the tetrakaidekahedron which he thought repre-
derived an equation for estimating Wadell’s spheri- sented a better aproximation to natural particle
city from measurement of the three orthogonal axes shape. Also it was relatively easy to handle math-
of a particle. The principal assumption is that the ematically. He took true sphericity to be the ratio
rock particle approximates an ellipsoid, Krumbein’s of the surface area of the rock particle to the surface
intercept sphericity being a function of the volume area of the reference form, and derived a formula
ratio of the ellipsoid defined by the three axes to the that allowed sphericity to be calculated from the
circumscribing sphere. Whilst he regarded the three orthogonal axes, using the tetrakaidekahedron
intercept sphericity as an approximation to true as the reference form. However, he noted that it is
sphericity Krumbein (1941a, p. 65) had in fact not possible to reach a sphericity of 1.0 unless the
created a conceptually purer parameter than reference form is an orthotetrakaidekahedron.
Wadell’s sphericity, for intercept sphericity measures Aschenbrenner arbitrarily and perhaps regrettably,
form alone. This was the time for the term equantcy, set the formula for his ‘working sphericity’ half-
proposed recently by Teller (1976) for intercept way between the two (Table 2). Although he could
sphericity, to be introduced. derive a formula using the orthotetrakaidekahedron
Krumbein (1941a) recognized that lines of equal capable of yielding a sphericity of 1.0, the reference
intercept sphericity plot as hyperbolic curves on form would itself have a ‘true sphericity’ of only
Zingg’s diagram (Fig. 3), but it was left to Aschen- 90.1. He appears not to have recognized that the
brenner (1956) to recognize the need for a parameter difference in sphericity values results from a differ-
to describe variations in form for particles of equal ence in roundness of the reference forms.
sphericity. His shape factor F (Table 2) had a range Sneed & Folk (1958) suggested that the sphericity
from 0 to infinity, but Williams (1965) has provided of a particle should express its behaviour in a fluid.
a transformation to give the shape factor a range Noting that particles tend to orientate themselves
from $ 1 to -1 (Fig, 3). with maximum projection area normal to the flow,
Aschenbrenner’s (1956) main purpose, however, they proposed a maximum projection sphericity
was to develop a measure of sphericity that used a derived from the ratio of a sphere equal to the
volume of the particle to a sphere with the same
s/ I 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
I/L 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

i:
WILL AMS
SCALE
*
ASCHENBRENNER
SCALE
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
Fig. 3. Zingg’s diagram, showing the relationship between
long axis (Table 2). Thus the two formulae appear
the axial ratios Z/L and SJZ, Aschenbrenner’s working to be equally valid measures of sphericity from a
sphericity and Williams shape factor (from Drake, 1970). 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/Lagainst ( 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 ,
-
L-s
criteria, namely: (1) their effectiveness in discriminat-
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 Both are clearly better than either of the simpler
in the range 8-64 mm, collected and organized into ratios of axes. The value obtained from Wentworth
sets of 20-30 pebbles. Each set represents a particular flatness is the highest of all, though the effectiveness
rock type and sedimentary environment. The of the measure is reduced by not being paired with a
pebbles came from two areas, Hooker Glacier in complementary form measure.
the Southern Alps of New Zealand (20 sets and The shape of the frequency distribution for each
597 pebbles), where the rock types distinguished measure was examined by comparing the average
were quartz schist and pelitic schist, and Taylor skewness and kurtosis values for each suite of
Valley, Antarctica (29 sets and 706 pebbles), where pebbles (Fig. 5). Only two of the nine measures
the rock types sampled were granite, porphyry, vein have a skewness value clearly different from 0.
quartz, dolerite and basalt. The environments Aschenbrenner’s sphericity is strongly negatively
sampled in both areas include the subglacial (basal skewed, suggesting that the distribution may be
till), superglacial (talus and stream deposits), ice- constrained by some practical upper limit of
marginal streams and proglacial streams. sphericity. Wentworth flatness showed, on average, a
The total range of mean values was found for strong positive skewness. Most measures have a
each measure from both suites of pebbles, and peak that is broad compared with a normal dis-
divided by the average standard deviation (Table 3). tribution (platykurtic) but the values are not
Whereas most measures show some difference in extreme. Again Aschenbrenner sphericity and
the average mean between Hooker and Taylor Wentworth flatness stand out from the rest by
samples, representing differences in average shape, showing a pointed (leptokurtic) frequency dis-
the average standard deviations are all very similar, tribution, and they have a larger variability in
indicating a similar natural variability in the kurtosis values than the other measures. In every
measures regardless of their mean value. The case for both skewness and kurtosis there is no
two pairs of sphericity and oblate-prolate measures significant difference in the mean value from each
have a similar high ratio of range to standard suite of pebbles, suggesting that the features of the
deviation, showing that there is little to choose frequency distributions described above are of a
between them as effective discriminators of form. general nature. Of the measures examined therefore,

