Lynnclaire Dennis' polyhedra

After a near-death experience in 1987 Lynnclaire Dennis brought back "The Pattern", a mandala of life and love. Seeing the Pattern, she "was looking at life itself. It was light, it was time and space. It was the energy of all matter."
According to Robert W. Gray the "pattern" seems to be a trefoil knot which "lives" on a torus (see reference for an animation). Lynnclaire has told him that "she saw a polyhedron inside another polyhedron. The inner most polyhedron had 48 cones of light coming out of it at 48 vertices. This polyhedron had 144 triangular faces. The outer polyhedron had 120 triangular faces." She confirmed it to me personally.

the golden 120-polyhedron (first stellation of the rhombic triacontahedron)

This 120 faces polyhedron hides several outstanding properties:
  • the coordinates of its 62 vertices belong all to the set
    {-φ³, -φ², -φ, 0, φ, φ², φ³} where φ refers to the golden ratio,
  • all the platonic solids share their vertices with this 120-polyhedron (dodecahedron and icosahedron, cube and tetrahedron, octahedron) and so do the rhombic dodecahedron and triacontahedron,
  • the 62=20+5x6+12 vertices are those of a dodecahedron (which coincide in two ways with the vertices of five tetrahedra), five octahedra and an icosahedron,
  • the icosahedron's and cube's edges have same length which is also the distance from the center to the octahedra's vertices,
  • many different "jitterbugs" appear in different positions within this 120-polyhedron; five jitterbugs define all the 62 vertices during their expansion/contraction/rotation motion.

Remarks: This 120-polyhedron can by built by assembling 30 pyramids on the faces of a rhombic triacontahedron; there are no pyramids in the compound of five octahedra which also has 120=5x(6x4) triangular faces (but no four of the five vertices which define a "peak" are coplanar).
The first stellation of the icosidodecahedron (compound of a regular dodecahedron and a regular icosahedron) is an other 120-polyhedron with triangular faces.

two 144-polyhedra

starting with a Waterman polyhedron (or with the great rhombicuboctahedron)

Take the Waterman polyhedron with root 7 (48 vertices) and move the hexagonal faces toward the center until the rectangles become squares; now all 26 faces are regular (this Archimedean polyhedron is the great rhombicuboctahedron). Augment each face with an appropriate regular pyramid and you get the first 144-polyhedron.
starting with the cuboctahedron

Take the cuboctahedron with its circumsphere, project the 6+8=14 face's centers on the circumsphere and you get a polyhedron with 26 vertices and 48 identical triangular faces. Augment each face with an irregular tetrahedron (with its apex on the line joining the center of the polygon to the incenter of the face) and you get the second 144-polyhedron.

Remarks:
The two light blue polyhedra (before the final augmentations with pyramids) are dual.
The two 144-polyhedra have 144 triangular faces, 74 vertices and 216 edges, but their vertices don't have the same orders (they have different topologies). Both have octahedral symmetry (like the cube).
On the second 144-polyhedron the "48 cones of light" described by Lynnclaire can be defined with the center (common apex) and the incircles of the 48 triangles with red sides (bases of the tetrahedra).

references: •  Lynnclaire Dennis' near-death experience
•  The Pattern  by Lynnclaire Dennis, published by Entagram Productions Inc. 1997
    an animated gif of "The Pattern" (Robert W. Gray)
•  the 120-polyhedron; the cuboctahedron and CCP approaches of the 144-polyhedra by Robert W. Gray

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French 		convex polyhedra - non convex polyhedra - interesting polyhedra - related subjects May 2005
updated 25-06-2005