Dear all, I've also (as well as Jim Propp) been assembling something for the Gathering for Gardner next week. This is honoring John Conway, so the following, which may be distributed in any way one wishes, is dedicated to him, with the hope that he will complete The Triangle Book. It would be flattering if he can use any of this. Guyration is the combination of `quadration' and `twinning' with Conway's `extraversion'. It yields a striking generalization of the triangle, sometimes with as many as 32 items appearing for the price of one. Quadration grants the same status to the orthocentre of a triangle as it does to the vertices, so that each of the four points is the orthocentre of the triangle formed by the other three. It regards the triangle as an orthocentric quadrangle, which now has 4 vertices, 6 edges and 3 diagonal points. Twinning draws the perpendicular bisectors of each of the six edges, producing a quadrangle of circumcentres, congruent to the original quadrangle. Alternatively, the two twins are related by reflexion in, or rotation through 180 deg about, a common `centre' . Our triangle now has eight vertices, which are also both circumcentres, and orthocentres. It has six pairs of parallel edges which form three rectangles whose twelve vertices lie on the `50-point circle' [9-point? Euler-Feuerbach?] The other 38 points are the points of contact with the 32 touch-circles and six points of contact with the double-deltoid [Steiner, with the appearance of a Star-of-David]. There are 12 edge-circles, 24 medial circles, 32 touch-circles, 32 Gergonne points, 32 Nagel points, 144 Morley triangles, 64 Pavillet tetrahedra, 256 radpoints, 384 guylines [haven't checked if these are distinct; maybe they coincide in sets of four], and much more. In case you'd like to do-it-yourself, here are the makings of a picture of the 32 touch-circles touching the 12 edges at 8 points, and each touching the 50-point circle. Vertices V1 (36,51), V2 (-204,-77), V4 (132,-77), V8 (36,103) Ve (-36,-51), Vd (204,77), Vb (-132,77), V7 (-36,-103) [hexadecimal subscripts; the 12 lines are ViVj with i,j in {1,2,4,8} and in {e,d,b,7} = {14,13,11,7}.] The 32 circles are 16 \put(x,y){\circle{d}} and 16 with (-x,-y) d, and the following centres (x,y) and diameters d. (20,-21) 112, (-92,-525) 896, (180,19) 192, (-252,115) 384, (48,55) 24, (660,259) 1248, (20,99) 32, (88,-105) 104, (16,63) 40, (-484,363) 1040, (60,91) 48, (-120,-209) 312, (12,-5) 144, (216,63) 280, (-288,175) 504, (-84,-437) 720. 32 touch-circles touch the 12 edges at 8 points, and each touches the 50-point circle. There are 3 or 4 largish files which can be sent to interested individuals. R.