Journal of Archaeological Science 33 (2006) 847e861
http://www.elsevier.com/locate/jas
A crossbeam co-ordinate caliper for the morphometric analysis
of lithic nuclei: a description, test and empirical
examples of application
Stephen J. Lycett*, Noreen von Cramon-Taubadel, Robert A. Foley
Leverhulme Centre for Human Evolutionary Studies, University of Cambridge, Fitzwilliam Street, Cambridge, United Kingdom
Received 5 June 2005; received in revised form 25 October 2005; accepted 31 October 2005
Abstract
Over the last four decades, there has been surprisingly little advance in the quantitative morphometric analysis of Palaeolithic stone tools,
especially compared to that which has taken place in biological morphometrics over a comparable time frame. In Palaeolithic archaeology’s
sister discipline of palaeoanthropology, detailed quantitative morphometric, geometric morphometric, and even 3D geometric morphometric
analyses are now seen almost as routine. This period of relative methodological stasis may have been influenced by the lack of homologous
landmarks on many lithic tools (essential for any comparative analysis), especially core-based technologies of the Lower Palaeolithic. Archaeological field conditions may also prohibit the use of expensive and delicate precision instruments in certain cases. Here we present a crossbeam
co-ordinate caliper that e crucially e both geometrically locates and measures distances between morphologically homologous landmarks upon
lithic nuclei via a single protocol. Intra- and inter-observer error tests provide evidence that error levels associated with the instrument fall within
acceptable ranges. In addition, we present empirical examples of application in the form of a multivariate analysis of 55 discrete morphometric
variables, and a 3D geometric morphometric analysis of co-ordinate landmark configurations derived from Pleistocene lithic nuclei (i.e. ‘cores’
sensu lato). We also introduce to lithic studies some techniques for the study of shape variation that have previously been used with success in
biological morphometric analyses. We conclude that use of an instrument such as the crossbeam co-ordinate caliper may provide a useful adjunct
to traditional techniques of lithic analysis, particularly in developing a quantitative morphometric approach.
Ó 2005 Elsevier Ltd. All rights reserved.
Keywords: Lithics; Morphometrics; Landmarks; Geometric mean; Size-adjustment; Semi-landmarks; 3D geometric morphometrics
1. Introduction
An interest in the variation of stone tool form (morphology)
has long been a central theme of Palaeolithic archaeology.
Such concerns play a fundamental role both in the establishment of typological distinctions, and in assessments of their
validity and meaning. During the Plio-Pleistocene especially,
it is the variation in shape of cores and core-tools (i.e. polyhedrons, discoids, Acheulean handaxes, cleavers, etc.) that form
* Corresponding author. Leverhulme Centre for Human Evolutionary Studies, University of Cambridge, The Henry Wellcome Building, Fitzwilliam
Street, Cambridge, CB2 1QH, United Kingdom.
E-mail address: [email protected] (S.J. Lycett).
0305-4403/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.jas.2005.10.014
the basis of many typological and descriptive schemes (e.g.
[6,8,27]), and discussion of such variation in terms of cultural
traditions, hominin biomechanical and cognitive capacities,
raw material factors, etc. (e.g. [33,56,60 and references therein, 65]). However, since the seminal work of Bordes [6], Roe
[48,49] and Isaac [23] there has been surprisingly little
advance in the quantitative morphometric analysis of such
artefacts, especially compared to that which has taken place in
biological morphometrics (e.g. [4,5,20,29,42,43,67]). Others
[2,53] have suggested that quantitative techniques of shape
analysis employed in biology have moved on to such a great
extent in recent decades, that a veritable ‘revolution’ has taken
place. In recent years, developments have been made toward
the goal of a productive science of lithic shape analysis (e.g.
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S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
[10,18,32,34,40,54,55,65]). However, compared with other
academic arenas it may reasonably be averred that this goal
remains largely unrealised to date.
One potential factor that may have inhibited the development of quantitative approaches to lithic variation is the lack
of morphologically homologous landmarks that can be identified on hominin-modified stone nuclei. (The term ‘nuclei’
here refers to lithic pieces possessing negative flake scars, the
size of which, indicate the piece could at least potentially
have functioned as a ‘core’ for utilizable flake tools, held in
the hand or between fingers.) A lack of homologous landmarks
tends to restrict lithic quantitative variables to a small number
of basic measurements (e.g. length, width, and thickness) on
a limited number of regions; most commonly resulting in
a data set comprised of 11 primary variables (e.g.
[7,36,39]). Moreover, applications of these measurements
have generally been limited to bifacial technologies (e.g.
[10,23,32,49]) and rarely involve the simultaneous analysis of
a wide range of disparate lithic ‘morphs’. Conversely, most biological morphometric approaches can readily employ a range
of morphologically homologous points (e.g. suture junctions,
projections, foramina, etc.) that allow either Euclidean distances to be calculated between pairs of points, or for landmark
configurations to be determined. These data are then readily
amenable to a battery of multivariate statistical approaches,
geometric approaches and, most recently, 3D geometric morphometric techniques. Moreover, these techniques can be applied across a broad range of species variation, enabling both
intra- and inter-specific comparative analyses to be undertaken.
Considering the above issues, the approach taken by Dibble
and Chase [12] remains one of the most interesting regarding
potential solutions to the problems of lithic morphometrics.
Dibble and Chase presented a simple device for capturing a series of 22 bilateral outline measurements from artefacts such
as bifaces or flakes. However, while a high proportion of flake
and biface variability could potentially be described from outline data alone, it is worth emphasizing that lithic nuclei are
3D and important shape information may be under-utilized.
This latter point is particularly important if analyses are to
move away from the traditional trend of comparing bifacial assemblages alone, and a broader array of lithic forms are to be
incorporated within analyses.
Although instruments designed to accurately record the location of pre-defined and readily visible landmarks are now
available (e.g. Microscribe MXÔ, Immersion Corp., San
Jose, CA.), such instruments cannot, in themselves, locate
landmarks; the very problem that may currently be hindering
morphometric lithic research. Moreover, such instruments
present attendant problems associated with electrical circuitry
and delicate mechanics that may be an impediment to those
working under archaeological field conditions [35]. Here we
present a crossbeam co-ordinate caliper that e crucially e
both geometrically locates and measures distances between
morphologically homologous landmarks upon lithic nuclei
via a single protocol. In addition, we present empirical examples of application in the form of a multivariate analysis of
55 discrete morphometric variables, and a 3D geometric
morphometric analysis of co-ordinate landmark configurations
derived from hominin-modified Pleistocene lithic nuclei. We
also introduce to lithic studies some techniques for the study
of shape variation that have previously been used with success
in biological morphometric analyses.
