Simplex_for_Jones.nb
T. Ratliff, Wheaton College, Feb 4, 2016

RegionBoundaries[q_]:=Graphics3D[{
Simplex[{{0,0,1},{1,0,0},{0,1,0}}],
Line[{{1-q,0,q},{0,1-q,q}}], 
Line[{{q,1-q,0},{q,0,1-q}}],
Line[{{0,q,1-q},{1-q,q,0}}],
Line[{{1-q,0,q},{1-q,q,0}}],
Line[{{0,1-q,q},{q,1-q,0}}],
Line[{{q,0,1-q},{0,q,1-q}}]
}];
ShowSimplex[objects_]:=Show[objects,ViewPoint->{2,2,2},Boxed->False];

Let’s show the geometry of the Paradox of Redistribution from class.
	P1 corresponds to the system [10: 8, 5, 2]
	P2 corresponds to the system [10: 6, 6, 3]

P1= Graphics3D[{PointSize[ 0.025],Red,Point[{8/15, 5/15, 2/15}]}];
P2 = Graphics3D[{PointSize[ 0.025],Blue,Point[{6/15, 6/15, 3/15}]}];
ShowSimplex[{RegionBoundaries[2/3],P1,P2}]

Let’s show the geometry of the Paradox of a New Member from class.
	P3 corresponds to the system [144: 135, 81, 0]
	P4 corresponds to the system [160: 105, 63, 72]
	L1 is the line where the plane y=(3/5)x intersects the simplex

P3= Graphics3D[{PointSize[ 0.025],Red,Point[{135/216, 81/216, 0}]}];
P4 = Graphics3D[{PointSize[ 0.025],Blue,Point[{105/240,63/240, 72/240}]}];
L1=Graphics3D[{Blue,Line[{ {135/216, 81/216, 0},{0,0,1}}]}];
ShowSimplex[{RegionBoundaries[2/3],P3,P4, L1}]