Techniques and tools for measurement of fabricatory depth
Gordon Rugg, October 2004
draft in progress
Abstract (light version)
It’s clear that some technologies are more complex than others, but trying to put a number on that
has traditionally been viewed as tricky. One way that you can handle this is to use a branch of
maths called graph theory. For instance, you can use graph theory to represent the layers of tools
you need to make the tools to make the tools (and so forth) that you need to make a given item.
You can then do some simple but powerful things like counting the number of layers, or the
number of tools within a given level. This then lets you do interesting things (well, some people
find them interesting) such as seeing whether hominids such as Neanderthals had technologies
with fewer levels than modern-style humans contemporary with them. You can also assess how
stable a culture’s technology is, and how dependent it is on a given material or tool; you can do
various other things, such as seeing which technologies pre-adapt a society for adopting a given
You can also do something pretty similar with the layers of explanation required to explain
abstract concepts (such as technical or subjective terms), to assess the conceptual sophistication of
a given discipline; this is known as elucidatory depth. If you’re feeling ambitious, you can also
measure the ripple effects from a given restructuring of concepts within a discipline (no, they
don’t ripple out forever) as a more sophisticated way of representing “paradigm shifts”. Neither
of these is described in any detail in this paper, but it should be fairly clear how you map the
concepts across from this paper to these concepts. The Rugg & McGeorge paper on laddering, in
“Expert Systems”, gives some more detail on this, but it’s a brute of a paper, so be warned… I’m
working on a more user-friendly paper about elucidatory depth and paradigms.
Abstract (heavy version)
Although the concept of complexity and hence of development of technologies is of central
importance to the study of innovation and technology, methods for describing technologies have
historically been qualitative rather than quantitative.
This paper describes the application of graph theory to the description of technologies, to produce
a formal, rigorous methodology which produces both quantitative and qualititative results in a
form which is computationally tractable and amenable to the formulation of testable hypotheses.
The method is applied to an extended case study (the manufacture of axes) to demonstrate its
use and implications.
It is concluded that this approach provides a useful tool for archaeology, cultural anthropology,
and studies of technological change and innovation.
Although the concept of technological complexity is at the heart of the idea of technological
development, previous attempts to describe technologies in terms of their complexities have been
primarily qualitative. The problems encountered in describing this area have led to
understandable caution among researchers: for instance, the concept of the "primitive society" has
long been abandoned, and instead terms such as "technologically primitive society" are used,
though often with visible unease. Given the difficulty of defining what is "technologically
primitive" in qualitative terms, this unease is understandable.
One way out of the existing impasse is to import methods from other areas. One suitable
approach, described here, is graph theory. Graph theory provides a formal mathematical
framework for describing graphs, in the sense of graphical network diagrams.
Figure 1: a simple graph
This graph has three levels. At the top level is a single node, shown by a rectangle. This node is
joined by two arcs (the technical term, even if the lines are actually straight) to the two nodes
below it. Each of these nodes is in turn joined to two more nodes on the bottom level of the graph.
Graph theory can be readily applied to the description of technologies because of the way in
which manufactured items are produced. An item is produced using techniques, tools and
materials, each of which may in turn require the use of other techniques, tools and materials, and
so on recursively until the sequence "bottoms out" in raw materials which do not require any
previous tools, materials or processing. This type of sequential decomposition is extremely
amenable to a graph theory representation, which then provides a variety of mathematical
metrics to describe the resulting graph.
A simple example is the concept of "fabricatory depth" (Rugg & McGeorge, 1995). This consists of
counting the layers of tools (or materials, or processes) needed to make the tools to make the tools
etc. to make the item. If there are two layers, then the item has a fabricatory depth of 2; if there are
seven layers, then the item has a fabricatory depth of 7. The implications of this are obvious: it is
possible to describe two or more technologies in terms of the fabricatory depth of the items in
those technologies, and then to compare the results in quantitative terms. A society can then be
described, for instance, as having several level 3 technologies and one level 4 technology. The
advantage of this method over previous definitions of levels of technology is that it is descriptive
rather than prescriptive: it does not depend on an a priori definition of what constitutes a "higher"
or "lower" technology. Comparison with the famous debate about what constitutes a civilisation is
instructive: instead of arguing about whether or not production of grape wine or barley beer is an
indicator of civilisation, researchers can simply count the levels of fabricatory depth in the
technologies available to that society.