2. Crossbeam co-ordinate caliper design description
and specifications
The crossbeam co-ordinate caliper (hereafter, CCC) essentially consists of a base with marked scales and axes, two upright supports and rulers, and a crossbeam linking the two
uprights, which also has a marked scale along its length
(Fig. 1). All the materials employed are readily available.
Base: the base we used had scales 300 mm in total length,
which together form the four sides of a grid to be utilized as
a co-ordinate system (Fig. 2). These were marked as zero at
the distal and proximal ends of the length axis running down
the centre of the ‘grid’, as well as at the left and right lateral
ends of the width axis. Hence, the intersection of the length
and width axes at the very centre of the base occupies
a zeroezero co-ordinate position. Axes of 45 were also
marked on the base (Fig. 2). The CCC base shown here was
made on a plain background for clarity of illustration, but could
also be manufactured on graph-lined paper to assist with the positioning and measuring of nuclei. Lamination of the base also
aids positioning of nuclei and adds strength for extended use.
Uprights: the uprights consist of two magnets, of a type designed for use in metal-welding (Fig. 1). Within these were set
two steel rules (300 mm in length and 40 mm in width) obtained
from engineer’s squares. The use of magnetic bases enables
metal plates to be placed beneath the base for extra stability during use of the CCC. Slots (4 mm wide and 280 mm long) were
cut into the upright scaled rulers in order to accommodate attachment of the adjustable, sliding crossbeam.
Crossbeam: the crossbeam consists of two half-square, angled alloy strips, each of which measures 10 mm wide, 10 mm
deep and 400 mm long. Placed together as they would in position during use of the CCC, they form a T-shape in cross-section.
Fig. 1. The crossbeam co-ordinate caliper.
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
150mm
150mm
0
0
0
150mm
150mm
0
Fig. 2. Schematic illustration of base with length, width and 45 axes depicted.
Note that the four sides of the base are each 300 mm in length and form the
basis of a grid that can be utilized as a co-ordinate system. The scales are
marked as 0 mm in the centre and 150 mm at the extremities of their range.
The intersection of the axes occupies a zeroezero co-ordinate position.
Holes (4 mm in diameter) drilled into either end of the crossbeam allow it to be bolted to the uprights. The bolts themselves
are 4 mm in diameter and 35 mm in length. Thin nylon washers
were placed directly on either side of the uprights upon the bolts,
so as to aid smooth operation of the CCC when moving the
crossbeam. Springs (18 mm long) placed on either side of the
bolts prior to tightening, ensure the two sections of the crossbeam are firmly squeezed together which assists in accurate
measuring when steel rulers are placed between them (see below). A millimetre scale (300 mm) was placed on the top portion
of one of the crossbeam elements. As with the scales on the base,
this was marked as zero at the centre and 150 mm at either end.
When all the elements are together (i.e. base, uprights, crossbeam) the entire CCC weighs 917 g.
2.1. Protocol for orientation of nuclei prior
to use of CCC
Prior to the recording of measurements using the CCC it is
essential that all nuclei are orientated in a standard fashion,
A
Width
axis
Height axis
B
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such that morphologically homologous features may be compared across different specimens. The protocol employed is
essentially a three-stage process and proceeds as follows.
2.1.1. Orientation and positioning protocol stage 1:
standard orientation of nuclei
Each nucleus is hierarchically oriented firstly by its
‘height’, secondly by its ‘width’ and thirdly by its ‘length’,
each being orthogonal to each other and intersecting at the
centre of the nucleus (Fig. 3A, B). The height of a nucleus
is defined as the minimum distance in any orientation passing
through the centre of its volume. In most cases this primary
axis is readily identifiable. In a minority of cases (e.g. certain
polyhedral type nuclei) this may need to be verified instrumentally via the use of spreading calipers normally applied in craniometric analyses. The width axis of a nucleus is orthogonal
to its height and is defined as the next most minimal distance
through the centre of its volume in the correct orthogonal orientation. In turn, the length of a nucleus is defined as the axis
orthogonal to both the height and width axes.
2.1.2. Orientation and positioning protocol stage 2:
identification of superior and inferior surfaces, and
identification of distal and lateral portions of nuclei
To identify the superior surface of the nucleus, it is held
with one end of the height axis facing directly upward, and
the ‘poles’ of the width and length axes all held in a level
plane. With the nucleus positioned in this manner and a line
of sight taken directly over the top of the nucleus, a view
can be seen which is directly analogous to that which would
appear in a 2D photographic reproduction, with the (hypothetical) end of the height axis pole appearing in the centre of the
photograph. This procedure is repeated for each end of the
height axis. The superior surface is defined on the basis of
the view that has the least amount of cortex visible in the overhead view. Where both sides have approximately equal cortex
remaining, or where no cortex is present, the superior surface
is defined as the most intensely flaked surface. This is identified by counting the number of all visible negative flake scars
(1 cm in length 0.5 cm in width) removed on each surface, whether complete or truncated by other flake scars.
The upper surface is that with the highest number of visible
flakes removed. The logic underlying both these criteria is
that, on average, the most intensely flaked surface will be
Height axis
Length
axis
Fig. 3. Standard orientation of nuclei. Each nucleus is hierarchically orientated first by minimum height, second by width (A), and third by length (B). Each axis is
orthogonal and intersects at the centre of the nucleus’ volume.
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
850
that more extensively modified by hominin agency, and forms
the focal surface for the example analyses detailed below.
Once the superior surface of the nucleus has been identified
the opposing surface is referred to as the inferior surface. To
assist the labelling of metric variables the left side of the nucleus will be referred to as the left lateral portion, and the right
side referred to as the right lateral (Fig. 4A). The position of
maximum width (directly parallel to the width axis) on the
correctly orientated nucleus defines its proximal end if it falls
toward one end of the nucleus away from the width axis, with
the opposing end of the nucleus referred to as the distal end
(Fig. 4B). If maximum width happens to fall directly on the
width axis line (as it might in a perfectly ovate biface) the
proximal end is defined as that which has the greatest sum
of all left and right lateral measurements taken on either
side of the length axis line (see below, and Fig. 6). In a minority of cases where this is not readily appreciable by eye, this
may prohibit firm identification of the proximal and distal aspects of the nucleus until the first series of lateral measurements have been determined. However, since these are
always the first measurements to be taken, it is then a simple
matter to define the left/right lateral portions and proximal/distal portions of the nucleus on the observations derived from
these first measurements.
