Family Day update

Task/Sierpinski-triangle-Graphical/Mathematica/sierpinski-triangle-graphical-1.math

@@ -1,8 +1,6 @@
-Sierpinski[n_] :=Nest[Flatten[Table[{{
-       #[[i, 1]], (#[[i, 1]] + #[[i, 2]])/2, (#[[i, 1]] + #[[i, 3]])/
-        2}, {(#[[i, 1]] + #[[i, 2]])/2, #[[i,
-        2]], (#[[i, 2]] + #[[i, 3]])/2}, {(#[[i, 1]] + #[[i, 3]])/
-        2, (#[[i, 2]] + #[[i, 3]])/2, #[[i, 3]]}}, {i, Length[#]}],
-    1] &, {{{0, 0}, {1/2, 1}, {1, 0}}}, n]
+Sierpinski[n_] :=
+ Nest[Join @@ Table[With[{a = #[[i, 1]], b = #[[i, 2]],  c = #[[i, 3]]},
+   {{a, (a + b)/2, (c + a)/2}, {(a + b)/2, b, (b + c)/2}, {(c + a)/2, (b + c)/2, c}}],
+ {i, Length[#]}] &, {{{0, 0}, {1/2, 1}, {1, 0}}}, n]
 
-Show[Graphics[{Opacity[1], Black, Map[Polygon, Sierpinski[8], 1]}, AspectRatio -> 1]]
+Graphics[{Black, Polygon /@ Sierpinski[8]}]

Task/Sierpinski-triangle-Graphical/Mathematica/sierpinski-triangle-graphical-2.math

@@ -1,3 +1,10 @@
-sierpinski[v_, 0] := Polygon@v;
-sierpinski[v_, n_] := sierpinski[#, n - 1] & /@ (Mean /@ # & /@ v~Tuples~2~Partition~3);
-Graphics@sierpinski[N@{{0, 0}, {1, 0}, {.5, .8}}, 3]
+cf = Compile[{{A, _Real, 2}},
+   With[{a = A[[1]], b = A[[2]], c = A[[3]]},
+    With[{ab = (a + b)/2, bc = (b + c)/2, ca = (a + c)/2},
+     {{a, ab, ca}, {ab, b, bc}, {ca, bc, c}}]],
+       RuntimeAttributes -> {Listable}
+   ];
+
+n = 3;
+pts = Flatten[Nest[cf, N@{{{0, 0}, {1, 0}, {1/2, √3/2}}}, n], n];
+Graphics[Polygon /@ pts]