Table 3. Mean (istandard

?), deviation (s) and the ratio of range in mean to standard deviation for pebbles
from Hooker Valley (20 sets) and Taylor Valley (29 sets) to express the effectiveness of each measure in
discriminating aspects of form
~ ~~~ ~~ ~~

Average Range Total Average Range in R

Measure R in 3 range S S

Max. projection Hooker 0-62 0.47-0-72

0.31 0.10 3.1
sphericity Taylor 0.69 0.53-0.78 0.10
Oblate-prola te Hooker -0.54 - 6.59-3.05 .54 5.71 2.5
index Taylor - 0.32 4.82
SIL Hooker 0.42 0.28-0.54 0.32 0.11 2.9
Taylor 0.49 0.35-0.60 0.11
( L - I)/ (L- S ) Hooker 0.49 0.37-0.61 0.45 0.21 2.1
Taylor 0.50 0.33-0.78 0.22
Aschen brenner Hooker 0.83 0.70-0.89 0.22 0.07 3.4
working Taylor 0.87 0.77-0.92 0.06
sphericity
Williams shape Hooker 0.16 0.01-0.41 0.62 0.28 2.3
factor Taylor 0.11 -0.21-0.40 0.25
IlL Hooker 0.72 0.64-0.77 0.17 0.13 1.4
Taylor 0.75 0.66-0.81 0.12
SII Hooker 0.59 0.40-0.71 0.46 0.1 5
3.1
Taylor 0.67 0.46-0.86 0.15
Wentworth Hooker 2.3 1 1.71-3.75 2.26 0.70 3.8
flatness index Taylor 1.91 1.49-2.8 I 0.49
298 P. J . Burrett

index
I I 11
-- 7
.. I I 1
I -
-
n

1
,,,

maximum projection sphericity and oblate prolate affecting the number of sides, though not beyond
index are the most satisfactory for describing form, the point when an entire side is lost. Several practical
in that they both approximate the normal dis- problems not yet overcome are: (1) being con-
tribution, and are relatively efficient discriminators. sistent in deciding on the number of (planar) sides
The measures reviewed above are all based on for many rock particles with irregular forms; ( 2 )
three orthogonal axes, and are not satisfactory distinguishing the number of sides objectively on
discriminators of some forms. In particular they do particles with a large number of sides and corners;
not separate particles with triangular, rectangular and (3) resolving planar sides on rounded par-
and pentagonal cross-sections. Perhaps this is why ticles.
Holmes (1960) used verbally defined categories A measure that takes some account of this aspect
instead of ratio scale measures for describing of form is that of Lees (1964), who proposed the
pebbles from Pleistocene till. following measure for the degree of angul-
The forms above can clearly be distinguished on arity:
the number of sides (or angles). This aspect of
shape seems therefore to be a n aspect of form, and
not roundness, at least when the number of sides is
small. Rounding of the corners can vary without
 Shape of' rock particks 299

with a range from 0-co where a~ is the angle of Roundness

each corner and X Q is the distance of the corner The claim that roundness of rock particles is a
from the centre of the maximum inscribed circle property independent of sphericity was first made by
(radius r i ) for each of the three sections through the Wadell (1932), and was immediately attacked by
long, intermediate and short axes of the rock Wentworth (1933). Wentworth argued that the
particle. distinction defied common usage of the terms, on
Defined this way angularity is increased by (1) the grounds that roundness is a property best
increase in acuteness of the corners; (2) increase in displayed by a sphere. Wadell (1933) responded with
number of corners; (3) increase in relative distance points from both the dictionary and common usage,
of corners from the centre of the particle. Although supporting his use of roundness, which incorporated
(1) and (2) are clearly aspects of form (the angle of a a sense of smoothness or lack of angularity. He
corner being geometrically independent of its observed that whilst roundness is an essential
roundness), (3) includes elements of both form and property of a sphere many other forms could be
roundness, for particles with rounded corners will equally well rounded. Wadell's view prevailed.
have a lower x/r ratio than particles with sharp Measurement of roundness has always posed
corners but the same acuteness of angle and number difficulties quite different from those of form
of corners (Fig. 6). For angularity to be a measure of measurement. Although roundness is clearly a
form alone, the distance x', and not x, should be three-dimensional (3D) property all methods of
measured. measuring roundness to date have begun with 2D

