Auxiliary materials for “Computing and Visualizing 3-D Quasicrystals”
by Josh Nolan and Cliff Reiter, January 2018
 
Check that M and P give an orthogonal coordinate system for R^6

gr=(1+Sqrt[5])/2

(M={{gr,0,1},{gr,0,-1},{1,gr,0},{-1,gr,0},{0,1,gr},{0,-1,gr}})//MatrixForm

grc=Simplify[-1/gr]

(P={{grc,0,1},{grc,0,-1},{1,grc,0},{-1,grc,0},{0,1,grc},{0,-1,grc}})//MatrixForm

(MP=Transpose[Join[Transpose[M],Transpose[P]]])//MatrixForm

Simplify[Transpose[MP].MP]//MatrixForm

Checked that M and P give an orthogonal coordinate system for R^6

Check that R and S generate icosahedral symmetry

(R={{0,0,0,0,0,1},{0,0,0,-1,0,0},{1,0,0,0,0,0},{0,0,1,0,0,0},{0,0,0,0,1,0},{0,-1,0,0,0,0}})//MatrixForm

MatrixPower[R,5]//MatrixForm

S={{0,-1,0,0,0,0},{-1,0,0,0,0,0},{0,0 ,-1,0,0,0},{0,0,0,-1,0,0},{0,0,0,0,0,1},{0,0,0,0,1,0}}

(S.S)//MatrixForm

T=S.R

T.T.T//MatrixForm

Checked that R and S generate icosahedral symmetry

Check that E is stable under icosahedral symmetry

(R . M)//MatrixForm

c=LinearSolve[M,R.M]

(R.M)==(M.c)

Simplify[c]//MatrixForm

(cc=1/2{{-grc,-gr,1},{gr,1,-grc},{-1,-grc,gr}})//MatrixForm

Simplify[cc-c]

d=LinearSolve[M,S.M]

(S . M)==(M.d)

Checked that E is stable under icosahedral symmetry