page_title: 120-cell
section_title: Intertwining rings
hierarchical_section_title: 120-cell / Visualization / Intertwining rings
caption_attribution_description: English: Two intertwined rings of the 120-cell.
context_page_description: In geometry, the 120-cell is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is also called a C₁₂₀, dodecaplex, hyperdodecahedron, polydodecahedron, hecatonicosachoron, dodecacontachoron and hecatonicosahedroid.
  The boundary of the 120-cell is composed of 120 dodecahedral cells with 4 meeting at each vertex.
  It can be thought of as the 4-dimensional analog of the regular dodecahedron. Just as a dodecahedron can be built up as a model with 12 pentagons, 3 around each vertex, the dodecaplex can be built up from 120 dodecahedra, with 3 around each edge.
  The Davis 120-cell, introduced by Davis, is a compact 4-dimensional hyperbolic manifold obtained by identifying opposite faces of the 120-cell, whose universal cover gives the regular honeycomb {5,3,3,5} of 4-dimensional hyperbolic space.
context_section_description: The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration.  Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells.  Five such 10-cell rings can be placed adjacent to the original 10-cell ring.  Although the outer rings "spiral" around the inner ring (and each other), they actually have no helical torsion.  They are all equivalent.  The spiraling is a result of the 3-sphere curvature.  The inner ring and the five outer rings now form a six ring, 60-cell solid torus. One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first.  The 120-cell, like the 3-sphere, is the union of these two (Clifford) tori.  If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle.
  Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed.  It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint great circles.
caption: Two intertwining rings of the 120-cell.
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