2.1.3. Orientation and positioning protocol stage 3:
positioning and zeroing a nucleus upon the
CCC for data collection
The nucleus is positioned on the base of the CCC with the
superior pole of the height axis facing directly upward, and the
poles of the width and length axes all held in a level plane. It is
important that the imaginary height pole of the nucleus be kept
directly vertical with at least one point of the inferior surface
of the core in contact with the base of the CCC, even if this is
not the point at which the imaginary height pole would exit the
nucleus itself. In this position the poles of both the width and
length axes should be equidistant from the flat surface of the
CCC base. This orientation is then held firmly in position
and the nucleus is secured to the base of the CCC with plasticine, in a manner analogous to that employed routinely by
A
researchers undertaking 3D geometric morphometric analyses
of biological material (e.g. [63]). It is important that the plasticine does not protrude from the maximum extremities of the
nucleus (when viewed from directly overhead) in a manner
that may interfere with the taking of measurements around
the perimeter of a nucleus.
The nucleus is then ‘zeroed’ such that the distances between the extremities of the distal/proximal and left/right lateral portions of the nucleus are minimized exactly. This is
achieved by positioning the CCC over the nucleus and ensuring, with the aid of steel rulers (Fig. 5), that the distal and
proximal extremities of the nucleus are equidistant from the
intersection of the length/width lines on the CCC base. This
is repeated for the left/right lateral extremities of the nucleus
(Fig. 5). This procedure may have to be repeated several times,
switching between the width and length axes of the base, until
the extremities of the distal/proximal and left/right lateral portions of the nucleus are minimized to an accuracy of the nearest millimetre. It obviously assists the zeroing procedure if this
stage is achieved as far as possible by eye, prior to the final
stage of instrumental zeroing. It is important that the ends of
the imaginary length/width poles identified in the orientation
phase (step 1) of the protocol are kept equidistant from the
base of the CCC, and the imaginary height pole is kept exactly
vertical during the zeroing phase.
2.2. Taking measurements with the CCC
The recording of data with the CCC operates on the principles of cartesian co-ordinate geometry. That is, distances between points of known geometric location can be recorded
and related to the overall dimensions and shape of a lithic nucleus. With a nucleus correctly orientated and secured on the
base, it is possible to position the crossbeam over the nucleus
and take a series of bilateral measurements from the length
line, which bisects the nucleus, to its lateral margins
(Fig. 6A, B). This is achieved with the aid of a steel ruler
B
Distal end
Distal
portion
Left lateral
Right lateral
Proximal
portion
Proximal end
Fig. 4. Standard morphological terminology is applied to each orientated nucleus. The left half of the nucleus is referred to as the left lateral portion, and
the right half as the right lateral (A), distal and proximal portions of the core
are defined accordingly (B) (see text for further explanation).
Fig. 5. Zeroing a nucleus by width. Use of steel rules and the scale along the
crossbeam ensures that extremities of the nucleus are equidistant from zero.
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
A
851
B
Fig. 6. Bilateral measurements taken from the length axis of the base. A total
of 26 measurements were taken at percentage points left (A) and right (B) of
this line.
inserted between the sections of the crossbeam (Fig. 5). The
crossbeam should be set at an arbitrary level height (this can
be kept precise throughout using the scaled uprights) such
that the crossbeam has ample clearance as it passes down
the length of the nucleus. The steel ruler should be set upright,
flush with the extremity of the nucleus’ margin at a pre-designated percentage point along the length line. Thereafter, the
distance from the centre (zero) line to the left and right lateral
margins of the nucleus may be read and recorded using the
millimetre scale along the crossbeam. It is essential that the
end of the ruler is flat with the base and hence at 90 during
this procedure. This can be repeated at a series of percentage
points along the length line (Fig. 6A, B). It should be noted
that bilateral measurements can be taken at any number of percentage points if deemed necessary for a particular research
design. In the example analyses that follow, we take all measurements at the percentage points illustrated in the appropriate
diagrams for each measurement description. Since zero occupies the mid-point of the scale on the crossbeam, it is necessary only to calculate percentages for one half of the length,
as these will be replicated on the opposing portion of the nucleus. The crossbeam is positioned using the scales along the
sides of the base, against which, the uprights are positioned
(Fig. 7). The use of setsquares assists and lends greater precision to this procedure (Fig. 7). With a repositioning of the
crossbeam, this entire procedure may be repeated for a series
of longitudinal measurements taken from the width line of the
base, thus capturing data regarding the extremities of the distal
and lateral portions of the nucleus (Fig. 8A, B). All measurements are rounded up to the nearest whole millimetre.
It is also possible to obtain data regarding the surface morphology of a nucleus, thus enabling substantial data from all
three of its dimensions to be gained. Surface data is collected
by aligning the crossbeam along one of the four major axis
lines drawn on the base, and lowering the beam to a level
height in contact with the surface of the nuclei in at least
one place, whereupon the height of the crossbeam is recorded.
Fig. 7. Using setsquares to accurately align and position uprights along the
scales of the base.
Thereafter, ‘depth’ measurements may be obtained with specially modified steel rules (Fig. 9) at percentage points along
the length of the particular axis under examination
(Fig. 10A). A small metal block is used to ensure that the steel
rule is kept at the 90 angle required for accurate measurements (Fig. 9). For ease, measurements are read from the
top of the bar and rounded up to the nearest whole millimetre
(Fig. 11A). Reading from the top of the bar requires that
10 mm (the thickness of the bar) be removed from the raw
data prior to analysis. At the edges of the core, depth measurements are taken to the lowest point that would be in contact
with a steel ruler if held vertically aside its edge (Fig. 11B).
In the example analyses that follow, measurements were taken
at the same percentage points indicated in Fig. 10A, B. Depth
measurements were also taken along the 45 e225 axis and
135 e315 axis (Fig. 10B). When taking the latter series of
measurements, the nucleus remains secured in its original position, but the crossbeam is placed (with the aid of steel rules)
A
B
10 20 25 30 40 50 60 70 75 80 90
10 20 25 30 40 50 60 70 75 80 90
Fig. 8. Longitudinal measurements taken from the width axis of the base. A
total of 22 measurements were taken at percentage points distal (A) and proximal (B) of this line.
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
852
Fig. 9. Taking depth measurements using modified steel rules. Note that one
corner of steel rule has been removed to enable easier access to any depressions in the nucleus surface. Also note the use of a small steel block to ensure
the steel rule is kept at 90 throughout measurement procedure in order to increase precision.
such that the zero mark of the scale along the top of the crossbeam is located centrally along the ‘length’ of the nucleus in
that particular axis.
