600-cell

Since the tetrahedra are made of shorter triangle edges than the octahedra (by a factor of ⁠1/φ⁠, the inverse golden ratio), the 600-cell does not have unit edge-length in a unit-radius coordinate system the way the 24-cell and the tesseract do; unlike those two, the 600-cell is not radially equilateral.

[ab] The vertex chords of the 600-cell are arranged in geodesic great circle polygons of five kinds: decagons, hexagons, pentagons, squares, and triangles.

Each √1 chord is the long diameter of a face-bonded pair of tetrahedral cells (a triangular bipyramid), and passes through the center of the shared face.

[27] The √2.𝚽 = φ chords form the legs of 720 central isosceles triangles (72 sets of 10 inscribed in each decagon), 6 of which cross at each vertex.

[af] Each fiber bundle of Clifford parallel great circles[ap] is a discrete Hopf fibration which fills the 600-cell, visiting all 120 vertices just once.

[ac] The 12 great circles and 30-cell rings of the 600-cell's 6 characteristic Hopf fibrations make the 600-cell a geometric configuration of 30 "points" and 12 "lines" written as 302125.

The densely packed helical cell rings[38][39][32] of fibrations are cell-disjoint, but they share vertices, edges and faces.

Thorold Gosset discovered the semiregular 4-polytopes, including the snub 24-cell with 96 vertices, which falls between the 24-cell and the 600-cell in the sequence of convex 4-polytopes of increasing size and complexity in the same radius.

[f] The icosahedra are face-bonded into geodesic "straight lines" by their opposite yellow faces, bent in the fourth dimension into a ring of 6 icosahedral pyramids.

[ad] The three helices are geodesic "straight lines" of 10 edges: great circle decagons which run Clifford parallel[af] to each other.

One can begin by picking out an icosahedral pyramid cluster centered at any arbitrarily chosen vertex, so there are 120 overlapping icosahedra in the 600-cell.

[k] There is another useful way to partition the 600-cell surface, into 24 clusters of 25 tetrahedral cells, which reveals more structure[55] and a direct construction of the 600-cell from its predecessor the 24-cell.

Six more triangular dipyramids fit into the concavities on the surface of the cluster of 5,[br] so the exterior chords connecting its 4 apical vertices are also 24-cell edges of length √1.

The 600-cell can also be partitioned into 20 cell-disjoint intertwining rings of 30 cells,[34] each ten edges long, forming a discrete Hopf fibration which fills the entire 600-cell.

The 30-cell ring is the 3-dimensional space occupied by the 30 vertices of three cyan Clifford parallel great decagons that lie adjacent to each other, 36° = ⁠𝜋/5⁠ = one 600-cell edge length apart at all their vertex pairs.

[cg] The 30 magenta edges joining these vertex pairs form a helical triacontagram, a skew 30-gram spiral of 30 edge-bonded triangular faces, that is the Petrie polygon of the 600-cell.

[ch] Five of these 30-cell helices nest together and spiral around each of the 10-vertex decagon paths, forming the 150-cell torus described in the grand antiprism decomposition above.

[67][68] For example, a full isoclinic rotation of the 600-cell in decagonal invariant planes takes each of the 120 vertices through 15 vertices and back to itself, on a skew pentadecagram2 geodesic isocline of circumference 5𝝅 that winds around the 3-sphere, as each great decagon rotates (like a wheel) and also tilts sideways (like a coin flipping) with the completely orthogonal plane.

, we can construct: The regular convex 4-polytopes are an expression of their underlying symmetry which is known as SO(4), the group of rotations[71] about a fixed point in 4-dimensional Euclidean space.

[dc] Disjoint 24-cells are related by a ⁠𝜋/5⁠ isoclinic rotation of an entire fibration of 12 Clifford parallel decagonal invariant planes.

)[de] Non-disjoint 24-cells are related by a ⁠𝜋/5⁠ isoclinic rotation of an entire fibration of 20 Clifford parallel hexagonal invariant planes.

