This page records the tight-frame enumeration behind the equal-strut 6-cube projection gallery.

The question is: for which cube dimensions can the coordinate directions be projected into 3D so that all projected cube edges are zomeable, use a single vZome color, and have the same abstract strut length?

Setup

For each vZome color, take the finite orbit of its unoriented direction axes under icosahedral symmetry. A choice of d axes gives an orthographic projection of the d-cube exactly when the axes form a tight frame:

sum_i v_i v_i^T = c I.

The enumeration is exact in the golden field. Antipodal directions are identified, and classes are quotiented by the full icosahedral symmetry group, including mirror symmetries.

The nondegenerate convention used here is that the projected coordinate directions are nonzero and distinct. This excludes lower-dimensional degeneracies, such as reusing three blue axes to obtain an ordinary cube.

Cube dimensions with same-color equal-strut projections

The entries below count symmetry classes of nonzero, distinct projected coordinate directions. Dimensions not shown have no same-color equal-strut class in these vZome direction orbits.

cube dimension red yellow blue green
3     1  
4   1    
6 1 1 1 2
9     1 1
10   1    
12     1 12
15     1 2
18       12
21       1
24       2
30       1

The gallery page shows exactly the five dimension-6 classes.

Full representative enumeration

The tables below make the enumeration self-contained. Each row gives one representative tight-frame class, including the actual golden-field direction coordinates.

  • Rotation group size: 60
  • Full group size used for rotation/mirror quotient: 120
  • Subsets use unoriented axes, so antipodal directions are counted once.
  • Coordinates are unnormalized golden-field vectors; all directions within one color orbit have a common norm.
  • In this document, an indecomposable tight frame means a tight subset that cannot be partitioned into two nonempty disjoint tight subsets spanning R^3.

Yellow

  • Axes: 10
  • Distinct induced symmetry permutations on axes: 60
  • Raw tight subset counts by size: {'4': 5, '6': 5, '10': 1}
  • Symmetry class counts by size: {'4': 1, '6': 1, '10': 1}
  • Indecomposable raw counts by size: {'4': 5, '6': 5}
  • Indecomposable symmetry class counts by size: {'4': 1, '6': 1}
size representative coordinates
4 (1+φ, 1+φ, 1+φ)
(1+φ, -1-φ, -1-φ)
(-1-φ, -1-φ, 1+φ)
(-1-φ, 1+φ, -1-φ)
6 (1+φ, 1+φ, 1+φ)
(1+φ, -1-φ, -1-φ)
(-1-φ, -1-φ, 1+φ)
(0, -φ, 1+2φ)
(φ, -1-2φ, 0)
(1+2φ, 0, φ)
10 (1+φ, 1+φ, 1+φ)
(1+φ, -1-φ, -1-φ)
(0, φ, 1+2φ)
(-1-φ, -1-φ, 1+φ)
(φ, 1+2φ, 0)
(0, -φ, 1+2φ)
(-1-φ, 1+φ, -1-φ)
(φ, -1-2φ, 0)
(1+2φ, 0, φ)
(-1-2φ, 0, φ)

Yellow indecomposable classes

size representative coordinates
4 (1+φ, 1+φ, 1+φ)
(1+φ, -1-φ, -1-φ)
(-1-φ, -1-φ, 1+φ)
(-1-φ, 1+φ, -1-φ)
6 (1+φ, 1+φ, 1+φ)
(1+φ, -1-φ, -1-φ)
(-1-φ, -1-φ, 1+φ)
(0, -φ, 1+2φ)
(φ, -1-2φ, 0)
(1+2φ, 0, φ)

Red

  • Axes: 6
  • Distinct induced symmetry permutations on axes: 60
  • Raw tight subset counts by size: {'6': 1}
  • Symmetry class counts by size: {'6': 1}
  • Indecomposable raw counts by size: {'6': 1}
  • Indecomposable symmetry class counts by size: {'6': 1}
size representative coordinates
6 (0, 1+2φ, 1+φ)
(1+2φ, 1+φ, 0)
(-1-φ, 0, 1+2φ)
(1+2φ, -1-φ, 0)
(1+φ, 0, 1+2φ)
(0, 1+2φ, -1-φ)

