# Mathematics in action: “Skin groups”

There are three main (though closely related) branches of mathematics – the study of number, the study of shape, and the study of relationships. An interesting ethnomathematical example of the latter is the “skin group” system of the Lardil people of Mornington Island in Australia.

Lardil children

Similar systems (often with two groups, called moieties) can be found around the world (I first discovered the concept as a child, in the classic young adult science fiction novel Citizen of the Galaxy). Among the Lardil, there are eight groups, each associated with particular totemic creatures or objects:

Skin Group Name Totem
1 Burulungi Lightning
2 Ngariboolungi Shooting star
3 Bungaringi Turtle
4 Yugumari Seagull
5 Gungulla Grey shark
6 Bulunyi Crane
7 Bulyarini Sea turtle
8 Gumerungi Rock

Membership of a “skin group” implies a complex set of tribal obligations and taboos, but the most significant is that only certain kinds of marriages are permitted. Members of group 1 must marry people from group 2 (and vice versa), and similarly for the pairs 3/4, 5/6, and 7/8. All other marriages are considered to be incestuous.

We can define a mathematical function, the spouse function σ, that maps each person’s “skin group” to the “skin group” that their spouse must have: σ(1) = 2, σ(2) = 1, σ(3) = 4, σ(4) = 3, etc. For each of the eight kinds of valid marriage, there is also a rule for determining the “skin group” of the children:

Father Mother Children
1 2 8
2 1 3
3 4 2
4 3 5
5 6 4
6 5 7
7 8 6
8 7 1

We can define two mathematical functions, the father-of function φ and the mother-of function μ, that map the “skin group” of a father or mother to the “skin group” that their children must have: φ(1) = 8, μ(1) = 3, φ(2) = 3, μ(2) = 8, etc.

This is all much clearer when displayed visually. In the diagram, two-part black arrows →→ indicate valid marriages. The arrows run from the “skin group” of the wife to the “skin group” of the husband. Red arrows run from each marriage arrow to the “skin group” of the children. Together, the arrows form an octagonal prism:

Following a single black arrow and then a red arrow () gives the mother-of function μ, with μ(1) = 3, etc. It can be seen that this function has a four-generation cycle: μ(μ(μ(μ(x)))) = x or, as it is often expressed, μ4(x) = x. In other words, each person’s “skin group” is the same as that of their great-great-grandmother in the female line (the maternal grandmother of their maternal grandmother).

Following a single black arrow backwards (from the head end) and then a red arrow () gives the father-of function φ, with φ(1) = 8, etc. It can be seen that this function has a two-generation cycle: φ(φ(x)) = x or, as it is often expressed, φ2(x) = x. In other words, each person’s “skin group” is the same as that of their grandfather in the male line (their paternal grandfather).

The combination of the two cycles makes the Lardil “skin group” system a very effective way of shuffling genes within a small population, thus avoiding inbreeding. It also highlights the fact that mathematics can be found in some surprising places. And there are even more patterns to be found in this example. Among others, σ(x) = φ(μ(x)). Also, μ(φ(μ(x))) = φ(x), which some readers may recognise as indicating a dihedral group.