Fabricatory depth is the maximum number of levels of fabrication needed to produce the item.
Fabricatory width is the maximum number of entities involved at any single level of fabricatory
Fabricatory size is the number of fabricatory entities needed to produce the item.
The distinction between breadth, width and size is useful for capturing the difference between
e.g. a simple process which uses a large number of raw materials, and a complex process using
the same number of entities but with much greater fabricatory depth.
• process (public domain or expert);
Expert process: a process involving skills or knowledge which are not known to everyone within
Public domain process: a process involving skills or knowledge which are known to everyone
within the society.
Primitive: an entity which cannot be further decomposed. (Note that an entity may be a primitive
in one technology or a composed entity in another. For instance, in handaxe manufacture "flint" is
a primitive, whereas for the manufacture of polished flint axes it is a composed entity, obtained
by mining using specialist techniques and tools.)
Composed entity: an entity which can be further decomposed.
Black box entity: a composed entity which is produced by one technology and then used "as-is"
in another technology.
Necessary entity: an entity which is a necessary part of the manufacture of the end product (e.g.
tin for the manufacture of bronze).
Contingent entity: the entity which happens to be used as the instantiation of a necessary entity
within a particular technology. For example, shaft mining involves the necessary entity of a
means of letting miners get in and out of the shaft; contingent ways of achieving this include
notched log ladders, runged ladders, rope ladders, etc.
Pre-adaptation: an entity originally produced for one purpose which can also be used for other
It is in principle possible to produce a complete description of all the techniques, tools and
materials used in a society's technology. This allows the use of more complex metrics such as
technological size (the number of entities in the society's technology) and technological
interdependence (the number of entities used in the manufacture of more than one end product).
Thus one society, for example, may have a relatively large number of tools, none of which shares
raw materials, processes or precursor tools with any of the others, whereas another society may
have a relatively small number of tools which are all dependent on the same set of materials,
processes and precursor tools. The latter is clearly likely to be more sensitive to perturbation than
the former if there is any change in availability of any of the entities involved.
Although in principle it is possible to represent an entire technology on a single graph, in practice
this produces complex graphs which are hard to understand and modify and which require a
large amount of paper. (This approach is eminently amenable to computerisation, which has
many advantages over manual representation. For clarity, the description here concentrates on
The simplest approach is to draw a separate graph for each tool and material used in the society,
showing the tools, techniques and materials used to produce it. Each tool or material and
corresponding graph should be uniquely labelled. When manufacture of a tool or material
requires use of another tool or material, then that tool or material can be identified on the graph
by its name and its own graph number. This keeps individual graphs down to a tractable size.
The completed graphs can be analysed in various ways. The metrics described above can be used
to provide a quantitative analysis of the technology. For a more impressionistic overview, the
graphs can be attached to a suitably large surface, with cross-links between them indicated by
coloured wool. This makes it possible to see at a glance to what extent any given entity is used
across the technology. For instance, in modern western society, copper and gold would be have
numerous links because of their use in electronics and electricity, whereas bone would have very
few links. In mediaeval western society, although all three materials were known, bone would be
much more widely linked to different materials and products than either copper or gold.
The case study examines the manufacture of four different types of stone artefact: handaxes,
tranchet axes, polished flint axes made from surface flint, and polished flint axes made from
mined flint. For present purposes the difference in use of handaxes and later true axes is of
secondary importance; the primary focus is on means of manufacture. The method will be
extended to metal axes, to demonstrate the principle.
Handaxes were not used in the same way as modern “true” axes: analysis of wear patterns on
handaxes has shown that were used as multi-purpose tools, usually for cutting rather than
hacking. They’re particularly good for butchering large game.
