!-Created June 2019.—>
Fractal curves provide convenient templates for walkable mathematical structures. The Hilbert curve is particularly easy to work with due to the face that it is based on a square grid. The curve is basically an algorithm for connecting the squares in a grid to form a non-intersecting path. This is illustrated at left in the following figure. The curve is covered with smaller squares at right that define a path equal in width to the spacing between the rows of squares. If each of these squares is imagined to be a stepping stone, a walkable tower can be constructed. A version that can be manipulated in 3D is below.
The Gosper curve is a plane-filling curve that provides an algorithm for connecting a tiling of hexagons, as shown at left here. It can be covered with hexagons in a similar manner to that used for the Hilbert curve with squares. This results in a chain of hexagons, a polyhex, with gaps that are long chains of hexagons, as shown at right for the second iteration of the curve. If the hexagons are treated as stepping stones, the tower-like structure below results.
Sierpinski Carpet Curve
A third example is based on a path I discovered a few years ago that connects the squares comprising a Sierpinski Carpet, shown at left here. The large open square in the center of this curve brought stepwells to mind. Making each square in the curve a square step results in the structure at right. These sorts of broad square steps do not evoke actual stepwells very strongly, so I decided to add small flights of stairs between the large squares, which become landings, as shown below.
All images copyright Robert Fathauer
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