!-Created June 2019.—>
Walkable Knots and Links
The crossings in links and knots create challenges and opportunities for more complex and interesting walkable designs. The Borromean rings are commonly shown in two dimensions as three flat interlocking circular rings. This structure is not possible as drawn, with distortion from perfect circles being required in a 3D object.
The (5,2) torus knot has five crossings and five-fold symmetry. This layout employs a pentagram inset in a pentagon.
(5,3) Torus Knot
Using torus knots with more crossings, which do not in general exhibit an alternating weave, requires more levels. This creates potential for more dramatic designs. The (5,3) torus knot has ten crossings, requiring three levels as constructed here.
(8,3) Torus Knot
This knot has an over-over, under-under weave like the (5,3) torus knot, but 8-fold symmetry. An octagram layout is used here, where the path acts as stairs and walkways accessing the ramparts of a fortified structure.
Trefoil Knot with Handrails
The trefoil is the simplest knot, and the structure shown here is more realistic than the above structures in terms of what would need to be built for our modern world. It's compactness would keep the price tag and footprint reasonable for a public park.
In this version every other staircase is replaced with a slide, making a playground object.
(6,5) Torus Knot
With a more complex knot like the (6,5) torus knot, with 18 crossings, it becomes more challenging to fit all the crossings in with bridges that allow adequate clearance. A spiral staircase allows a large elevation gain in a small footprint. Descending on slides would be more fun than walking stairs, and the slides allow for graceful smooth curves.
This knot layout, which contains a square grid of alternating weave, comes from an iterated knot I designed previously. More architectural detail is added in this example. The cupolas supported by pillars are consciously patterned after the sort of architectural details Escher used in prints like Belvedere. Since the crossings are laid out as a square weave, they could obviously be extended to an arbitrarily large structure.
All images copyright Robert Fathauer
Robert Fathauer's Fractal Diversions home
Fractal Tiling Fractal Knots Gasket Fractals Fractal Trees Hyperbolic & Folded Fractals Polyhedra Papers etc.