Early handaxes were made using hard hammer technique, as shown in figure 2. The raw material
was a flint nodule, probably found in a riverbank, as opposed to being mined. The only other
tool needed was a roundish pebble of some stone other than flint. This pebble is usually hard
(hence “hard hammer technique” but may be quite soft. Soft hammers (usually antler or dense
wood billets, but sometimes soft stone pebbles) allow the removal of finer flakes, and therefore
the production of more finely-shaped artefacts. Many modern knappers also use a piece of leather
to protect their hands (freshly broken flint is sharper than a razor), but many others don’t,
claiming that they get better control that way. The cuts aren’t usually deep, and don’t hurt much.
If we disregard the leather, this gives us one layer underneath the final artefact “handaxe”; the
layer contains one material (“flint nodule)”and one tool (“round pebble”).
Later stone axes, such as tranchet axes and polished flint axes, were used similarly to modern
axes. They tended to be made of better quality flint than handaxes, since otherwise they were
prone to shattering. Flint varies in quality; flint picked up in a field will probably have been
exposed to frost, which makes it prone to shattering. Flint freshly exposed in a riverbank
probably won’t have been exposed to frost, but there are obvious limitations to this as a source of
material. The best British flint is found some distance (about twenty feet) underground, and was
mined by Neolithic people, who dug straight through layers of less deep, but less high quality,
flint to get to the best material. They were made by knapping the basic shape, and then finishing
them. Tranchet axes were finished with a single, highly-skilled transverse blow to create a cutting
edge; polished flint axes were finished by grinding them against sand and/or a rock for hours
(about two to twenty, depending on how fine a finish you wanted and how good the basic shape
(end of light version)
Figure 2: Fabricatory graph for lower Palaeolithic handaxe
flint nodule round rock
Figure 3: Fabricatory graph for polished flint axehead
flint polishing hard antler
nodule stone hammer
Note that the hard hammer is used both in its own right (to rough out the flint nodule into the
right general shape) and also to make the soft hammer (by breaking the tines off an antler, so the
remaining antler shaft can be used as a soft hammer).
Figure 4: Partial fabricatory graph for mined flint nodule
(first two levels, and part of third level)
pick ladde rope shove basket lamp
Note that this only goes three levels down, because of clarity requirements. I’ve fully unpacked
only the pick: the other second level entities can also be unpacked. I’ve indicated this with small
boxes with small arcs underneath them, but have not filled in the details, because of space
constraints. The lamp, for instance, would be unpacked into a chalk nodule, a burin to work it
with, a knife or scraper to cut up fat or tallow, and a wick; the wick in turn would also need to be
unpacked, for instance into a stripped length of rush. The main point about this diagram is that
it’s clearly much larger and deeper than the graph for a handaxe. There are other interesting
points, such as a loop involving the ladder; attested examples of ladders include tree trunks
worked to shape with an axe (so an axehead will be needed to make the ladder which is needed
to extract the flint to make the axhead). I haven’t represented that in this diagram, for clarity.
Figure 5: Fabricatory graph for bronze axhead
(To be completed… will contain mining for copper; smelting copper; bellows; forge; moulds for
The examples above demonstrate how graph theory can be used to represent fabricatory depth.
The representations used above have deliberately been kept simple, and anyone wanting to use
this approach will probably want to use more sophisticated representations.
For instance, the graphs above show the tools and materials which were used, but do not show
the processes and procedures which were used. In the case of tranchet axe manufacture, for
instance, an essential process was the blow which detached a transverse flake, giving the axe its
cutting edge. It is fairly easy to represent processes via graphs which show activities, sub-
activities and so forth: for instance, the process of roughing out a blank for a polished flint axhead
can be broken down into the sub-processes of hard hammer roughing out and soft hammer
Similarly, the graphs above do not make any distinction between tools and materials which are
archaeologically attested (such as the stones used as hard hammers), and those which are
hypothesised (such as rush pith for lamp wicks). This can be represented using different formats
for the arcs in the graph, such as different colours, different thickness of lines, and dashed lines
rather than continuous lines. This can be useful for “what-if?” representations to check the
feasibility of a hypothesis about how something was manufactured. For instance, this makes it
possible to check whether the hypothesis would involve lower-level tools or materials which were
not available in that area, such as a particular type of ore or clay. Coloured graphs can also be
useful for showing which parts of a technology depend on tools or materials coming from outside
the local area (and are therefore vulnerable to disruption if that source of supply is interrupted).