Red indecomposable classes

size representative coordinates
6 (0, 1+2φ, 1+φ)
(1+2φ, 1+φ, 0)
(-1-φ, 0, 1+2φ)
(1+2φ, -1-φ, 0)
(1+φ, 0, 1+2φ)
(0, 1+2φ, -1-φ)

Blue

  • Axes: 15
  • Distinct induced symmetry permutations on axes: 60
  • Raw tight subset counts by size: {'3': 5, '6': 10, '9': 10, '12': 5, '15': 1}
  • Symmetry class counts by size: {'3': 1, '6': 1, '9': 1, '12': 1, '15': 1}
  • Indecomposable raw counts by size: {'3': 5}
  • Indecomposable symmetry class counts by size: {'3': 1}
size representative coordinates
3 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(0, 2+2φ, 0)
6 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(1+φ, 1+2φ, φ)
(0, 2+2φ, 0)
(φ, -1-φ, 1+2φ)
(-1-2φ, φ, 1+φ)
9 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(1+φ, 1+2φ, φ)
(0, 2+2φ, 0)
(φ, -1-φ, 1+2φ)
(1+φ, -1-2φ, -φ)
(1+2φ, φ, 1+φ)
(-1-2φ, φ, 1+φ)
(φ, 1+φ, -1-2φ)
12 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(1+φ, 1+2φ, φ)
(0, 2+2φ, 0)
(φ, -1-φ, 1+2φ)
(1+φ, -1-2φ, -φ)
(1+2φ, φ, 1+φ)
(-φ, 1+φ, 1+2φ)
(-1-2φ, φ, 1+φ)
(φ, 1+φ, -1-2φ)
(1+2φ, -φ, 1+φ)
(1+φ, 1+2φ, -φ)
15 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(1+φ, 1+2φ, φ)
(0, 2+2φ, 0)
(φ, -1-φ, 1+2φ)
(1+φ, -1-2φ, -φ)
(1+2φ, φ, 1+φ)
(-φ, 1+φ, 1+2φ)
(-1-2φ, φ, 1+φ)
(φ, 1+φ, -1-2φ)
(1+2φ, -φ, 1+φ)
(-1-2φ, -φ, 1+φ)
(φ, 1+φ, 1+2φ)
(1+φ, 1+2φ, -φ)
(1+φ, -1-2φ, φ)

Blue indecomposable classes

size representative coordinates
3 (2+2φ, 0, 0)
(0, 0, 2+2φ)
(0, 2+2φ, 0)

Green

  • Axes: 30
  • Distinct induced symmetry permutations on axes: 60
  • Raw tight subset counts by size: {'6': 25, '9': 20, '12': 260, '15': 40, '18': 260, '21': 20, '24': 25, '30': 1}
  • Symmetry class counts by size: {'6': 2, '9': 1, '12': 12, '15': 2, '18': 12, '21': 1, '24': 2, '30': 1}
  • Indecomposable raw counts by size: {'6': 25, '9': 20, '12': 130, '15': 20}
  • Indecomposable symmetry class counts by size: {'6': 2, '9': 1, '12': 8, '15': 1}
size representative coordinates
6 (1, 2+3φ, 1+φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
6 (-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
9 (-φ, 1+3φ, 1+2φ)
(2+3φ, 1+φ, -1)
(1+2φ, -φ, -1-3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+2φ, -φ, 1+3φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, 2+2φ, 0)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(-1-3φ, -1-2φ, -φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, 2+2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(1, 2+3φ, 1+φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(-1-φ, -1, 2+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
12 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1+φ, 1, 2+3φ)
(1+3φ, -1-2φ, φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
12 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(1, -2-3φ, 1+φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (-φ, 1+3φ, 1+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1+2φ, -φ, -1-3φ)
(-1-3φ, -1-2φ, -φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(-2-3φ, 1+φ, 1)
12 (2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(1, 2+3φ, 1+φ)
(1+2φ, -φ, -1-3φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
12 (-1-2φ, -φ, 1+3φ)
(1, 2+3φ, 1+φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
15 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
15 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, -1-3φ)
(1+φ, 1, 2+3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
18 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1+2φ, -φ, -1-3φ)
(-1-3φ, -1-2φ, -φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
18 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(1+2φ, -φ, -1-3φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
18 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-2-2φ, 0, 2+2φ)
(0, 2+2φ, 2+2φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, 2+2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1+2φ, -φ, -1-3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, 2+2φ, 0)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(1+φ, 1, 2+3φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-2-3φ, 1+φ, 1)
18 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, -1-3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, -2-2φ, 0)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-2-3φ, 1+φ, 1)
18 (2+2φ, -2-2φ, 0)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(1, -2-3φ, 1+φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
18 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1, 2+3φ, 1+φ)
(1+φ, 1, 2+3φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
18 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
21 (2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(0, 2+2φ, -2-2φ)
(-1-φ, -1, 2+3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
24 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(-1, -2-3φ, 1+φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
24 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(-1, -2-3φ, 1+φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, -1-3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
30 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(0, 2+2φ, -2-2φ)
(-1, -2-3φ, 1+φ)
(2+3φ, -1-φ, 1)
(1+φ, -1, 2+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, -1-3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)

