In 1356, Narayana posed a question in his book Ga?ita Kaumudi: "A cow gives birth to a calf every year. In turn, the calf gives birth to another calf when it is three years old. What is the number of progeny produced during twenty years by one cow?" This is now known as Narayana's cows sequence. The Narayana's cows sequence constant, cow=1.4655712318767680266567312252199391080255775684723, is the limit ratio between neighboring terms.
LinearRecurrence[{1, 0, 1}, {2, 3, 4}, 21]
NestList[Round[# Root[-1 - #1^2 + #1^3 &, 1]] &, 2, 20]
Either gives {2, 3, 4, 6, 9, 13, 19, 28, 41, 60, 88, 129, 189, 277, 406, 595, 872, 1278, 1873, 2745, 4023}. This turns out to be a good constant to use for a Rauzy fractal. The outer fractal triangle can be divided into copies of itself
r = Root[-1 - #1^2 + #1^3 &, 3]; iterations = 6;
cowed[comp_] := First /@ Split[Flatten[RootReduce[#[[1]] + (#[[2]] - #[[1]]) {0, -r^5, r^5 + 1, 1}] & /@ Partition[comp, 2, 1, 1], 1]];
poly = ReIm[Nest[cowed[#] &, #, iterations]] & /@ Table[N[RootReduce[r^({4, 1, 3, 5} + n) {1, 1, -1, 1}], 50], {n, 1,14}];
Graphics[{EdgeForm[{Black}], Gray, Disk[{0, 0}, .1], MapIndexed[{Hue[#2[[1]]/12], Polygon[#1]} &, poly]}]
The ratio of areas for the triangles turns out to be cow. Try Area[Polygon[poly[[1]]]]/Area[Polygon[poly[[2]]]] and you'll see.
If you want to laser cut that, it's handy to get a single path.
cowpath[comp_] := First /@ Split[Flatten[RootReduce[#[[1]] + (#[[2]] - #[[1]]) {0, -r^5, r^5 + 1, 1}] & /@ Partition[comp, 2, 1], 1]];
path = ReIm[Nest[cowpath[#] &, N[Drop[Flatten[Table[r^({4, 1, 3} + n) {1, 1, -1}, {n, 1, 16}]], -1], 50], iterations]]; Graphics[{Line[path]}]
What else can be done with cow? With some trickier code I put together the pieces this way. Notice how order 5 spokes appear.
The opening gave an order 3 infinite spiral. Is there an order 5 infinite spiral? It turns out there is. Behold the cow-nautilus!
It can be made with the following code:
r=Root[-1-#1^2+#1^3&,3]; iterate=3;
cowed[comp_]:= First/@Split[Flatten[RootReduce[#[[1]]+(#[[2]]-#[[1]]){0,-r^5,r^5+1,1}]&/@Partition[comp,2,1,1],1]];
base={{r^10,r^7,-r^9,r^11},{-r^12,-r^9,r^11,-r^13},{r^8,r^5,-r^7,r^9},{-r^7,-r^4,r^6,-r^8}}+{-r^10,r^11,-r^6,r^4+r^8};
naut=RootReduce[Join[Table[base[[1]] (-r)^n,{n,0,-4,-1}],Flatten[Table[Drop[base,1](-r)^n,{n,-8,0}],1]]];
poly=ReIm[Nest[cowed[#]&,#,iterate]]&/@N[naut,50];
Graphics[{EdgeForm[{Black}],MapIndexed[{ColorData["BrightBands"][N[Norm[Mean[#1]]/2]],Polygon[#1]}&,poly]},ImageSize-> 800]