The graphs above show that representing all the entities in all the levels of the graph can quickly
produce very cluttered diagrams. One simple solution is to use large sheets of paper, or cards for
the entities and wool for the arcs, on pinboards. Although this may sound like a flippant
suggestion, there are in fact considerable practical and methodological advantages in doing this
compared to the usual formats of A4 or a computer screen. I’ve seldom had much success in
persuading people of this, though: there’s a widespread implicit belief that a computer-based
representation is always more sophisticated. With the A4/computer screen format, graphs can be
de-cluttered by using “black box” entities: the node shows an entity, such as “lamp”, and ends
there; the entity “lamp” is unpacked in a separate graph. The advantage of this is clarity. The
disadvantage is that it can easily lose the lower-level interconnections between entities which are
an important advantage of this approach – for instance, discovering which low-level entities are
used in the production of a large proportion of higher-level entities, as opposed to those which
are only used in the production of a few higher-level entities.
Implications of using mined flint
There is a large increase in fabricatory depth and width associated with the change from the
handaxe to the polished flint axehead using mined flint. For instance, fabricatory width increases
from 2 to 6, even for the first level of decomposition. The number of non-primitive entities also
increases from 1 to 4 for the mining alone. The technologies used for working the flint are very
similar between handaxe and polished flint axehead manufacture - the only difference is the
addition of the polishing stone, which is usually a lump of suitable sedimentary stone or a layer
of sand on a hard stone. The mining process, however, involves a set of entities larger than those
used for the flint working itself.
One surprising result of the analysis above is the number of items concerned with producing the
raw material: flint mining, for instance, is clearly a significant activity, rather than an insignificant
add-on to the process of producing the axe. When we come to describe the production of a metal
axehead using machine tools, this becomes even more apparent, and the attraction of using "black
box" representations to represent e.g. the ships used to transport iron ore is obvious. (Once an
item such as a ship has been described once, the description can be re-used as a black box when
describing other products, thus keeping the task down to a feasible size).
There has been considerable and heated debate about the relative importance of diffusion and
independent development in technology. The metric of technological interdependence, and the
concept of pre-adaptation, make it possible to quantify the extent to which a given technological
product is likely to be the result of diffusion or of independent development. A pre-adaptation in
this context is a pre-existing technology or tool which can be used as part of a new process. (There
is debate in the evolutionary ecology community about suitability of the term “pre-adaptation”
in that discipline, since it has connotations of a feature being consciously evolved in anticipation
of future needs. This is a cogent argument in that field, in the context of debates with religious
creationists who advance arguments for deliberate pre-adaptation, but this is less of an issue in
cultural anthropology, so the discussion here uses the term “pre-adaptation” on the grounds that
it is clearer than alternatives such as “exadaptation”.)
For example, several of the main tools involved in Neolithic flint mining are attested as already in
use for the erection of monuments involving significant amounts of earth moving (ditches, banks,
barrows, etc.). The development of mining for societies with such tools would not require many
technological developments (if any) to make mining possible, and such societies would be pre-
adapted for mining. It would therefore not be surprising if two different societies already
engaged in earth-moving for monuments independently began mining.
Perhaps significantly, early farming does not necessarily require major earth moving - slash and
burn agriculture, combined with the use of the digging stick, is well attested. For such a society
with no history of defensive or monumental earthworks, mining would require the development
of several new entities, and it would be unlikely that two societies of this sort would
independently begin mining.