These metric data, which in their raw form are essentially
distances (i.e. dimensions) between cartesian co-ordinate
points, may be adjusted and manipulated (see Section 4.1.2)
such that they more effectively express the shape characteristics of lithic nuclei specimens.
3. Intra- and inter-observer error assessment
Methodologies with low intrinsic error and high degrees of
replicability are a fundamental goal of quantitative analyses.
An assessment of intra- and inter-observer error was achieved
following the protocol of White and Folkens [64, p. 307],
which operates on the coefficient of variation (CV) statistic.
A CV is calculated by dividing the standard deviation of
a data set by its mean, and multiplying the result by 100 in order to express deviations from the mean in terms of a percentage [59]. Firstly, to assess intra-observer error levels one of us
(SJL) measured all the attributes described in the previous section on three different lithic nuclei a total of three times, with
one week elapsed between each measuring session. Thereafter,
CVs were calculated from these three measuring sessions for
each individual measurement. For the purposes of this error assessment a lava polyhedral core from Bed II, Olduvai Gorge
(Tanzania), an unprovenanced quartzite Acheulean handaxe,
and a chert Clactonian polyhedral core from Lion Point, Essex
(England) were analysed. These three nuclei were specifically
chosen for their disparity of shape, morphology and raw material. The measurement procedure for these nuclei was repeated
once more by a further observer (NvCT) for comparison with
each of those taken by the first observer. In this case, a CV for
each measurement was calculated for the mean observation of
the first observer and the measurement recorded for that particular attribute obtained by the second observer. By convention, error rates 5% are generally deemed to be acceptable
in such assessments.
CVs obtained by the first observer for the lava core from
Olduvai ranged from 0 to 4.56, with a mean CV of 2.47 for
the total data. When data were compared with the observations
of the second observer this produced CVs ranging from 0.49 to
4.95, with a mean CV of 2.59. Although the mean interobserver error was higher than mean intra-observer error, differences between the two series of CV values were not found
to be significant when CV groups were compared statistically
by Wilcoxon’s signed ranks test (exact p ¼ 0.716) [14,26].
CVs obtained by the first observer for the unprovenanced handaxe ranged from 0 to 4.88 with a mean CV of 2.39. When
these data were compared with observations by the second
observer this produced CVs ranging from 0 to 4.76, with
a mean of 2.42. Although the mean inter-observer error was
again higher than mean intra-observer error, differences between the two series of CV values were not significant when
subjected to Wilcoxon’s signed ranks test (exact p ¼ 0.822).
CVs obtained by the first observer for the Clactonian polyhedral nucleus ranged from 0 to 4.92 with a mean CV of 3.03.
When these data were compared with observations by the second observer this produced CVs ranging from 1.03 to 4.95,
with a mean of 2.89. Hence, in contrast to the previous nuclei,
A
B
0
10
20 25 30 35 40
50
60 65 70 75 80
90 100
0 10 20 25 30 40
50
60 70 75 80 90 100
Fig. 10. (Schematic) Depth measurements taken from the crossbeam to the superior surface of nucleus. A total of 15 measurements were taken at percentage points
along the length axis of the base (A) and 13 depth measurements were taken for the additional three axes of the base (B).
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
A
853
B
Fig. 11. (Schematic) (A) Depth measures are taken with the crossbeam level and in contact with superior surface of nucleus. Measurements are read from the top of
the crossbeam. (B) Taking depth measurements at the edges of nuclei. Measurements are taken at the lowest point in direct contact with steel ruler held vertically
against this edge.
mean inter-observer error was found to be lower than mean
intra-observer error. However, differences between the two
series of CV values were again found not to be significant
when assessed statistically (exact p ¼ 0.10). In sum, both
intra- and inter-observer error fell within acceptable limits,
and even when inter-observer error comparisons produced
higher mean CVs than intra-observer assessments, there is
no evidence that such differences produce statistically significant effects. Such consistency is perhaps not surprising when
the simplicity of operating the CCC is taken into consideration. All an observer needs to do is to be able to co-ordinate
two metric rulers at one time, ensure some basic controls are
adhered to (e.g. that rulers are always at 90 ), and read-off
the measurements accordingly.
4. Discriminant function analysis of morphometric
variables derived from the crossbeam co-ordinate caliper
In order to illustrate some of the potential utility of variables captured by the CCC for the comparison of lithic nuclei
variability, we undertook two separate analyses.
4.1. Materials and methods
A total of 85 hominin-modified lithic nuclei dating from the
Pleistocene were analysed from three different localities. Thirty
nuclei were bifacial handaxes made on quartzite cobbles and
large flakes from Attirampakkam, Tamil Nadu, southeast India
[11,45]. A further 30 handaxes, made on chert, came from
Saint Acheul, France. Typologically, both the Attirampakkam
and St. Acheul material would be assigned to the Mode 2 technocomplex if Clark’s [8] terminology were employed. The
remainder of the sample was comprised of 25 non-handaxe
specimens consisting of ‘chopper’ cores (n ¼ 4), polyhedral
cores (n ¼ 16) and crude discoidal-type cores (n ¼ 5). All
were made on quartzite cobbles and collected from the
Soan River Valley (Siwalik Hills), northern Pakistan [11].
Employing Clark’s [8] terminology, this assemblage may collectively be assigned to a Mode 1 techno-complex.
The limited sample sizes, and perhaps even the collection
history of these lithic artefacts, place constraint upon the nature of interpretations that may be made in the light of the following analyses. However, they do allow statements to be
made regarding the comparative morphologies of the nuclei
that have been measured and the groups concerned, at least
as we have partitioned them for the analyses (i.e. by locality).
The main issue is whether these analyses suggest that employment of the CCC provides data that may be useful in future
analyses of lithic nuclei of the general type examined here.
All the material analysed is under the care of the Cambridge
University Museum of Archaeology and Anthropology, Cambridge, UK.
4.1.1. Discriminant function analysis (DFA)
DFA is a multivariate technique that is used to provide a set
of weightings (i.e discriminant functions) that most effectively
discriminate between groups that have been defined a priori;
these weightings are linear combinations of the independent
variables [21,22,46]. The weightings maximise the probability
of correctly assigning individuals within each group to their
respective groups, thus potentially allowing individuals of unknown group assignation to be grouped with a probability estimation of accuracy. It is also possible to test the effectiveness
of the discriminant function in producing significant differences between the groups, using the Wilks’ lambda statistic [26].
