#Desargues' two triangle theorem represented as a graph #The red and blue nodes represent points and lines. #See file for more info. #by # Desargues graph laid on 4-dim regular 5-tope (not shown here, see desarg5.rot). # Cyan nodes are on mid-edges, and yellow nodes on mid-faces of the 5-tope. # # The graph shows the points and lines (the two color nodes) in # the 5th Axiom of Incidence (Proj. Geometry). # # Graph by Xah Lee, 4dim. model by # # Not all the symmetry is visible in 3-d, for instance the links that # persist in being shorter than the rest are simply mainly in the # w-direction and have only a short projection in x,y,z. # edges of 5-tope points # # 0 0 0 0 0 # #+6 +6 -6 3 to 1 #+6 -6 +6 3 to 2 #-6 +6 +6 3 to 3 #-6 -6 -6 3 to 4 # 0 0 0 3 to 0 # #+6 -6 +6 3 to 2 #-6 -6 -6 3 to 4 #+6 +6 -6 3 to 1 #-6 +6 +6 3 to 3 # 0 0 0 3 to 0 # midedges nodes -6 0 0 -1 43 R' -6 0 0 0 -4 +1 -1 2 to q -6 0 0 0 -4 -1 +1 2 to p -6 0 0 0 -5 0 0 2 to a3 0 -6 0 -1 42 Q' 0 -6 0 0 -1 -4 +1 2 to p 0 -6 0 0 +1 -4 -1 2 to r 0 -6 0 0 0 -5 0 2 to a2 0 0 -6 -1 41 P' 0 0 -6 0 +1 -1 -4 2 to r 0 0 -6 0 -1 +1 -4 2 to q 0 0 -6 0 0 0 -5 2 to a1 +6 0 0 -1 21 A3 +6 0 0 0 +5 0 0 2 to r' +6 0 0 0 +4 -1 -1 2 to r +6 0 0 0 +4 +1 +1 2 to s 0 +6 0 -1 32 A2 0 +6 0 0 0 +5 0 2 to q' 0 +6 0 0 -1 +4 -1 2 to q 0 +6 0 0 +1 +4 +1 2 to s 0 0 +6 -1 13 A1 0 0 +6 0 0 0 +5 2 to p' 0 0 +6 0 -1 -1 +4 2 to p 0 0 +6 0 +1 +1 +4 2 to s -3 -3 -3 -1 04 S -3 -3 -3 0 -3.5 -1.5 -1.5 2 to a3 -3 -3 -3 0 -1.5 -3.5 -1.5 2 to a2 -3 -3 -3 0 -1.5 -1.5 -3.5 2 to a1 -3 +3 +3 -1 03 R -3 +3 +3 0 -1.5 +3.5 +1.5 2 to q' -3 +3 +3 0 -1.5 +1.5 +3.5 2 to p' -3 +3 +3 0 -3.5 +1.5 +1.5 2 to a3 +3 -3 +3 -1 02 Q +3 -3 +3 0 +1.5 -1.5 +3.5 2 to p' +3 -3 +3 0 +3.5 -1.5 +1.5 2 to r' +3 -3 +3 0 +1.5 -3.5 +1.5 2 to a2 +3 +3 -3 -1 01 P +3 +3 -3 0 +1.5 +3.5 -1.5 2 to q' +3 +3 -3 0 +3.5 +1.5 -1.5 2 to r' +3 +3 -3 0 +1.5 +1.5 -3.5 2 to a3 # midfaces nodes +2 -2 -2 -3 421 r +2 -2 -2 0 +1 -1 -4 2 to P' +2 -2 -2 0 +1 -4 -1 2 to Q' +2 -2 -2 0 +4 -1 -1 2 to A3 -2 +2 -2 -3 413 q -2 +2 -2 0 -4 +1 -1 2 to R' -2 +2 -2 0 -1 +1 -4 2 to P' -2 +2 -2 0 -1 +4 -1 2 to a2 -2 -2 +2 -3 432 p -2 -2 +2 0 -1 -4 +1 2 to Q' -2 -2 +2 0 -4 -1 +1 2 to R' -2 -2 +2 0 -1 -1 +4 2 to a1 +2 +2 +2 -3 321 s +2 +2 +2 0 +4 +1 +1 2 to A3 +2 +2 +2 0 +1 +4 +1 2 to A2 +2 +2 +2 0 +1 +1 +4 2 to A1 +4 0 0 -3 021 r' +4 0 0 0 +3.5 -1.5 +1.5 2 to Q +4 0 0 0 +3.5 +1.5 -1.5 2 to P +4 0 0 0 +5 0 0 2 to A3 0 +4 0 -3 013 q' 0 +4 0 0 -1.5 +3.5 +1.5 2 to R 0 +4 0 0 +1.5 +3.5 -1.5 2 to P 0 +4 0 0 0 +5 0 2 to A2 0 0 +4 -3 032 p' 0 0 +4 0 +1.5 -1.5 +3.5 2 to Q 0 0 +4 0 -1.5 +1.5 +3.5 2 to R 0 0 +4 0 0 0 +5 2 to A1 -4 0 0 -3 430 a3 -4 0 0 0 -5 0 0 2 to R' -4 0 0 0 -3.5 +1.5 +1.5 2 to R -4 0 0 0 -3.5 -1.5 -1.5 2 to S 0 -4 0 -3 420 a2 0 -4 0 0 0 -5 0 2 to Q' 0 -4 0 0 +1.5 -3.5 +1.5 2 to Q 0 -4 0 0 -1.5 -3.5 -1.5 2 to S 0 0 -4 -3 410 a1 0 0 -4 0 0 0 -5 2 to P' 0 0 -4 0 +1.5 +1.5 -3.5 2 to P 0 0 -4 0 -1.5 -1.5 -3.5 2 to S