How to choose your relations

Abstract

Mathematicians are fascinated by relatios and have worked our extensive theories about them. The treatments tend to be abstract and sometimes the basic ideas are lost in the abstraction. Here we investigate some common ground between mathematical relations and down-to-earth genealogy, the study of family relations.Family relations have been studied by mathematicians, perhaps none more playfully than Thomas P. Kirkman (the nineteenth century expert in combinatorial mathematics) in his little puzzle rhyme he sent to the Educational Times [1, p.117]:Baby Tom of Bay HughThe nephew is and uncle too;In how many ways can this be true

Ubiquitous cousins and parental slides

“…In [1] we showed that family trees exist in which a pair of first cousins can also be second cousins. In what follows we classify such family trees (up to isomorphism), and then we go on to deal with non-blood relations.…”

“…In a previous article [1] we examined the concept of blood relations in a family tree and we used mathematics to highlight and simplify the descriptions involved, such as child, parent, great-uncle, second cousin, third cousin once removed.…”

96.24 Genealogical algebra

“…I thought it might be interesting to apply this system to the Kirkman puzzle discussed in the article 'How to choose your relations' by Tony Crilly and Colin Fletcher [1] (Notice that, contrary to the more usual convention, the order of the two relationships is read from left to right. In ordinary functional notation, you might want to write Hugh's uncle as bro(far(Hugh)), and represent 'uncle' as bro 0 far.…”

“…So the equivalent of (2) is H = Hpar sibh par sibh, (5) but this cannot be simplified into a form similar to (3). This can be represented by Finally, we need to check (as Crilly and Fletcher did in [1]) that the solution does not conflict with the requirement that children must be younger than their parents. If arrows are used to indicate the relation 'is younger than', there are eight requirements:…”