An Investigation into Viking Mathematics

Ola Nordmann

April 1st, 2013

Creative Commons License


  1. Home

  2. 1. Introduction

  3. 2. Preliminary Oddities

  4. 3. The Fragments of Chain Mail

  5. 4. Conclusion


Many ancient civilisations are well-known for their advanced mathematical knowledge. Traditionally, the list of such does not include the Vikings. However, evidence gathered from archaeological findings shows that the Vikings had a sophisticated grasp of geometry, which they put to typically gruesome use.

1 Introduction

This article was inspired by a visit that the author made to a museum in the Norwegian region of Ingenmannslandfjorden. The museum has a display of items found at a nearby Viking grave. While these items seem like ordinary Viking weaponry and armour, closer inspection reveals a design that is far more subtle and displays a remarkable understanding of topology and geometry.

In 1922, the archeologist Professor Brumm led an expedition to the northern parts of Scandinavia to investigate possible areas for further archeological investigation. A truncated account of the journey was published in [BMNR26 ]. Several such areas were discovered and investigated, but for the majority then the finds were not of significant interest and ended up furnishing exhibits in small local museums. The dig at Ingenmannslandfjorden was no exception to this. There were certain aspects to the discovery that were regarded as unusual — of which more anon — but no explanation for the oddities was ever put forward and so there the matter lay.

On a recent trip to Ingenmannslandfjorden the author visited the local museum and noticed some rather curious aspects of the armour on display. At first sight it appears to be ordinary chain mail, albeit of a superior workmanship than is typical for the time. On second sight, though, some of the examples display a curious structure which suggest a mathematical knowledge heretofore unknown in the Vikings, and a cunning that was all too well known.

Photography being not allowed in the museum, the author made copious drawings of the exhibits which have been rendered in graphic form.

2 Preliminary Oddities

As already mentioned, the burial site has some oddities that have not, until now, been satisfactorily explained. The first, and most unusual, is the person who was buried. At the time that this burial took place such ornate burials were only afforded to the nobility or to great warriors. But this grave was that of an armourer or smith. Not only was he buried with many examples of his work, but also his skeleton showed the characteristics of a life of crafting weapons rather than wielding them. The workmanship of the armour was of a high quality, but even so to bury an armourer with such high honour shows that there must have been something unique about his work.

Armourers of the time used marks to denote their work. This armourer appears to have had two marks, closely related but clearly distinct. The mark is based on the Valknut, a common symbol of the time. The two variants are reproduced in Figure 1. Since the two marks are similar the original theory as to the existence of two was that they date from different periods of the smith's life and the mark's form evolved as he grew older. However, the marks are found on a variety of the samples of the smith's work that were found in the burial site and from the quality of the work involved it would appear that they were both in use concurrently for a large period of the smith's working life.

Figure 1: The two Valknut marks

There were also found some scraps of writing which were sent for translation to a scholar in Oxford with particular expertise in old Scandinavian languages. The translations, together with details on the etymology and theories as to the meaning of the writings, were eventually published in [LT54 ]. Of particular note are the two following fragments (written here in modern Norwegian).

Tre ringer for alvene under det blå;
syv er for dverger i saler av sten;
ni for menn som all kjødets gang mågå.

Én ring skal samle dem, én ring finne dem,
én ring betvinge dem og i mørket binde dem.

3 The Fragments of Chain Mail

Chain mail is a popular form of armour still in use today (albeit no longer in military situations). It provides protection particularly against stabbing weapons and affords greater flexibility than plate armour. There are many designs of chain mail. A standard design was based on linking each ring to four others as in Figure 2. A segment of the resulting mail is shown in Figure 3.

Figure 2: Basic chain mail pattern

Figure 3: Segment of chain mail

Some of the samples of chain mail are of this type, but some are based on a different pattern. The alternative pattern still has each ring being linked to its four neighbours, but the rings are deformed in such a manner that their linking is more complicated. The rings are pulled into ellipses and then bent back over themselves. The basic pattern is shown in Figure 4 and a segment in Figure 5.

Figure 4: Advanced chain mail pattern

Figure 5: Segment based on the advanced chain mail pattern

It was originally thought that this was to make them easier to connect together: the majority of the rings can be connected without breaking and reforming. In addition, the doubling of the rings provides more protection.

However, closer examination of the design reveals a more startling and typically Viking reason for this type of chain mail. The method of connecting the rings together is designed so that if one of the rings is cut then the chain mail falls apart.

The implications of this are astounding. The armour that a warrior wore into battle then becomes their weak spot. A single well-aimed blow leaves them defenceless just at the moment that they need protection.

This is quite a claim. To back it up, let us consider what happens when a link is broken in the chain mail. Let us first redraw the segment in a more schematic manner, replacing the round rings with square ones, as in Figure 6.

Figure 6: Angular and curved rings

Now let us consider a segment of the chain mail drawn with the square components, see Figure 7. This is only a segment so we have to imagine it extending in all directions. Consider what happens if the component at the middle of the top row is broken (marked in blue in Figure 7). This partially releases the two red components below it. Each of these in turn partially releases the two green components below it, but this completely releases the green component directly below the blue. The process continues, fanning out as in Figure 8. The unlinking would have been aided by the additional force of the cut that broke the first link.

Figure 7: Segment of chain mail

Figure 8: Segment of chain mail with unlinkings marked

There is evidence for the fact that this feature was a deliberate part of the design in the way in which the armour was constructed. The disintegration of the chain mail has a definite direction: rings above the one that was cut are not free. The solution is to ensure that the mail is made so that the direction of the unlinking goes around the body, thus continuing to maximum effect. This is seen in the items exhibited in the museum. Moreover, the direction is consistently chosen so that a blow from a right-handed attacker has the most effect.

4 Conclusion

The design is clearly no accident and shows both great skill and ingenuity on the behalf of the smith who was buried at Ingenmannslandfjorden. Further evidence for the deliberate nature of the work comes from closer examination of the smith's marks. On those pieces where the mark can still be seen, it is plain that the more complicated mark (on the left in Figure 1) is on the more complex mail and the simpler mark on the more usual mail. Thus those in the know could easily distinguish between the two types of chain mail. The more complicated mark has this same property: that cutting one of its components releases the others.

In honour of the archeologist who first discovered this grave we propose using the term Brummian to denote this property.

With this understanding of the armour, the fragments of writing can be given new interpretations. The first would appear to be a list of noblemen to whom gifts of armour were given, divided into nationality by their physical characteristics. The third line would seem to indicate that these nine lords were to be given the Brummian armour. Perhaps these were clans that the local lord was currently at peace with but who he fully expected to be at war with in the near future.

The second fragment may be part of the instructions to the junior smiths on how to make the new style of armour. There is clear indication of both the method: "Én ring skal samle dem", and the necessity of keeping this a secret: "og i mørket binde dem."

In conclusion, we see that the smith buried at Ingenmannslandfjorden was truly a master of his craft and deserved the rich burial that he was given. The translator of the fragments had great foresight in giving him the title Ringenes Herre.


  1. [BMNR26]O. Brumm, A. Milne, N. Nøff, K. Robin. En ekspedisjon til Nordpolen. 1926.

  2. [LT54]B. Lommelun, F. Lommelun, J. Tolkien. Ringens brorskap. 1954.