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**The (192+192) lines
& broken lines of the 64 hexagrams symbolize the 192 rotations of the
tesseract and its 192 rotations+reflections**

The point has one symmetry, namely, the reflection of it into itself in a mirror in which it is
embedded. The straight line has two symmetries: it is unchanged by a rotation of 180° about an axis passing through
its centre perpendicular to it and by a reflection in a mirror passing through its centre perpendicular to its
length. The square has eight symmetries: four rotations of 0°, 90°, 180° & 270° and four reflections: two
across its diagonals and two across vertical or horizontal lines passing through centres of opposite pairs of
lines. The cube has **48** symmetries that define the octahedral symmetry group O_{h}. Its 24
proper rotations fall into four classes consisting of:

1. the identity, or rotation of 360°;

2. nine rotations of 90° (×3), 180° (×3) & 270° (×3) about axes passing through the centres of opposite
faces;

3. six rotations of 180° about axes passing through the *centres* of opposite pairs of edges;

4. eight rotations of 120° (×4) & 240° (×4) about the four body diagonals.

The cube has also 24 proper rotations combined with reflections that leave it unchanged. The four-dimensional version of the cube (3-cube) is the 4-cube, or tesseract. It has 384 symmetries. The easiest way of seeing this is to note that each of its 16 vertices is the intersection of four edges, which can be permuted in (4!=24) ways, generating 24 independent orientation, so that there are (16×24=384) possible ways of changing the 4-cube.

The 384 symmetries* consist of 192 rotations and 192 combinations of rotations and reflections. Here are animated examples of its rigid rotations. The three-dimensional projections of the rotations of a tesseract can also be viewed here. The 192 rotations comprise the following rigid rotations in a plane:

1. 6 planes containing four 3-cube-centres and four face-centres, each with three rotations (90°,
180°, 270°), for a total of **18 rotations**;

2. 24 planes containing two 3-cube-centres, two face-centres, and four edge-centres, each with one rotation (180°),
for a total of **24 rotations**;

3. 16 planes containing six face-centres, each with two rotations (120° and 240°), for a total of **32
rotations**;

4. 12 planes containing four face-centres and four vertices, each with one rotation (180°), for a total of
**12 rotations**.

There are 105 rotations that act in more than one plane, that is, 104 rotations, apart from the
inversion, in which the point (x_{1}, x_{2}, x_{3}, x_{4}) goes to (−x_{1},
−x_{2}, −x_{3}, −x_{4}). Together with the identity, there are **87** proper
rotations in the 4-cube, which consists of 16 vertices (0-cubes), 32 edges (1-cubes), 24 squares (2-cubes) & 8
cubes (3-cubes), i.e., **80** 0-, 1-, 2- & 3-cubes. **87** is the number value
of *Levanah*, the Mundane Chakra of Yesod, and **80** is the number value of this Sephirah. If
one is willing to make the reasonable judgement that the appearance of two numbers associated with the
*same* Sephirah is *not* a matter of coincidence, it indicates that *the
tesseract is a holistic system*, its 384 symmetries being symbolized by the 384 lines & broken lines
in the **64** hexagrams of the I Ching table. The **48** lines & broken lines in its
diagonal would then symbolize the **48** symmetries of a cube, namely, the octahedral symmetries of O_{h}, the 24 lines & broken lines in the eight
upper trigrams in the diagonal denoting its 24 rigid rotations and the 24 lines & broken lines in the eight
lower, diagonal trigrams denoting its 24 rotations/reflections. The seven off-diagonal copies of the 24
lines & broken lines in the eight diagonal trigrams would refer to the **168** rotations of the
seven copies of the 3-cube (its "cells") that extend in the fourth dimension of space.

What is the significance of the 4-cube for superstrings? Hypercubes can fill 4-space, just as
cubes can be stacked to fill 3-space. According to E_{8}×E_{8} heterotic superstring theory, two
10-dimensional space-time sheets are separated by a finite gap extending along the tenth dimension of space
predicted by supergravity theories. It suggests that 3-space might be a cross-section of a 4-space tessellated
across this gap with tesseracts. If so, the eight diagonal hexagrams of the I Ching table would represent the
octahedral symmetry O_{h} of cubes filling ordinary space and the 56 off-diagonal hexagrams would represent
the extra 336 symmetries of hypercubes filling 4-dimensional space that are hidden from physical awareness because
they refer to a dimension of space that *normal* cognition is not programmed to detect, namely, the seven
cubic cells (3-cubes) of a tesseract that surround its central cube. Perhaps Charles Hinton was right
after all when he proposed in his book *A New Era of Thought* (1888) that space has four dimensions (see here)....

It is, of course, not just the **64** hexagrams that symbolize the tesseract
because, as this website proves, *all* sacred geometries embody the universal patterns:

384 = 192 + 192,

192 = 24 + **168**,

and

384 = **48** + 336

(see here). Nor, in view of remarks made in the footnote, is the
4-cube the only 4-polytope that possesses 384 symmetries. So does the 24-cell, which plays a central role in the
geometry of the 4_{21} polytope discussed in **4-d sacred
geometries**. Here is the reason why it embodies this global parameter of holistic systems.

* This is the hyperoctahedral group BC_{4} of 384 symmetries possessed by the tesseract
and two other 4-dimensional, regular polytopes called the "16-cell" (the dual of the tesseract) and the
"24-cell," which is self-dual. See here. The 192 symmetries are those of the hyperoctahedral group D_{4}, a subgroup of
BC_{4}. The demitesseract, the 4-dimensional version of demihypercubes, has 192 symmetries
(see here). D_{4} contains the tetrahedral group T_{d} with symmetry order 24.
The 192 symmetries are symbolised by the 192 lines & broken lines in each diagonal half of the 8×8 matrix
array of **64** hexagrams, the 24 lines & broken lines in the eight diagonal trigrams in each
half symbolising the 24 symmetries of T_{d}. The I Ching table represents the 384 symmetries of the
tesseract, the 192 symmetries of the demitesseract, the **48** symmetries of the octahedral group
and the 24 symmetries of the tetrahedral group.

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