Additionally, DFA ranks variables according to their relative
effectiveness in discriminating between groups. Hence, DFA
can be employed to identify which independent variables are
most important in assigning individuals to groups. The DFA
was undertaken using SPSS v.12.0.1.
4.1.2. Compilation of the metric data set
Metric measurements were recorded as described in Section 2.2 and compiled into a data set composed of 55 variables.
This data set contained 49 size-adjusted Euclidean distance
variables, four ‘coefficients of surface curvature’, a ‘coefficient
of edge point undulation’ and an ‘index of symmetry’ for each
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
854
of the 85 lithic nuclei analysed. The computation and adjustment of these variables are described in the following sections.
4.1.2.1. Size-adjustment of Euclidean distance variables. The
left/right lateral and distal/proximal measurements taken either side of the length and width axes of the CCC base
(Figs. 6 and 8), created a total of 49 Euclidean distance variables. These data were size-adjusted in order to emphasize (allometric) shape differences between individual specimens.
Previous work by Crompton and Gowlett [10,18,19] has
shown that many Acheulean bifaces exhibit allometric (i.e.
size-related) shape differences. An important corollary of
this is that attempts to examine shape differences in lithic
products should attempt to account for the effects of isometric
size prior to the analysis of shape [10,33]. Although lithic archaeologists have given relatively little concern to such matters, attempts to remove the effects of isometric scaling in
order to compare allometrically-scaled shape differences are
now routine in biological morphometric analyses [15,24].
However, some popular methods for size-adjustment have received criticism. Dividing each variable by a single measurement designated to represent ‘size’ (such as maximum
length or height of an individual) has been criticized since it
has been shown not to fully remove correlations with size
[47]. McPherron [33] highlighted the problems of such a method as employed by Wynn and Tierson [65] during their study
of handaxe morphology. Regression-residual based methods
that observe deviations from an allometric regression line
based upon a proxy ‘size’ variable, have also been heavily criticized on the grounds that they are biased by the composition
of the data set [3], adversely affected by outliers [15] and by
the particular line-fitting technique employed to obtain the initial best-fit regression line [3,31].
Given the problems associated with these methods, geometric mean size-adjustment was selected here. This method isometrically corrects for size enabling direct comparison of
allometric shape variation yet, unlike regression based methods, is not dependent upon the composition of the data set.
Size-adjustment via this method has become increasingly
popular in biological morphometric analyses of shape (e.g.
[1,9,28,62]). The geometric mean is one of the Mosimann
family of size variables [37,38]. Like the arithmetic mean,
the geometric mean provides a measure of central tendency,
but it is not as strongly influenced by outliers or deviations
from the modal data. The geometric mean ðGMx Þ may be
computed as:
sffiffiffiffiffiffiffiffiffiffi
n
Y
n
GMx ¼
xi
ð1Þ
i¼1
where xi ¼ individual variables to be size-adjusted, and
n ¼ number of individual variables to be size-adjusted. Simply,
the geometric mean is the nth root of the product of all n variables [24,59]. Size-adjustment of the data proceeds on a specimen-by-specimen basis, dividing each variable in turn by the
geometric mean of all variables for that individual specimen.
This procedure effectively equalizes the volume of all specimens in a sample, creating a dimensionless scale-free variable
while maintaining the original shape information of the data
[15,24]. Size-adjusted Euclidean distance data were supplemented by additional variables labelled and calculated as
described in the following sections.
4.1.2.2. Coefficient of surface curvature. A ‘coefficient of surface curvature’ was calculated by taking the standard deviation
of the depths along an axis, and dividing this by the length of
that axis. Hence, the coefficient of surface curvature emphasizes relative variation over the length of each axis. This measure is necessary since the raw individual depth measures
along each axis are not necessarily correlated with the size
of each nucleus as in the case of the inter-landmark Euclidean
distances. This is readily appreciable conceptually if one visualizes the case of a large flat nucleus with a high geometric
mean yet which, due to its flat superior surface, has relatively
small depth measures. Such a situation can be contrasted with
the case of a far smaller nucleus, which is highly domed on its
superior surface and hence has relatively high depth measures.
Four coefficients of surface curvature were calculated here for
the length, width, 45 e225 and 135 e315 axes.
4.1.2.3. Coefficient of edge point undulation. A ‘coefficient of
edge point undulation’ was determined by computing the standard deviation of the depths at the endpoints of the length,
width, 45 e225 and 135 e315 axes (i.e. eight variables
in total) and dividing this by the geometric mean of the lengths
of the four axes. (Note that any differences between the height
settings of the bar when taking the initial endpoint depth measurements across each axis (see Section 2.2) need to be determined and accounted for prior to calculation of the
coefficient of edge point undulation.)
4.1.2.4. Index of symmetry. Symmetry is not a major focus
here, for which rigorous methodologies (at least for biface outlines) are now available [54]. However, it is possible to create an
‘Index of Symmetry’ from the data collected here that, as part of
a multivariate analysis, allows relative bilateral symmetry to be
assessed. The index of symmetry may be computed as:
0qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
2
n
X
ðXi Yi Þ
@
A
Xi þ Yi
i¼1
ð2Þ
where Xi ¼ the width value left of the length line taken at a particular percentage point, Yi ¼ the width value right of the
length line taken at the corresponding percentage point, and
n ¼ the number of percentage points at which Xi and Yi are
taken. Hence, a value of zero would correspond to perfect
bilateral symmetry.
4.2. Results of DFA
for
Fig. 12 shows a plot of the DFA scores (functions 1 and 2)
individual lithic nuclei from the Soan Valley,
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
a quantitative manner. These results suggest that important aspects of shape variation in lithic nuclei are not being quantitatively analysed by traditional methodologies.
DFA (along with other related multivariate techniques such
as principal components analysis and canonical variates analysis) is the type of methodology that, despite being widespread in
other sciences and morphometric analyses, has infrequently
been applied to morphometric variables of lithic nuclei. This
may be due to a paucity of such variables captured by traditional
techniques, especially outside of bifacial assemblages. However, the DFA performed here employing the CCC, provides
more than enough hints that future analyses of these types of
data would be fruitful. It is important to note that data obtained
from the CCC could also be supplemented by data taken by traditional methods (e.g. flake scar counts, flake scar size, position
and percentage of cortex, etc.). The introduction of new metric
variables should be seen as useful adjuncts to traditional data,
not necessarily as direct replacements [12].