@ AZD=O

///
A2~=720
A2~-390

,,--.
\ ,J
b A
Az~=508

\
,--.
. * I
23
A~~-508

'

A ~ =I6
.'
D 25

A2~=1115 A ~ =I340
D
Fig, 6. (a) Construction and formula for determining 2D angularity of a particle by the method of Lees (1964). (b)
Values for degree of angularity (A2D)for various regular figures (mainly from Lees, 1964, Fig. 8, with a corrected value
for the octagon).
300 P. J . Burrett

projections of the particle. A truly 3D roundness inscribed circle to provide a measure of average
measure would involve fitting 3D reference surfaces roundness (Table 4). Swan (1974) has suggested
(such as spheres of varying radii) to all corners on that the limitation on the size of corners was a
the pebble surface. Wadell (1932) believed that matter of convenience to get a measure that would
measurement of roundness from a 2D projection not exceed 1.0, but Wadell’s limitation on what
did not cause serious bias, and everyone since has constitutes a corner is essential for separating the
implicitly agreed, though perhaps largely because corners from the rest of the particle outline. If the
of the impracticality of measuring true 3D round- limit were a straight part of the outline, all pebbles
ness. with convex outlines would consist entirely of
Wadell’s procedure, as well as his concept of corners, some of which would have a radius of
roundness, has survived with little change. The key curvature of almost infinite size. This would cer-
to it is the corner, which he defined as ‘every such tainly be out of keeping with the common notion
part of the outline of an area (projection area) of a corner. Because of the time-consuming nature of
which has a radius of curvature equal to or less than the procedure, Krumbein (1941a) prepared a set of
the radius of curvature of the maximum inscribed pebble images of predetermined Wadell roundness
circle of the same area’. Each corner of the maxi- from 0.1 in steps of 0.1 to 0.9 for faster roundness
mum projection outline is measured with a template determination. The price of faster measurement is
of concentric circles of known diameter by finding the higher subjectivity of the values obtained,
the largest circle that will fit. Most pebbles have because an entire image is being compared, rather
between two and six corners, as defined above; the than a set of single corners one at a time. Another
diameters are averaged and divided by the maximum source of error derives from Krumbein’s images

Table 4. Parameters for estimating roundness. L =longest axis, Z=interrnediate axis, S, =short axis in maximum
projection plane, D,(D,,, Dsz)=diameter of circle fitting sharpest corner (two sharpest corners), D , =diameter of
pebble particle through D,, Di =diameter of inscribed circle, D k =diameter of circles fitting corners, n =number of
corners, C -circumference of Darticle
Author Formula Name or description Range
Roundness of sharpest corners
Wentworth (1919) DdDx Shape index 0-1
Wentworth (1922) D, 0-1

( L+ Sm)P
Cailleux (1947) Ds x 1000 Cailleux roundness index 0-1000
-
L
Kuenen (1956) Kuenen roundness index 0-1
-
DS
Z
Modified Wentworth roundness
Dobkins & Folk (1970)
-
D, 0- 1
Di
Swan (1974) (&I +D,7.)/2 0- 1
Di
II
Average roundness of corners (z Dk)/n Wadell roundness 0-1
Wadell (1 932) lc - = I
Di
[note also pebble comparison charts in Russell & Taylor (1937), Krumbein (1941) and Powers (1953), all of which are
keyed to Wadell roundness]
Average roundness of outline
Szadeczsky-Kardoss Length of convex parts of C
x 100 0-100
(in Krumbein & Total length of C
Pettijohn, 1938)
 Shape of rock particles 301

