All for One and One for All

by Nicholas Mee on February 10, 2018

In days of old, when knights were bold, it was essential that a knight should bear an elegant mathematical symbol on his coat of arms. Well, perhaps not, but at least the Borromeo family used a design that is well known to mathematicians.

The coat of arms of the Borromeo family of merchants and bankers from northern Italy bore a device symbolising the motto

All for one and one for all.

The Borromean rings, as shown in the illustration here, are arranged such that no two rings are linked, but the rings cannot be separated without cutting one of the rings.

The Walknot

The Borromeo were not the first to use this motif, however. For 1000 years, from the 4th Century onwards, the Norse people of Scandinavia carved magical runes and symbols on wood and stone. One such symbol is the Valknutr or Walknot. This triangular representation of the Borromean Rings was the sign of the slain. It is known from cremation urns from East Anglia and an early English ring, and is also shown on the woodcarvings and tapestries of the Oseberg ship burial. The Walknot is also found on the picture-stones of Gotland, a Baltic island off the southeast coast of Sweden. The image below is a cleaned up version of part of the Stora Hammars 1 picture-stone located in Lärbro parish in Gotland and it depicts a human sacrifice to Odin, the hanged god. In Odin’s words from the Old Norse poem Hávamál translated by Carolyne Larrington:

I know that I hung on a windy tree
nine long nights,
wounded with a spear, dedicated to Odin,
myself to myself,
on that tree of which no man knows
from where its roots run.
No bread did they give me nor a drink from a horn,
downwards I peered;
I took up the runes, screaming I took them,
then I fell back from there

The image shows the ravens Huginn and Muninn who keep Odin abreast of all that takes place in Middle Earth, while Odin himself hangs from the cosmic tree Yggdrasil. The Walknot is prominent in the centre of the image above the sacrificial offering to the god.

It has been suggested the association between the walknot and Odin symbolised the God’s power to bind and unbind the mind of the Viking warrior. Odin could make men helpless in battle, but he could also loosen the tensions of fear and dread with his inspiration and his intoxicating gift of battle-madness.

Symbolic Sculpture

The Borromean rings design has featured in many artworks over the centuries. John Robinson created a number of abstract sculptures based on the theme of the Borromean rings. One example of these is Creation, which consists of three interlocked squares. The image below is a frame from the ray traced animation that I created with John Robinson almost twenty years ago.

Creation by John Robinson and Nicholas Mee

John Robinson also devised a sculpture called Genesis formed of three rhombi interlocked to form the Borromean rings. The short diagonal of each rhombus is slotted into the long diagonal of one of the other two rhombi. The six outer vertices of Genesis are the vertices of a regular octahedron, one of the five Platonic solids.

Genesis by John Robinson and Nicholas Mee

 

Brunnian Links

There are also links of more than three rings in which, just like the Borromean Rings, no two rings are joined. They are known as Brunnian links after Hermann Brun who drew examples of them over a century ago.

The image on the right shows a pentagonal Brunnian link formed of five nonagons inscribed within a decagon. The link is in the form of a stained glass mandala set into a wall decorated with a Penrose tiling. I came across this image many years ago and I would be interested to know who it was created by.

Apparently this Brunnian mandala is also found in the Principia Discordia, a satirical counter-culture text written in 1963 by Greg Hill (Malaclypse the Younger) and Kerry Wendell Thornley (Lord Omar Khayyam Ravenhurst).

Further Information

The connection between the walknot and the Borromean Rings was pointed out by Peter Cromwell in The Mathematical Intelligencer, Winter 1995, Vol 17, No.1.

 

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