4
2
0
Function 2
855
-2
-4
-6
-6
-4
-2
0
2
4
6
8
Function 1
Fig. 12. Discriminant functions plot for 55 variables recorded on material from
Soan Valley, Pakistan (B), Attirampakkam, India (>) and St. Acheul, France
(*). Differences between centroids (-) are significant (Wilks’ lambda ¼ 0.21,
df ¼ 108, p < 0.0001) on DF 1. The six variables most highly correlated with
DF 1 were the coefficient of surface curvature along the width axis, the index
of symmetry, the left lateral measure at 40% of width, left lateral measure at
35% of width, the coefficient of edge point undulation, and the coefficient of
surface curvature for the 135 e315 axis.
Attirampakkam and St. Acheul. Under the DFs produced,
98.8% of the nuclei would be correctly assigned to their
a priori defined group. On the first DF (88.4% of variation)
the centroid of the Soan Valley Mode 1 cores is clearly differentiated from those of the Attirampakkam and St. Acheul
Acheulean bifaces. Differences between centroids on this DF
are significant (Wilks’ lambda ¼ 0.21, df ¼ 108, p < 0.0001).
Differences between centroids on the second DF (11.6% of
variation) were not significant (Wilks’ lambda ¼ 0.338,
df ¼ 53, p ¼ 0.231). The six variables most highly correlated
with DF 1 were the coefficient of surface curvature along
the width axis, the index of symmetry, the left lateral measure
at 40% of width, left lateral measure at 35% of width, the coefficient of edge point undulation, and the coefficient of surface curvature for the 135 e315 axis.
4.3. Discussion of DFA
It is notable that at least three of the variables out of the top
six that load most highly on DF 1 are precisely the type of variables that have not previously been quantitatively examined
in lithic nuclei (i.e. the coefficient of surface curvature along
the width axis, the coefficient of edge point undulation, and
the coefficient of surface curvature for the 135 e315 axis).
Even an attribute such as symmetry, the second most highly
correlated variable with DF 1, has not previously been examined between Mode 1- and Mode 2-type assemblages in
5. 3D geometric morphometric analysis of surface
morphology
Given the protocol for acquiring Euclidean distance measures from lithic nuclei described above, it is possible to extend
the methodology to derive 3D cartesian co-ordinates from the
raw data collected using the CCC. The dearth of landmarks
available on lithic nuclei is particularly problematic in the
case of 3D co-ordinate analyses that crucially depend on such
data. Even a traditional measurement, such as ‘maximum
width’, which may be considered a homologous feature when
employed as a Euclidean distance, does not provide two discrete
homologous landmarks at each of its termini if the relative position of the measurement can vary widely, or can occur at different points along a nuclei and hence is not represented by a unique
pair of landmarks [4 (p. 2),66 (pp. 174, 178, 179)]. In such circumstances, only geometrically defined landmarks that remain
morphologically homologous to each other regardless of shifting morphology can be employed in 3D analyses. Such geometrically defined landmarks would be termed ‘semi-landmarks’
under Bookstein’s [5] revised landmark terminology.
Fig. 13 illustrates 51 landmarks on the lithic nuclei for
which information on their location in 3D space can be derived. The principle of deducing (x, y, z) co-ordinates (hereafter referred to as ‘landmarking’) using the CCC is based on the
understanding that the origin (0, 0, 0) lies at the intersection of
the orientation lines on the base of the CCC. The X-axis follows the width line of the CCC base (i.e. the 90 e270 line), which bisects the nucleus into its proximal and distal
portions The Y-axis follows the length line of the base (i.e.
the 0 e180 line), dividing the nucleus bilaterally into its
left and right halves. The Z-axis projects orthogonal to these
from the base and is assumed to bear positive values when
landmarking the superior surface of the lithic nucleus. The
X- and Y-axes bear both positive and negative values, as in traditional geometric axis (see Fig. 13). The 45 e225 line on
the base also yields geometric landmarks and is read from
the right distal quadrant to the left proximal one (following
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
856
Positive
Conversely, landmark co-ordinates are zero-dimensional and
only contain information on form when presented and analysed as a configuration, where relative positioning of landmarks holds useful ‘shape’ information.
5.2. Materials and methods
Negative
Y 0.00
ve
ati
g
Ne
Z
ive
sit
Po
0.0
0
Negative
0.00
X
Positive
Fig. 13. Configuration of 51 landmarks employed for the 3D analysis of nuclei
surface morphology. All landmarks are located in 3D space along four lines of
CCC base orientation, where X ¼ 0 (width), Y ¼ 0 (length) and two axes where
X ¼ Y (45 e225 ; 315 e135 ). Landmark 8 occupies the X, Y origin (0, 0,
Z ) and is derived from measurements taken along the 0 e180 length axis.
The core can be divided into four 2D quadrants, based on segments left and
right of the Y-axis, and proximal and distal to the X-axis.
nucleus orientation rules). Similarly, the 135 e315 line is
read from the left distal quadrant to the right proximal one.
5.1. Geometric morphometrics
Geometric morphometrics is an analytical approach to
shape analysis that operates on co-ordinate data in a nonEuclidean shape-space [25] for which the geometric and statistical properties are well defined ([43] and references therein).
Co-ordinate based shape analysis has already gained widespread support and recognition amongst biologists and palaeontologists. Therefore, a detailed explanation of the
methodologies involved will not be given here. Those unfamiliar with geometric morphometrics should find the following
publications of interest [4,13,29,30,42e44,50e52,67]. Geometric approaches offer advantages in terms of statistical robusticity [50,51] over other methodologies including the
multivariate analysis of morphological variables such as outlines (Elliptical Fourier analysis) and Euclidean distances.
However, data of the latter varieties contain information on
the dimensions of an object and therefore represent discrete
morphological properties. Discrete data may be especially important in some types of analysis if specific elements of nuclei
shape need to be compared directly in a particulate manner.
The 85 nuclei employed in the previous DF analysis were
utilized for the 3D analysis (see Section 4.1). Employing the
same positional and depth data as that collected for the previous analysis, x, y and z co-ordinates were determined for each
of the 51 landmarks illustrated in Fig. 13 using the protocol
described in Appendix. The raw (x, y, z) co-ordinates for the
85 lithic nuclei were compiled into a single input file for the
geometric morphometrics package Morphologika, available
online at http://www.york.ac.uk/res/fme/resources/software.htm
[41,42,44]. This software minimizes scaling, translational and
rotational differences by Generalised Procrustes Analysis
(GPA) [4,16,17,44,52] and conducts a Principal Components
Analysis (PCA) on landmark residuals. PCA is a multivariate
procedure that identifies and extracts uncorrelated variables
(i.e. principal components) from a set of inter-correlated raw
variables, such that the major variation between specimens
can be described and quantified [14,46,57]. Of the principal
components extracted, each consecutive PC explains less of
the original variation.