having corners all of similar curvature, whereas objection. Kuenen modified the Cailleux formula by
many natural pebbles have corners with a range of replacing the long axis with the intermediate axis,
curvatures. This is particularly evident in the three whereas Dobkins and Folk suggested using the
classes of least rounded pebbles. However, as long largest inscribed circle, and Swan proposed a
as the operator compares corners, and not the whole modification of the Dobkins and F o l k procedure
shape, the roundness values obtained should be as by averaging the diameters for the two sharpest
accurate as other limitations of the procedure will corners (Table 4).
allow (Krumbein, 1941a, p. 72). A concept of roundness independent of the
Another pebble comparison chart developed from character of corners was proposed by Szadeczky-
the idea that particle roundness can best be described Kardoss in 1933 (see Krumbein & Pettijohn, 1938;
as a series of five stages of development, i.e. from Miiller, 1967). The measure, p, is the percentage of
angular to very rounded, each stage being charac- convex parts along the circumference of a rock
terized by different features (Russell & Taylor, 1937). particle, and is obtained from an enlarged image
They transformed the data from ordinal to ratio using a measuring wheel, or by measuring the total
scale by keying the stage boundaries to Wadell’s angular distance about the centre of the inscribed
average roundness values, and since then further circle subtended by convex parts of the circum-
modifications have been made. Pettijohn (1949) ference. A complementary measure of angularity
noted that by moving stage boundaries slightly the (sum of angles subtended by plane sides 360”) was
intervals represented a geometric progression. proposed independently about this time by Fischer
Powers (1953) added another class (very angular), (see Krumbein & Pettijohn, 1938).
and F o l k (1955), noting the geometric progression In contrast to Wadell’s roundness, which estimates
in interval size, proposed a logarithmic transforma- degree of curvature, p compares the direction or
tion to give a roundness scale from 1 to 6 (the rho sense of curvature. A visual comparison chart of
scale). However, it is difficult to see how distinct images with values in the range 1 6 9 0 % has been
natural classes of rounding can develop from con- prepared by Sames (1 966). However, roundness
tinuous or even episodic abrasion. If this point is measured in this way is rarely reported in the
conceded and it is agreed that roundness forms a English-language literature. In practice, it is difficult
continuum, then the most useful comparison chart to locate the boundary between convex, planar and
will be the one with the largest number of classes, concave segments of the circumference of the particle
as long as adjacent classes can be distinguished. because they grade into each other. Miiller (1967)
This makes Krumbein’s chart the most satisfactory reported a precision of 3%, but this relates only to
of those available, although the verbal classes of measurement and appears not to take account of
roundness are still useful for purposes of dis- differences in locating the limits of convex
cussion. segments.
The importance of the sharpest point on the out- Each of the three groups of measurements in
line of a particle was first recognized by Wentworth Table 4 estimates a different aspect of roundness.
(1919), who proposed as a measure of shape Roundness estimated from corners has a clearer
the ratio of the diameter of curvature of the sharpest physical meaning than that based on convexity of
corner to the diameter of the particle through that the outline, and while the validity of the latter
point (not the longest diameter, as stated by Dobkins approach is not denied, it is not considered further
& Folk, 1970, among others). He later changed the here. Of the others only Wadell’s procedure mea-
divisor to the average of the long and short diameter sures average roundness, and of the various charts
of the particle in the plane of projection (Went- available for visual estimation Krumbein’s (1941a)
worth, 1922b). Cailleux (1947) proposed a similar is preferred. Of those measures estimating roundness
measure of roundness, namely the diameter of the from the sharpest corner, the Modified Wentworth
sharpest corner to the longest diameter of the roundness (Dobkins & Folk, 1970) is the most
particle(1ongaxis). This measure has beencriticized by desirable. It uses as denominator a length (diameter
Kuenen (1956), Dobkins &Folk (1970), Swan (1974) of the largest inscribed circle) that can be taken at
and Folk (1977), because it confuses both roundness the same time as that of the sharpest corner from
and form in the same measure. It is difficult to under- the image of the rock particle, whatever its scale.
stand the continued use of Cailleux’s measure Swan’s (1974) modification of the measure (by
(e.g. Briggs, 1977) in the face of this substantial averaging the two sharpest points), is unsatisfactory
302 P.J. Barretl

