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Roelofs polyknots are intricate knots that exhibit polyhedral symmetries or asymmetries. Inspired by the mathematical artistry of Rinus Roelofs—particularly their 2018 Bridges Math Art Conference paper, Weaving Double-Layered Polyhedra—as well as Johannes Kepler’s annuloid studies, these polyknots appear in chiral pairs, though only one from each pair is typically displayed. Each polyknot forms a knot at the vertices of its base polyhedron, with the type of knot determined by the vertex number, which corresponds to the vertex configuration of the base polyhedron.
The crossing number of a polyknot is determined by multiplying the number of vertices of the base polyhedron by its vertex number. For instance, a cubic polyknot, based on a cube with 8 vertices and a vertex number of 3, has a crossing number of 8 × 3 = 24.
This section showcases over 150 Roelofs polyknots, including those based on the five Roelofs-Platonic, thirteen Roelofs-Archimedean, thirteen Roelofs-Catalan, five Roelofs-Prism, five Roelofs-Antiprism, and ninety-two Roelofs-Johnson polyknots. Below is a collection of Kepler-Roelofs polyknots.
Explore more of Rinus Roelofs' mathematical art here: Rinus Roelofs - Index .
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs cubic polyknot
components: squares, hexagons
structure: Kepler-Roelofs cubic polyknot
components: squares, pentagons
structure: Kepler-Roelofs tetrahedral polyknot
components: triangles
structure: Roelofs tetrahedral polyknot
components: triangles, pentagons
structure: Roelofs tetrahedral polyknot
components: triangles
structure: Kepler Roelofs cubic polyknot
components: octagons
structure: Roelofs cubic polyknot
with nested cube
components: squares
structure: Kepler-Roelofs octahedral polyknot
components: squares, hexagons
structure: Kepler-Roelofs tubular polyknot
components: squares, hexagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs tetrahedral polyknot
components: triangles, hexagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, pentagons
structure: Kepler-Roelofs dodecahedral polyknot
components: triangles, squares, pentagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs dodecahedral polyknot
components: squares, hexagons
structure: Kepler-Roelofs cubic polyknot
components: octagons
structure: Roelofs cubic polyknot
components: squares, octagons
structure: Roelofs cubic polyknot
components: squares
structure: Roelofs cubic polyknot
components: squares, octagons
structure: Kepler-Roelofs dodecahedral polyknot
components: pentagons
Kepler-Roelofs dodecahedral polyknot
components: pentagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs octahedral polylink
components: squares, hexagons
structure: Kepler-Roelofs tetrahedral polyknot
components: squares, hexagons
structure: Kepler-Roelofs tetrahedral polylink
components: squares, hexagons
structure: Roelofs tetrahedral polyknot
components: triangles, pentagons
structure: Roelofs tetrahedral polyknot
components: triangles, squares, pentagons
structure: Roelofs tetrahedral polyknot
components: triangles, pentagons, hexagons
structure: Roelofs tetrahedral polyknot
components: triangles, squares, pentagons
structure: Kepler Roelofs star dodecahedral polyknot
components: triangles, squares, pentagons
structure: Kepler Roelofs star dodecahedral polyknot
components: triangles, squares, pentagons
structure: Kepler Roelofs star dodecahedral polyknot
components: triangles, squares, pentagons
structure: Roelofs star icosahedral polyknot
components: triangles, squares, pentagons
structure: Roelofs star cubic polyknot
components: triangles, squares
structure: Kepler Roelofs star dodecahedral polyknot
components: triangles, squares, pentagons
structure: Roelofs star octahedral polyknot
components: triangles, squares
structure: Roelofs star tetrahedral polyknot
components: triangles, squares
structure: Kepler Roelofs star dodecahedral polyknot
components: triangles, squares, pentagons
structure: Kepler-Roelofs octahedral polylink
components: triangles
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