5.3. Results of 3D geometric morphometric analysis
Fig. 14 shows the plot of the first two principal components
(PCs) from the 3D geometric analysis of the Soan Mode 1 assemblage, Attirampakkam handaxes and the St. Acheul handaxes. The wireframe diagrams at the termini of the PC axes
illustrate the ‘shape’ occupying the extreme position in PC
shape-space and do not necessarily refer to any specific nucleus
morphology. By moving along an axis it is possible to visually
detect the shape changes that characterise the particular PC. In
PC1 (36.8% of the total variation), the extreme negative terminus of the axes is characterised by domed and more circularshaped nuclei, while the extreme positive end is occupied by
very flat and pointed nuclei. PC2 (17.5% of total variation)
also contains considerable information about the shape differences amongst these assemblages. The negative end of this
range is characterised by flat, rounded nuclei and is dominated
by specimens belonging to the Mode 1 Soan assemblage,
while the positive end is dominated by domed and pointed individuals belonging to the handaxe assemblages. It is interesting to note that the two bifacial assemblages differ more on
PC2 than on PC1, indicating a slight separation between individuals that are relatively more ovate and flat, compared with
individuals pointed in outline and more domed in crosssection.
Fig. 15 illustrates PC1 against PC3 from the same analysis.
PC1 contains exactly the same information as in the previous
figure, but PC3 (8.3% total variation) appears to exhibit differential bilateral asymmetry. The negative terminus of PC3
expresses lateral flattening towards the left side of nuclei,
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
857
0.18
0.16
PC2
0.14
0.12
0.10
0.08
0.06
0.04
PC1
-0.28
-0.24
0.02
-0.20
-0.16
-0.12
-0.08
-0.04
-0.02
0.04
0.08
0.12
0.16
0.20
-0.04
-0.06
-0.08
-0.10
-0.12
-0.14
-0.16
-0.18
Fig. 14. Plot of PC1 and PC2 for the 3D geometric morphometrics analysis of 51 surface landmarks. Soan Mode 1 (B), Attirampakkam handaxes ( ), St. Acheul
(A). Superior and lateral views of wireframe diagrams representing the extreme shapes at termini of axes are included to aid interpretation of shape differences
between groups.
whereas the positive end describes lateral flattening on the right
side of nuclei. While both the Soan assemblage and the bifacial
assemblages occupy some space on PC3, the Soan assemblage
occupies a much greater range, indicating a greater tendency for
these nuclei to exhibit bilateral asymmetry. Although the bifacial assemblages also exhibit some degree of asymmetry, they
do so to a lesser degree indicated by their clustering around
the low positive and low negative values of PC3. However,
PC3 contains additional information regarding edge points
that would account for the source of this asymmetry that could
not be expressed by outline analysis alone. This is visible in the
side-view wireframe diagrams for PC3 (Fig. 15), which indicates that the left and right lateral flattening of outline appears
to correlate with relatively lower edge points.
5.4. Discussion of 3D geometric morphometric analysis
The changes along each of the PCs derived from the 3D
geometric morphometric analysis essentially reflect the morphological variables that best distinguish individual nuclei
and, ultimately, assemblages. Hence, the Soan Mode 1 nuclei
are broadly distinguishable from the bifacial assemblages by
being more domed in terms of their surface convexity and being round in their general outline profile on PC1. In contrast,
bifaces are generally much flatter and more elongate. The handaxe assemblages appear to vary between relatively more
ovate and flat surfaces at one extreme of their morphological
range, to pointed outlines and more rounded surface morphologies at the opposite end of their shape variation (PC2). PC3
appears to account for bilateral asymmetry due to a flattening
of outline profile, that also appears to be explained by a correlation between flatness of outline and relatively lower edge
points along the sides of the nuclei. Such quantification procedures, both within and between assemblages of these types,
may have implications in more detailed analyses, which
attempt to investigate potential relationships between shape
variation in lithic nuclei and factors such as raw material,
reduction intensity, regional differences, etc.
In future studies, this protocol could be extended in a number of ways. For instance, it is possible to acquire more information along any of the four base axes described here by
further sub-division into more percentage points. Moreover,
by recording the edge-point depths of the lateral measurements from the length and width lines, it would be possible
to gain more detailed outline information. Such additional
landmarks may prove useful to researchers interested in outline symmetry, outline regularity and/or undulation across
specimens. Finally, by recording Euclidean distances on the
inferior side of the nucleus, it is possible to combine data
from the superior and inferior surface in a cross-platform morphometrics software program such as Morpheus [58] to create
an entire 3D representation of any lithic artefact. One such
representation, employing the latter two of these potential
methodological extensions, was created for 118 landmarks
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
858
0.12
0.10
PC3
0.08
0.06
0.04
0.02
PC1
-0.28
-0.24
-0.20
-0.16
-0.12
-0.08
-0.04
-0.02
0.04
0.08
0.12
0.16
0.20
-0.04
-0.06
-0.08
-0.10
-0.12
-0.14
-0.16
-0.18
Fig. 15. Plot of PC1 and PC3 for the 3D geometric morphometrics analysis of 51 surface landmarks. Soan Mode 1 (B), Attirampakkam handaxes ( ), St. Acheul
(A).
recorded on both the superior and inferior surfaces of
a handaxe from St. Acheul, and is illustrated in Fig. 16. Although this is included here for purely graphical purposes, extension of the geometric methodology could result in full-scale
analyses of the 3D morphology of lithic nuclei. Moreover,
with this type of methodological extension it would be
Y
0.00
Z
X
0.00
Y
X
Z
Fig. 16. Full-scale 3D representation of a St. Acheul biface, combining a total
of 118 landmarks on both the superior and inferior surface of the lithic
nucleus.
possible to conduct a 3D investigation of biface (a) symmetry,
which has hitherto been impossible.
6. Discussion and conclusions
D’Arcy Thompson [61, p. 269], one of the pioneers of biological morphometrics, once stated that in science ‘‘we begin
by describing the shape of an object in the simple words of
common speech: we end by defining it in the precise language
of mathematics; and one method tends to follow the other in
strict scientific order and historical continuity’’. We suggested
at the outset of this paper that a fully developed and productive
science of quantitative lithic shape analysis was still far from
realised. While there are encouraging recent signs of development in lithic circles, it is doubtful whether the same degree of
revolutionary advancement has taken place in our own field as
that of other disciplines. Indeed, it may even be argued that
much of lithic shape analysis remains rooted in the potentially
ambiguous language of qualitative description (i.e. the earliest
stage of D’Arcy Thompson’s sequence of scientific
development).