This implies that pre-adaptations facilitating the development of mining were not a necessary
consequence of farming, but were an attested contingent consequence of the construction of large
defensive and ceremonial structures. A further implication is that since large defensive structures
require both significant manpower to construct, and a significant threat to guard against, there
was neither means nor need to construct them until the population growth associated with the
spread of farming. This produces the testable hypothesis that mining would be considerably more
likely to occur after farming and warfare were common than before them.
This is in agreement with the archaeological record, where there are a few cases of pre-Neolithic
mining, but the earliest large scale evidence for mining comes from the Neolithic.
Another interesting point is the fabricatory loop in which ladders are used in the manufacture of
axes, and axes are used in the manufacture of ladders.
Such loops can be reached by an evolutionary process in which, for instance, axes made from
surface flint are used to make the ladders used for mining flint to make axes. Once sufficient axes
have been made with mined flint, the process of using mined flint axes to make ladders for
mining flint becomes self sustaining.
Fabricatory loops of this sort have implications for technological resilience. As described above, a
loop can be self sustaining. If, however, one part of the loop is removed (for example, because the
supply of flint in the mines becomes exhausted), then the process can no longer be self-sustaining.
If there are alternative ways of achieving the same end process (e.g. using surface flint, or trade,
or making polished stone axes using surface igneous rock), then the technology of the society can
continue relatively unchanged. If, however, there is no alternative path, then the repercussions
can be considerable. It is likely that some technological breakdowns can be traced to the
breakdown of such loops, and research into this area promises to be fruitful.
An advantage of the graph approach described here is that it can be used to trace the implications
of a breakdown in any given item, process or material.
Other relevant approaches
There are numerous methods which apply top-down decomposition to different disciplines.
Particularly salient examples include laddering, task decomposition and JSD.
Laddering is a technique for eliciting hierarchically arranged knowledge from people: for
instance, finding out the tasks and sub-tasks involved in producing an artefact. It was invented by
Hinkle, in an unpublished PhD thesis, and was subsequently developed and extended by other
researchers, including myself. There is a brief article on laddering elsewhere on my website.
Laddering is well suited to gathering the information needed to draw up graphs of the type
described here. It appears superficially similar to a simple structured interview, with a few
questions being used to unpack successive layers of explanation. It’s useful if you have to extract
knowledge from a human expert, rather than observing the process at first hand. The main thing
to remember is to keep going down to the final levels of detail, rather than stopping when you
think you know what a particular term means – the same term may be used in different ways
across different fields.
Although laddering is good at eliciting this type of knowledge, it’s wise to complement it by first-
hand observation. Experts frequently forget to mention things that they take for granted, and
these things can be essential details (in fact, the most basic things are the most likely to be
overlooked). That’s why this draft doesn’t contain a section on metal axeheads yet: I’ve made and
used handaxes, tranchet axeheads and polished flint axeheads, but I don’t have first-hand
experience of replica bronze-age metalworking.
Task decomposition is a well established technique in ergonomics, and deals with tasks, rather
than technologies. It has a well established set of representations and techniques; it also has
experience of the problems involved in trying to represent complex tasks.
JSD (Jackson Structured Design) deals with software design. It was fashionable some years ago,
and there are other approaches which are similar to it. Their strength is that they have had to
tackle a full representation of tasks, in a form which can be translated into software, and this
leads into complex representational issues which cannot be fudged in the context of software
development. For example, if a task contains several different optional ways of handling various
sub-tasks, then the software developer needs to work through all the permutations of options, to
check that none of them interfere with each other. Similarly, software developers need to spell out
explicitly which sub-tasks need to be done in sequence, and which can be done in parallel with
each other. From the viewpoint of cultural anthropology, most of this debate is probably not
relevant, but some of the underlying notational concepts from software engineering can be useful.
Graph theory can be used as a representation of the complexity of a technology, in a way which
helps understand a culture’s technology in context – for instance, in terms of dependencies on
raw materials, or in terms of activities for which a culture will be pre-adapted. It has clear
applications in modelling early societies, but can also be used to model aspects of modern