Die Links and Sequences in Numismatics

By David Yoon for the American Numismatic Society (ANS) ……
 

One of the essential methods of numismatics is the die study. This involves comparing coins carefully, to see whether similar coins were struck using the same dies. The dies for premodern coins were individually handmade, so each one is slightly different, even from the hand of the same practiced engraver. When a die is used to strike coins, the individual handmade blanks will differ, as will the placement of the handheld dies, but the design features transferred to the coin from the die will be the same on each example (varying only in how deeply the hammer blow pushed the die into the metal blank).

Identifying the dies with the coins they struck has two major uses in numismatics.

The information is often used to provide rough estimates for the relative intensities of production of different types. And in some areas of numismatics where the quantity produced was small but the proportion of surviving specimens is relatively high (such as medals, early American coins, or ancient Greek coins, for example), die identifications can be used to arrange coins into emission sequences. This relies on what are called die links.

Since usually each coin bears the impressions of two dies–one on the obverse and one on the reverse–it is possible that two coins may have been struck with the same die on one side, but with two different dies on the other. This would occur, for example, if one die cracked and was replaced. By noting which obverse dies were paired with which reverse dies for as many related coins as possible, it is possible to compile a list of all the die pairings that exist.

Here are two coins of Gundobad, King of the Burgundians (ANS 2013.40.68, above, and ANS 2014.44.52, below). The obverses of both coins were struck with the same die; if the two images are overlaid in an image-editing program–allowing for how the die was placed differently relative to the two blanks–the features transferred from the die line up perfectly. The reverses, on the other hand, were struck using different dies. Some of the obvious areas of difference have been indicated with small arrows.

In this situation, the two reverse dies are said to be linked to each other through the obverse die with which they are both paired. They were used in conjunction with the same obverse die, so they were used, either contemporaneously or sequentially, close together in time. If enough such links can be put together, they may represent an extended sequence of dies entering use and then failing and being replaced by others. In other words, the order in which the dies were used can, in principle, be discovered.

Let’s take an example of a simple set of pairings between obverse and reverse dies:

This is a relatively well-behaved example. Most real die studies are plagued with pairs or small groups of dies that do not link to others. The die links in this group would normally be presented arranged as a sequence, such as this one:

This poses a question: When constructing a sequence of dies in a numismatic die study, how does one know whether a particular sequence is a good answer? The problem is that this sequence must be inferred by the researcher out of a mess of unordered connections and the dies can be arranged in various different orders. Some arrangements are more plausible than others, because they put closely linked dies closer together in the sequence, but it is rare that only one clear sequence emerges from the data.

This question is not a simple one, and thinking about it led me into the mathematics of die-link charts. These die-link charts are what mathematicians call graphs. More specifically, because the links in die-link charts always connect two distinct sets (obverse dies and reverse dies), this chart is what is called a bipartite graph. There is a branch of mathematics called graph theory that studies sets of linked objects. In the humanities, an applied form of graph theory is known as network analysis, which uses the mathematical tools of graph theory to study networks of connections.

However, the methods of network analysis have largely concentrated on different questions from those of numismatic die studies. In studies of human networks (social, economic, political), analyses try to measure connectedness, centrality, and other such questions about the connections represented. In die-link studies, by contrast, the focus is usually not on how well-connected dies were, but rather on which dies were used earlier or later.

To approach this question, we can look at another representation of graphs. The graphs above can also be represented as a matrix: a grid showing, for each obverse and each reverse, a 1 if they are linked and a 0 if they are not. This is what is called a biadjacency matrix in graph theory.

If we assume that dies are normally used for a short part of the sequence, then break and are destroyed and replaced, then when the numismatic biadjacency matrix is put in order by time, it should have the die links concentrated near a diagonal axis running from one corner to the opposite. This gives a mathematical and visual representation of how well the sequence works: a good sequence minimizes departures from the diagonal.

Of course, not all die sequences are well-behaved. Not only are there components of the sequence that do not link to the others, as mentioned before, there are also events that do not fit the assumptions. One example would be the situation in which an old die is retired but not destroyed, only to be brought back into use much later—something that will blemish the matrix with a 1 far from the diagonal. Various forms of splitting or pooling of dies–for example, if different mint workers share the same set of anvil dies but use distinct trussel dies–will also complicate the results.

A final point to mention is that the mathematical structure is symmetric. Without further information, a sequence of links can be read in either direction. In order to know which end of the sequence is the start and which is the end, it is necessary to bring in additional information, such as deterioration of dies, to indicate which end of the numismatic graph is earlier and which is later.

* * *

* * *

The post Die Links and Sequences in Numismatics appeared first on CoinWeek: Rare Coin, Currency, and Bullion News for Collectors.