because it yields a value somewhere between the the range 0.25-0.80, is approximately normal
least and the average roundness. If one is going to (Folk, 1972). The data presented here suggest that
measure more than the one sharpest corner one the measure should be used with care where
might as well measure the total of four or five roundness values are low.
needed to calculate Wadell sphericity, a long
established measure with a clear meaning. CONCLUSIONS
Dobkins and Folk (1970) implied that the Modi-
fied Wentworth roundness is more objective and Difficulties in using aspects of shape as geological
quantitative than any other procedure for measuring evidence remain considerable. Even the best of the
roundness, including Wadell’s, but in both respects commonly used procedures have their limitations
it is no better and no worse. The problem of recog- for both description and hypothesis testing because
nizing corners (one of their objections) is inherent their discriminating power is low compared with
in both methods, and it is in fact most vexing for most other sedimentological procedures (e.g. mea-
small angular coarse-grained particles, where it may surement of size, composition, orientation). In
be difficult to distinguish the sharpest corner from a addition, each measure depends at some point on
textural feature. Dobkins & Folk (1970, p. 1170) subjective assessment, whether it be the location of
argued that the sharpest corner provided the best the three orthogonal axes or the fitting of arcs to
measure of roundness, because it ‘best reflects the particle outlines. Further limitation arises from the
amount of rounding going on in the latest environ- observation that shape may be determined by a
ment.’ This is a good point for river pebbles, but in large number of natural variables interacting in
other situations, for example, beneath glaciers, the different ways (Krumbein, 1941b; Kuenen, 1956),
level of rounding attained, rather than roundness but can be assessed so far by only three independent
change since last breakage, may be of most measures of form (sphericity, oblate-prolate index,
interest. angularity) and one independent measure of
Wadell roundness using Krumbein’s visual com- roundness (all three types discussed above are
parison chart is now compared with Modified related).
Wentworth roundness, again using the glacial and The seductive ease with which measurements of
fluvioglacial pebbles from Hooker Valley and form and roundness can be made is in contrast to
Taylor Valley (p. 297). The discriminating power of the difficulties encountered in ascribing a geological
each measure is about the same (Table 5). However, meaning to them. Causal relationships between
whereas the average skewness and kurtosis values aspects of shape and the natural variables that lead
of the Wadell-Krumbein measures are approxi- to them can be established only where the variables
mately normally distributed, values for the Went- are controlled, where the operator error can be
worth measures show them to be consistently assessed and where a large number of measurements
positively skewed, with kurtosis highly variable. can be made (to narrow the confidence interval on
This is not so much a feature of the measure, but of mean values). Confidence in the inferences made
the restricted range in roundness of the particles, about ancient sediments on the basis of particle
close to the lower bound of the measure. The dis- shape studies must also depend to a considerable
tribution of Modified Wentworth roundness values degree on independent knowledge of the natural
for beach and river pebbles, which are typically in variables (rock type, particle size, flow direction,

Table 5. Comparison of Wadell roundness using Krumbein’s comparison chart and Modified Wentworth roundness for
suites of pebbles from the Hooker Valley (20 sets, 597 pebbles) and Taylor Valley (29 sets, 706 pebbles) ,?-mean.
s-standard deviation, Sk-skewness, K-kurtosis
Measure Suite Average A? Range in ,? Average s Total Average Sk Average K
range s ( =O for normal distribution)
Wadell by Hooker 0.41 0.16 4 . 6 3 0-10 4.8 0-06(rt 0.34) 0.18( & 0.43)
Krumbein Taylor 0.33 0.15-O.50 0.10 0-27( rt 0-23) -0-31 ( rt 0.39)

Modified Hooker 0.14 0.02-0.3 5 0.08 4.7 0.96( k 0.40) 1.25( 1.22)
Wentworth Taylor 0.09 0‘02-0.1 9 0.06 1-10(k 0.21) 0.78(? 0.63)
 Shape of rock particles 303

character of flow). Therefore unambiguous inter- KRUMBEIN, W.C. (1941a) Measurement and geological
pretations from shape studies of ancient sediments significance of shape and roundness of sedimentary
can be expected only from carefully planned particles. J. sedim. Petrol. 11, 6472.
KRUMBEIN, W.C. (1941b) The effects of abrasion on size,
studies of rather unusual geological situations. shape and roundness of rock particles. J. Geol. 49,
482-520.
ACKNOWLEDGMENTS KRUMBEIN, W.C. & PETTIJOHN, F.J. (1938) Manual of
Sedimentary Petrography. Appleton-Century Crofts,
Inc., New York.
This paper arose from work supported by the New KUENEN, Ph. H. (1956) Experimental abrasion of pebbles.
Zealand University Grants Committee a n d the 2. Rolling by current. J. Geol. 64, 336-368.
Victoria University of Wellington Internal Research LEES,G. (1964) A new method for determining angularity
Committee o n glacial debris in New Zealand and of particles. Sedimentology, 3, 2-21.
MULLER,G. (1967) Methods in Sedimentary Petrology.
Antarctica. I am grateful t o the Nuffield Foundation
Translated by H. U. Schminke, Hafner Publ. Co.,
and the Scott Polar Research Institute, Cambridge, New York.
for their assistance a n d facilities during preparation PETTIJOHN, F.J. (1949) Sedimentary Rocks. Harper &
of manuscript. Bros, New York.
POWERS,M.C. (1953) A new roundness scale for sedi-
mentary particles. J. sedim. Petrol. 23, 117-119.
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(Manuscript received 24 November 1978, revision received 4 August 1979)