I'm trying to see what sorts of algebraic structures can be built up in a Conway-style "genetic" way. I think I may have an approach to constructing the surcomplex numbers that's different from Conway's, but I'm stuck; maybe one of you can help me see what the next step is (or help me see that the enterprise is doomed). Some background may be helpful here. We say that one dyadic rational number is simpler than another if the first has smaller denominator, or, in the case where they have the same denominator, the first has smaller absolute value. Thus 1/4 is simpler than 3/4 but 1/2 is simpler than both of them. This gives a partial ordering on dyadic rationals. The simplest dyadic rational is 0; the next simplest are +1 and -1; the next simplest are +2, -2, +1/2, and -1/2; etc. Conway approached simplicity via birthday ordinals but this is not necessary. Every open interval I in the dyadic rationals contains a unique simplest number I*, and we have the addition rule I*+J* = ((I*+J) \cap (I+J*))* where the sum of a number and an interval is defined in the obvious way (shift the interval by the number, or if you prefer, take the Minkowski sum of the interval with the singleton set associated with the number). This formula is just Conway's definition of addition in disguise. To rip away the disguise, write I* as {x_L | x_R} where x_L and x_R are the left- and right-endpoints of I respectively. If we write I = (x_L, x_R) and I* = x and J = (y_L, y_R) and J* = y, then Conway's definition of addition tells us that x + y is the simplest dyadic rational that is bigger than both x + y_L and x_L + y and smaller than both x + y_R and x_R + y; that is, it is the simplest dyadic rational in the intersection of the interval (x + y_L, x + y_R) = x + (y_L, y_R) = I* + J and the interval (x_L + y, x_R + y) = (x_L, x_R) + y = I + J*. (Has anyone seen this way of reformulating surreal numbers before?) Now on to my real question. Define the dyadic Eisenstein rationals as the complex numbers of the form a + bw where a and b are dyadic rationals and w is a primitive cube root of 1; define its denominator as the maximum of the denominators of a and b. Every dyadic Eisenstein rational z can be written as a + bw, a + bw^2, or aw + bw^2 with a,b non-negative; accordingly define the "triangle norm" of z as a+b (so that the level sets of the triangle norm are triangles centered at 0 in the complex plane; it's not a true norm since the level sets aren't centrally symmetric). We say that one dyadic Eisenstein rational is simpler than another if the first has smaller denominator, or, in the case where they have the same denominator, the first has smaller triangle norm. The role played by intervals in the dyadic rationals will be played by open equilateral triangles that point to the right (like the "play" icons on many devices); hereafter when I write "triangle" I mean that kind of triangle. I can prove that every triangle T in the dyadic Eisenstein rationals contains a unique simplest number T*. What I haven't been able to prove, but believe to be true, is that for all triangles T,U we have (*) T*+U* = ((T*+U) \cap (T+U*))* If this is true, then there's probably a way to "genetically" build up the Eisenstein rationals (perhaps extending beyond the complex numbers into the surcomplex realm) starting from the empty set, much as Conway built up the surreals. (This leaves out the problem of defining multiplication genetically, but hey, you gotta start somewhere.) I've proved the conjecture for triangles that are symmetric about the real axis; in this case (*) follows from facts about surreal numbers. (I should mention that there's an alternative, simpler way to build up the surcomplex numbers genetically, using rectangles instead of triangles. But then all the theorems one needs to prove are trivialities, as all we're doing is looking at ordered pairs of surreal numbers and doing the obvious things with them. This is pretty close to Conway's construction of the surcomplex numbers by adjoining a formal square root of -1. But what interests me is the idea of genetic construction itself. The only things I've seen genetic constructions for are the surreals and the nimbers.) Jim Propp