∞ · the large numbers of Q6

Astronomical counts of the cube

The large numbers of Q6, with their sources

Reference page, for reading

The hypercube of the 64 hexagrams hides enormous numbers. Here are the ones that can be computed with a closed formula (and are verified) and the ones that are only cited because computing them is beyond reason. The rule of the laboratory is kept to the letter: no figure without a source.

Computed and verified

Quantity	Value	Formula
Vertices (hexagrams) computed	64	2⁶
Edges (single-line mutations) computed	192	6 · 2⁶ / 2
Diameter (maximum distance) computed	6	n
Automorphisms of the graph Q6 hyperoctahedral group Z₂ ≀ S₆ (Harary, Graph Theory)	46,080	2⁶ · 6!
De Bruijn sequences B(2,6) computed (De Bruijn's formula)	67,108,864	2^(2⁵ − 6) = 2²⁶
Maximal chains from Kun to Qian computed (lattice B6)	720	6!
Orbits under the Klein group computed	20	Burnside
Orbits under D4 (the square) computed	10	Burnside

Hexagrams at each distance (binomial coefficients of 6)

d=0

d=1

d=2

d=3

d=4

d=5

d=6

from any hexagram there are C(6,k) at distance k: 1, 6, 15, 20, 15, 6, 1 (adding up to 64). The only one at distance 6 is its opposite (dui).

Cited (too large to reproduce)

Cyclic Gray codes (Hamiltonian cycles) of Q6

The number of closed walks that visit the 64 hexagrams changing one line per step is finite, known and astronomical. We do not reproduce it: its exact value is tabulated in the source.

source: OEIS A003042 (cyclic Gray codes / directed Hamiltonian cycles of the n-cube)

Their number grows super-exponentially with the dimension of the cube: the exact values for each dimension are tabulated in the cited source. Q6 lives so many orders of magnitude up that it is cited rather than printed; the laboratory does not publish a figure without its source.