The example multivariate and 3D geometric morphometric
analyses we present here suggest that interesting aspects of
shape variation in Pleistocene lithic nuclei could be collected
and analysed by employing a device such as the CCC. Such
attributes are largely unanalysed via traditional methodologies
of data collection, yet remain important in testing ideas concerning the potential significance of lithic shape variation, be
it due to cultural differences, raw material, reduction intensity,
etc. Moreover, the instrument we describe is lightweight,
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
portable, inexpensive, easy to operate, and yet accurate
enough to recover meaningful and interpretable results. The
methodology for data collection is flexible enough to be adjusted to collect a greater or lesser quantity of data, or target
more attributes in detail, if specific research aims deemed
this appropriate. It is worth repeating that any data collected
by the CCC can be augmented by additional data collected
by traditional techniques of lithic analysis. It may even be possible for researchers working in different regions to build directly comparable quantitative databases of lithic attributes,
enabling greater scope for inter-regional or inter-continental
research programmes to be undertaken.
Acknowledgments
We are especially indebted to Norbert von Cramon-Taubadel for helping to turn our paper scribbles and ramblings into
material reality. Chris Clarkson, Mark Collard and Marta
Mirazón Lahr provided valuable conversations. We are also
grateful to Parth Chauhan and John Grattan for helpful and
perceptive comments. Access to lithic material and hospitability
during data collection was gratefully received from Anne
Taylor, Assistant Curator, Cambridge University Museum of
Archaeology and Anthropology. SJL is supported by a Trinity
College, University of Cambridge Research Scholarship.
NvCT is supported by St John’s College, University of Cambridge and a Gates Trust Scholarship.
Appendix A. Deriving landmarks
0 e180 Landmarks 1e15 lie along the length line of the base (0 e
180 ), which corresponds to the Y-axis (see Fig. 13). Hence,
they all have x-values of 0. The y-values are the actual percentage values along the length line and the z-values can be calculated from the depth measurements recorded below the
crossbeam at those percentage points. For example, if the overall length of a nucleus were 120 mm, then landmark 2 (corresponding to 10%) would lie at 12 mm along the line. This
landmark might have a depth of 3 mm below the crossbeam, positioned at 60 mm above the base. The x-value in this case is 0;
the y-value is 12 and the z-value can be computed as:
60 3 ¼ 57, giving a co-ordinate of (0, 12, 57). Landmark 8,
which lies at the X, Y intersection, has x- and y-values of 0,
and a z-value calculated from the depth measurement at that
point. Landmarks 9e15 differ by having negative y-values.
90 e270 Landmarks 16e27 lie along the width line of the base
(90 e270 ), corresponding to the X-axis (Fig. 13). They all
have y-values of 0. The x- and z-values for these landmarks
are calculated as before, with x-values corresponding to the
percentage values along the width line and z-values being
computed from depth measurements as above. In this case
landmarks 16e21 have negative x-values.
859
45 e225 and 135 e315 The 45 e225 and 135 e315 base line landmarks (28e
51) differ slightly from the other landmarks in their deduction
protocol. This is because the nucleus is zeroed relative to the
length (Y-axis) and width (X-axis) lines only (see Section
2.1.3). When the depths are taken along the 45 e225 and
135 e315 lines, the CCC crossbeam is moved relative to the
base, such that the zero of the crossbeam and the intersection
of the base axes no longer align. This movement of the CCC
along these base axes must be accounted for when deriving
the landmark co-ordinates. The following protocol is followed:
(1) The CCC is lined up along the 45 e225 (or 135 e315 )
line such that the zero of the crossbeam and the intersection of the base axes are aligned. The distance to both
edges of the nucleus edges along the 45 e225 (or
135 e315 ) line are recorded.
(2) The CCC is moved along the 45 e225 (or 135 e315 )
line such that the nucleus edges (corresponding to landmarks 28/39 or 40/51) are equidistant from the zero of
the crossbeam. Percentages along the total length are calculated and the depth values at these percentages are recorded.
(3) The x- and y-values for the distal endpoints (landmarks 28
and 40) are calculated first and then all other landmarks
along the line can be deduced. Any point which lies on
a line at a 45 angle to the X, Y plane, has equal x and y
co-ordinate values, assuming equal scaling of the axes.
The length of a line from a point (xi, yi) to the origin (0,
0) in 2D space is equal to:
D¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
ðxi 0Þ þðyi 0Þ
But since xi ¼ yi on the 45 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
2
D ¼ ðxi 0Þ þðxi 0Þ
qffiffiffiffiffiffiffiffiffiffiffiffi
D ¼ 2ðxi Þ2
2
D ¼ 2ðxi Þ
ðA1Þ
2
D2 ¼ x 2
i
2
rffiffiffiffiffiffi
D2
xi ¼
2
where D equals the distances from the zero of the crossbeam to the distal edge of the nucleus prior to equalizing
the distances (as in step 2 above).
Example. If the distal edge corresponding to landmark 28, is
60 mm distant from the zero of the crossbeam prior to realigning
p
ffiffiffiffiffiffiffiffiffiffiffiffi the CCC, the x- and y-values of landmark 28 are
602 =2 ¼ 42:43. The z-value will be the depth measurement
taken at the edge. Therefore, if its value is a depth of 15 mm
taken from a crossbeam at a height of 63 mm the z-value will
860
S.J. Lycett et al. / Journal of Archaeological Science 33 (2006) 847e861
be 48. Given that X ¼ þY in the right distal quarter, the coordinate for this landmark is (42.43, 42.43, 48).
(4) Once the distal edge landmark has been calculated (where
xi ¼ yi), the (x, y) co-ordinates for all other percentage
points (x) along the 45 e225 (or 135 e315 ) line are
computed as:
sffiffiffiffiffiffiffiffiffiffiffiffi
ðaDÞ2
x¼
xi
2
ðA2Þ
where D ¼ total length of the 45 e225 (or 135 e315 )
line from endpoint to endpoint. a ¼ percentage point along
the axis line as a decimal (e.g. 10% ¼ 0.1)
As before, the z-values are calculated as the height of the
crossbeam in that orientation minus the depth taken at that
percentage point. The only other aspect of the protocol is to
assess whether X ¼ Y in that quarter (see Fig. 13), and
thereby assign negative or positive prefix signs to the x- and
y-values.
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