Nuclear

Established[E] icosian geometry on the unit S³. Shell-to-nuclide assignments are [M].

S³ 600-Cell Sections

The regular 600-cell on the unit 3-sphere has 120 vertices. Its parallels are an icosahedron, a dodecahedron, an inner icosahedron, and an icosidodecahedron. A nuclide placed on one of those sections is a modelled reading.

Last updated: October 2026

Reading this entry

The regular 600-cell on the unit 3-sphere supplies the vertex geometry behind the nuclear shell reading. Its 120 vertices, edge length 1/φ, and nine latitudes are icosian geometry, and the placement of a nuclide on one of those sections is the modelled shell bookkeeping that sits beside the V4 calculator.

What this is used for

Shell closures, the lead and oxygen counts, and the identification of group order 120 with the 8s pair all borrow these latitudes. The nuclide horizon and the superheavy maps keep their own cage labels and point back here for the 12, 20, and 30 vertex sections.

Walk through

The binary icosahedral group 2I has order 120 and is the vertex set of the 600-cell {3,3,5} on the unit S³, with 120 vertices, 720 edges, 1200 faces, and 600 cells. Every edge has length 1/φ, where φ = (1+√5)/2 and 1/φ = φ − 1. The nine heights are q₀ = cos χ in {±1, ±φ/2, ±1/2, ±1/(2φ), 0}, and the parallel radius is r = sin χ = √(1 − q₀²), with sin 36° = √((5−√5)/8) ≈ 0.5878 and sin 72° = √((5+√5)/8) ≈ 0.9511. Those nine counts sum as 1+12+20+12+30+12+20+12+1 = 120. The modelled closures then read 4×2 = 8, 7×4 = 28, 12×4+2 = 50, 20×4+2 = 82, and 120+6 = 126, and |2I| = 120 completes the pair [Og] 8s².

Structure package: The vertex count, the edge length

, the nine latitudes, and the subgroup-order obstruction are theorems of the 600-cell and of the binary icosahedral group. They do not fix a nuclear radius, a binding energy, or

and

remain inputs. See /status.

[M] Shell reading: Earlier shell lists chose a polyhedron for each observed closure. The 12-vertex, 20-vertex, and 30-vertex sections below are the parallels of the 600-cell, so those three polyhedra are fixed by the slice. The step from a section to a nuclide, a magic number, or a fission threshold is a modelled reading. Weber overlap integrals are not evaluated on this page. The calibrated binding model remains the V4 engine. This 600-cell section is not the Z ≥ 80 cage 12+30=42. That horizon is the nuclide horizon.

Five-fold axes and the lattice in ℝ³

The crystallographic restriction theorem limits the rotational symmetries of a two-dimensional or three-dimensional lattice to orders 1, 2, 3, 4, and 6. A rotation of order 5 is excluded. A regular icosahedron (12 vertices), a regular dodecahedron (20 vertices), or a regular icosidodecahedron (30 vertices) therefore cannot be the unit cell of a lattice in .

The same polyhedra appear as parallel sections of a regular polytope on the unit 3-sphere. That polytope is the 600-cell. The exclusion in and the existence of the 600-cell are separate theorems. The second does not repeal the first.

The 600-cell on the unit S³

The binary icosahedral group is a subgroup of the unit quaternions. Its order is 120. The orbit is the vertex set of the regular 600-cell on the unit sphere : 120 vertices, 720 edges, 1200 triangular faces, and 600 tetrahedral cells. Every edge has length , where and .

The 24 Hurwitz units sit inside this vertex set and form a copy of the binary tetrahedral group , a subgroup of order 24. Conjugation by an icosian outside that set carries the unit outside it, so this copy of is not a normal subgroup of .

The vertices are unit quaternions, so the combinatorics are scale-free. This page sets no nuclear radius. The elastic container named by the framework remains the single cosmological .

Nine latitudes

Take the pole . The real coordinate is the height . The nine heights are . The parallel at height is a 2-sphere of Euclidean radius in the orthogonal . Within one latitude the three-dimensional chord equals the four-dimensional chord.

and . Nearest-neighbour degrees on the layers of 12, 20, 12, and 30 vertices are 5, 3, 5, and 4. Those degrees are the icosahedron, the dodecahedron, the inner icosahedron, and the icosidodecahedron. The equator is the 30-vertex layer at . The dodecahedron sits at .

Layer				In-layer edge	Section
0	0°	1	0	—	North pole
1	36°	12			Icosahedron
2	60°	20			Dodecahedron
3	72°	12		1	Inner icosahedron
4	90°	30	1		Icosidodecahedron (equator)
5	108°	12		1	Inner icosahedron
6	120°	20			Dodecahedron
7	144°	12			Icosahedron
8	180°	1	0	—	South pole

The nine counts sum to . The in-layer minimum is on both outer icosahedra, both dodecahedra, and the icosidodecahedron. It is on both inner icosahedra, the layers at and . The 720 edges of the 600-cell itself all have length . The southern layers repeat the northern ones.

Orbits that this slice does not contain

A tetrahedral cell of the 600-cell has four vertices. No latitude in this polar slice has four vertices. The binary tetrahedral subgroup has order 24, which counts group elements, not a 4-vertex shell.

The six points and their coordinate permutations lie inside the equatorial 30 and are the vertices of a regular octahedron. The binary octahedral group has order 48. Lagrange's theorem on subgroup orders requires 48 to divide . It does not, so supplies no subgroup of .

A 10-vertex gyroelongated square bipyramid is not a latitude of this slice. Deleting two opposite vertices of an icosahedron leaves a pentagonal antiprism. The cuboctahedron and the jitterbug are not constructed here.

Modelled readings of the magic numbers

The integers 2, 8, 20, 28, 50, 82, and 126 are the familiar shell closures. The products below are a modelled bookkeeping of alphas, protons, and vertices. Each row says what the product counts. The V4 engine remains the binding calculation.

Closure	Product	What is being counted
2, helium-4	Pole pair	Two of the 120 vertices. One alpha is four nucleons.
8, oxygen-16		Four alphas are . The factor 2 selects the protons. No latitude has four vertices.
20, calcium-40	20, or	The dodecahedral latitude has 20 vertices. The 10-vertex count is a separate reading; that polyhedron is not a latitude.
28, nickel-56		No latitude has seven vertices. Nickel-56 is fourteen alphas, .
50, tin-100		Twelve alphas are 48 nucleons. The added poles already belong to the 120. Tin-100 has .
82, lead-208 protons		Twenty alphas are 80 nucleons and 40 protons. Lead-208 has and .
126, lead-208 neutrons		The six octahedral vertices already lie in the equatorial 30, hence inside the 120. The sum counts those six a second time.

The order 120 and the 8s pair

is the order of the binary icosahedral group and the number of vertices of one 600-cell. In the ordinary periodic table, element 118 is the closure and the pair 119, 120 completes . Setting the group order equal to a maximum proton number is a typed identification. This page derives no branch cut and no fission threshold above . The diagnostic superheavy map is the SHE dual-edge package.

Integrity

[E]

Geometry. Order 120, the 720 edges of length

, the nine latitudes and their radii

, the in-layer edges

and

, the non-normal subgroup of order 24, and the failure of 48 to divide 120.

[M]

Shell reading. The products in the magic-number table, the identification of

with

, and any energy assigned to a polyhedral cage.