How to turn a collection of contour lines into a printable object (mesh, etc)?

Question

Inspired by Henry Segerman's developing fractal curves, I decided to try to convince Mathematica to do something similar. The inspiration for cf, f, and g below came from How to make this Dragon Curve?.

cf = Compile[{{M, _Real, 2}, t, a}, 
   With[{A = M[[1]], B = M[[2]]}, 
    With[{P = (A + B + a t Cross[B - A])/2}, {{A, P}, {B, P}}]], 
   RuntimeAttributes -> Listable];

f[n_, a_] := Flatten[Nest[cf[#, 1, a] &, {{{0, 0}, {1, 0}}}, Floor@n], Floor@n];
g[n_, a_] := Flatten[cf[f[n, a], FractionalPart[n], a], 1];

iterations = 10;
ang = 1;
widthscale = GoldenRatio;
functionPoints3D[z_] := 
  Map[Append[(1.5)^(Sqrt[z])], Flatten[widthscale g[z, ang], 1]];
pointlist = Table[functionPoints3D[n], {n, 0, iterations, .1}];
Graphics3D[BezierCurve[Partition[Flatten[pointlist, 1], 2]]]

I haven't been able to figure out how to turn this into a mesh or extrude this into something that could be exported as a printable object. I've tried several things, but none of them have worked.

Bonus if there's a way to make the corners more curved (akin to c or s shapes instead of right angles).

Edit (2/29): Here's a cheated image of the "solid" using tubes spaced .01 apart to give the idea of what it will look like once done. Once I get it all finished, I will definitely post a picture of the printed object. For now, here's the teaser:

Answer 1

Here is an example how you can mesh the regular part of your curves:

n = 2^3;
m = 5;
h = 1/n;
newpointlist = Table[
   Subdivide[{0, 0, -i h}, {1, 0, -i h}, n],
   {i, 0, m}];
edges = Partition[#, 2, 1] & /@ 
   Partition[Range[(m + 1) (n + 1)], {n + 1}];
quads = Join @@ Join[
    Map[Reverse, edges[[;; -2]], {2}],
    edges[[2 ;;]],
    3
    ];
Graphics3D[
 GraphicsComplex[Join @@ newpointlist, Polygon[quads]]
 ]

Triangles can be obtained like this:

triangles = Riffle[quads[[All, 1 ;; 3]], quads[[All, {3, 4, 1}]]];
Graphics3D[
 GraphicsComplex[Join @@ newpointlist, Polygon[triangles]]
 ]

For the irregular part (where the number of edges doubles), you can try such a strategy:

n = 2^3;
m = 5;
h = 1/n;
newpointlist = Table[
   Subdivide[{0, 0, -i h}, {1, 0, -i h}, 2^i],
   {i, 0, m}];

triangles = Join @@ Transpose[{
     Transpose[{
       Flatten[idx[[1 ;; -2, 1 ;; -2 ;; 1]]],
       Flatten[idx[[2 ;; -1, 1 ;; -2 ;; 2]]],
       Flatten[idx[[2 ;; -1, 2 ;; -1 ;; 2]]]
       }],
     Transpose[{
       Flatten[idx[[1 ;; -2, 1 ;; -2 ;; 1]]],
       Flatten[idx[[2 ;; -1, 2 ;; -2 ;; 2]]],
       Flatten[idx[[1 ;; -2, 2 ;; -1 ;; 1]]]
       }],
     Transpose[{
       Flatten[idx[[1 ;; -2, 2 ;; -1 ;; 1]]],
       Flatten[idx[[2 ;; -1, 2 ;; -2 ;; 2]]],
       Flatten[idx[[2 ;; -1, 3 ;; -1 ;; 2]]]
       }]
     }
    ];

Graphics3D[{
  Point[Join @@ newpointlist],
  GraphicsComplex[Join @@ newpointlist, Polygon[triangles]]
  }]

I have to admit, it is a bit of fiddling. That is why I insisted on correct ordering. =)

After you have all triangles, put everything into a mesh region. You can round off averything nicely, e.g., with a few iterations of my function LoopSubdivide.

Afterwards, Export to STL or whatever format suits you. Usually, printing software can do the extrusion/thickening of the surface for you, but you can also find code for that job somewhere on this site...