For example, the Rho Quad tiling https://community.wolfram.com//c/portal/getImageAttachment?filename=RhoQuad.... is in the algebraic number field of rho, ρ, the plastic constant, 1.32472. I explain this in depth at https://blog.wolfram.com/2019/03/07/shattering-the-plane-with-twelve-new-sub... I list the 6 points for this tiling in terms of {ρ^0, ρ^1, ρ^2} as {{{4,0,0},{0,0,0}}, {{-1,2,-1},{-1,-2,3}}, {{-2,-1,1},{6,-3,-1}}, {{10,1,-5},{6,-3,-1}}, {{19,-7,-4},{-15,-1,8}}, {{2,1,-1},{-6,3,1}}} / 4 These translate to .. {{1.,0.}, {-0.162359,0.635452}, {-0.626466,0.260273}, {0.798488,0.260273},{0.822719,-0.755926},{0.626466,-0.260273}} The second point is real roots of {1+4 x-12 x^2+8 x^3 == 0, -19+56 x^2-48 x^4+64 x^6,-5+28 x^2-64 x^4+64 x^6 == 0}. But in terms of the plastic constant, it's {{-1,2,-1},{-1,-2,3}}/4 I have about a thousand examples of constrained geometric constructions that uses a particular algebraic number field ... with a twist. Values need to be *squared* with original signs preserved. And then everything works perfectly within the given algebraic number field. I have various functions at http://demonstrations.wolfram.com/SubstitutionTilings/ that allow automatic conversions from a set of points to an elegant representation within a particular algebraic number field. --Ed Pegg Jr On Thu, May 9, 2019 at 6:46 PM Fred Lunnon <fred.lunnon@gmail.com> wrote:

<< I've been parameterizing substitution tilings by their algebraic spaces ... >>

Further clarification of the statement above would be appreciated! See https://en.wikipedia.org/wiki/Algebraic_space

WFL

On 5/9/19, Ed Pegg Jr <ed@mathpuzzle.com> wrote:

...

I've been parameterizing substitution tilings by their algebraic spaces in my program at http://demonstrations.wolfram.com/SubstitutionTilings/ I've put a bunch of pictures at https://community.wolfram.com/groups/-/m/t/1679969

Additions, Comments, Corrections are all welcome.

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