While at the Joint Winter Meetings in Atlanta, I saw a rather curious object sitting on a table: a regular tetrahedron made of white cardboard, whose sides were labelled "A", "C", "G", and "T". (Does anyone know who created it, and why?) This sighting prompted a question which is perhaps more suited to Michael Kleber than to any other person alive, since it concerns both polyhedral models (an old interest of his) and nucleotides (a newer interest of his), but since the answer may interest lots of people besides the two of us, I thought I'd ask it in this forum: What are the biologically relevant group-actions on the set {A,C,G,T}? Leaving aside the symmetric group action ("they're all necleotides, aren't they?") and the trivial group action ("yeah, but they're DIFFERENT nucleotides"), there's the 4-element group action that stabilizes the set {A,G} and the set {C,T} ("sure, but the two purines are more like each other than they are like the two pyrimidines, and vice versa"), and there's the 8-element group action that stabilizes the partition {{A,T},{C,G}} ("okay, but what really matters is that A pairs with T and C pairs with G") and there's the 4-element group action that stabilizes the sets {A,T} and {C,G} ("fine, but don't forget that the two pairs behave differently vis-a-vis transcription of DNA into RNA (the whole uracil thing)"), and ... I'd be surprised if there were some biologically relevant action of the 2-element group that leaves T and G alone but acts like Sym({A,C}) on the other two elements (to give just one example of a group-action that probably isn't biologically meaningful). But, like most of you, I like to be surprised, which is why I'm asking. Also, do any linear representations (as opposed to permutation representations) of Sym({A,C,G,T}) play a role in genomics? Jim Propp