Each left or right fiber bundle of isoclines by itself constitutes a discrete Hopf fibration which fills the entire 600-cell, visiting all 120 vertices just once.

In addition, each distinct isoclinic rotation (left or right) can be performed in a positive or negative direction along the circular parallel fibers.

[g] The 600-cell has six such discrete decagonal fibrations, and each is the domain (container) of a unique left-right pair of isoclinic rotations (left and right fiber bundles of 12 great pentagons).

The √1 chords of the 30-cell ring (without the √0.𝚫 600-cell edges) form a skew triacontagram{30/4}=2{15/2} which contains 2 disjoint {15/2} Möbius double loops, a left-right pair of pentadecagram2 isoclines.

The 10 √1.𝚫 chords of each isocline form a skew decagram {10/3}, 10 great pentagon edges joined end-to-end in a helical loop, winding 3 times around the 600-cell through all four dimensions rather than lying flat in a central plane.

Each pair of black and white isoclines (intersecting antipodal great hexagon vertices) forms a compound 20-gon icosagram {20/6}=2{10/3}.

Each isocline is a skew {24/5} 24-gram, 24 φ = √2.𝚽 chords joined end-to-end in a helical loop, winding 5 times around one 24-cell through all four dimensions rather than lying flat in a central plane.

[94] The symmetries of the 3-D surface of the 600-cell are somewhat difficult to visualize due to both the large number of tetrahedral cells,[v] and the fact that the tetrahedron has no opposing faces or vertices.

A three-dimensional model of the 600-cell, in the collection of the Institut Henri Poincaré, was photographed in 1934–1935 by Man Ray, and formed part of two of his later "Shakesperean Equation" paintings.

Net
Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths [ c ] with angles of arc. The golden ratio [ q ] governs the fractional roots of every other chord, [ r ] and the radial golden triangles which meet at the center.
A 3D projection of a 600-cell performing a simple rotation . The 3D surface made of 600 tetrahedra is visible.
A 3D projection of a 24-cell performing a simple rotation . The 3D surface made of 24 octahedra is visible. It is also present in the 600-cell, but as an invisible interior boundary envelope.
Cell-centered stereographic projection of the 600-cell's 72 central decagons onto their great circles. Each great circle is divided into 10 arc-edges at the intersections where 6 great circles cross.
Triacontagram {30/12}=6{5/2} is the Schläfli double six configuration 30 2 12 5 characteristic of the H 4 polytopes. The 30 vertex circumference is the skew Petrie polygon. [ aw ] The interior angle between adjacent edges is 36°, also the isoclinic angle between adjacent Clifford parallel decagon planes. [ at ]
A regular icosahedron colored in snub octahedron symmetry. [ bi ] Icosahedra in the 600-cell are face bonded to each other at the yellow faces, and to clusters of 5 tetrahedral cells at the blue faces. The apex of the icosahedral pyramid (not visible) is a 13th 600-cell vertex inside the icosahedron (but above its hyperplane).
A cluster of 5 tetrahedral cells: four cells face-bonded around a fifth cell (not visible). The four cells lie in different hyperplanes.
100 tetrahedra in a 10×10 array forming a Clifford torus boundary in the 600 cell. [ by ] Its opposite edges are identified, forming a duocylinder .
Icosagram {20/6}=2{10/3} contains 2 disjoint {10/3} decagrams (red and orange) which connect vertices 3 apart on the {10} and 6 apart on the {20}. In the 600-cell the edges are great pentagon edges spanning 72°.
The Clifford polygon of the 600-cell's isoclinic rotation in great square invariant planes is a skew regular {24/5} 24-gram , with φ = √ 2.𝚽 edges that connect vertices 5 apart on the 24-vertex circumference, which is a unique 24-cell ( √ 1 edges not shown).
Vertex geometry of the radially equilateral 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.
Two Clifford parallel great circles spanned by a twisted annulus .