Green indecomposable classes

size representative coordinates
6 (1, 2+3φ, 1+φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(1+3φ, -1-2φ, -φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
6 (-1-3φ, -1-2φ, -φ)
(1+φ, -1, -2-3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)
9 (-φ, 1+3φ, 1+2φ)
(2+3φ, 1+φ, -1)
(1+2φ, -φ, -1-3φ)
(1+φ, 1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(1+2φ, -φ, 1+3φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, 2+2φ, 0)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(-1-3φ, -1-2φ, -φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, 2+2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(1, 2+3φ, 1+φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(-1-φ, -1, 2+3φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
(1+3φ, -1-2φ, -φ)
12 (2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(-1-φ, -1, 2+3φ)
(-1-3φ, -1-2φ, -φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1+φ, 1, 2+3φ)
(1+3φ, -1-2φ, φ)
(-φ, 1+3φ, -1-2φ)
(1, -2-3φ, 1+φ)
12 (2+2φ, -2-2φ, 0)
(-2-2φ, 0, 2+2φ)
(2+3φ, 1+φ, -1)
(0, 2+2φ, 2+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(1, -2-3φ, 1+φ)
(-φ, -1-3φ, 1+2φ)
12 (-φ, 1+3φ, 1+2φ)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(-1-2φ, -φ, 1+3φ)
(1+2φ, -φ, -1-3φ)
(-1-3φ, -1-2φ, -φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
12 (2+3φ, 1+φ, -1)
(1+2φ, φ, 1+3φ)
(-φ, -1-3φ, -1-2φ)
(1+3φ, 1+2φ, -φ)
(1, 2+3φ, 1+φ)
(1+2φ, -φ, -1-3φ)
(1+φ, -1, -2-3φ)
(1+3φ, -1-2φ, φ)
(1+2φ, -φ, 1+3φ)
(-φ, 1+3φ, -1-2φ)
(1+3φ, -1-2φ, -φ)
(-φ, -1-3φ, 1+2φ)
15 (2+2φ, 2+2φ, 0)
(2+2φ, -2-2φ, 0)
(2+2φ, 0, 2+2φ)
(-φ, 1+3φ, 1+2φ)
(-2-2φ, 0, 2+2φ)
(1+2φ, φ, 1+3φ)
(-1-2φ, -φ, 1+3φ)
(1, 2+3φ, 1+φ)
(-1-φ, -1, 2+3φ)
(1+2φ, -φ, 1+3φ)
(-1, 2+3φ, 1+φ)
(-φ, 1+3φ, -1-2φ)
(2+3φ, 1+φ, 1)
(-φ, -1-3φ, 1+2φ)
(-2-3φ, 1+φ, 1)