got interested in Seth Major's "spin network primer" gr-qc/9905020 and also "An efficient algorithm for the riemannian 10j symbols" by JD Christensen and Greg Egan gr-qc/0110045 calculations in 4D quantum geometry seem to depend on the 10j symbol and whether or not you can calculate the partition function depends on how good a 10j algorithm you have (that includes calculating a sum-over-histories amplitude for getting from one spatial geometry to another----feynmann style quantum spatial dynamic) it seems to excite people just now---a fair number of papers (ask for links if you want) so I want to gather together some equations and recipes for a few of the numbers that keep surfacing in the papers Ive been reading. Physics Forums format may defeat this. I cant draw graphs! There is a two vertex, three link graph called a "THETA" with the links labeled with integers m,n,o that sum to an even integer and satisfy the usual condition that a,b,c non-negative a = (m+n-o)/2 b = (n+o-m)/2 c = (o+m-n)/2 Then θ(m,n,o) = (-1)a+b+ca!b!c!(a+b+c+1)!/((a+b)!(b+c)!(a+c)!) Seth Major gives an example θ(N,N,2) for any N. Would you like to calculate it? Note that a=N-1, b=1, c=1. Then there is TET which is a tetrahedron number. The 6 edges of the tetrahedron are labeled a,b,e,c,d,f Here's where I should draw a picture but imagine a square with the sides labeled clockwise a,b,c,d starting on the left. So the left side is a and the top is b. Then make the rising diagonal e and the falling diagonal f. The vertices are where these threesomes meet: abf ade bce cdf The pairs of opposite edges are ac, bd, ef. These pairs are not in contact. Edge a touches all save c, and so on. So Seth defines 4 vertex numbers a1 = (a+d+e)/2 a2 = (b+c+e)/2 a3 = (a+b+f)/2 a4 = (c+d+f)/2 And he defines 3 edgepair numbers b1 = (b+d+e+f)/2 b2 = (a+c+e+f)/2 b3 = (a+b+c+d)/2 Then he says N = Π (bj - ai)!/(a!b!c!d!e!f!) The product is of 12 terms, all possible i and j. This is what computers are good for in quantum gravity obviously. Finally there is the Tet symbol, a number written: Tet[(a,b,e), (c,d,f)] actually Seth arranges it so it looks like Tet of 2 row-vectors or Tet of a 2x3 matrix. But other places I've seen it typed like I just did. So this doesnt get too long I will put the actual definition of the Tet in the next post. It uses this number N just defined.