Alternative representations: Knight's move, Swastika and BaGua ?

AWOL: American Way Of Life: Assumptions -- justifying worldwide imposition of democratic imperialism (Part #7)

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Pattern of distinct Knight's moves: The variety of Knight's moves is usefully represented within a 3x3 matrix as shown below. Two sets of patterns may be distinguished: those passing around the central cell and those passing through the central cell (without starting or finishing there). Both bear a resemblance to the traditional yantra.

Pattern of Knight's moves in a 3x3 Matrix Note that a move cannot start or finish in the central cell in a 3x3 matrix
8 Moves passing through the central cell	8 Moves passing around the central cell

Formation of Swastika from Knight's moves: Complementary (or mirror image) paths, taken by two Knight's moves through the centre (left-hand image above), may be combined -- thereby to form opposite branches of a Swastika. This can be represented through the following animations.

Swastika emergent (by optical illusion) from the dynamics of Knight's moves across a 3x3 Matrix of cells (note that any Knight's move goes from white to black or from black to white, through the centre)
Left-facing Swastika	Right-facing Swastika

A similar approach can be taken with the "Contextual" set of Knight's moves -- those avoiding the centre.

"Avoidance container" emergent (by optical illusion) from the dynamics of Knight's moves across a 3x3 Matrix of cells (note that any Knight's move goes from white to black or from black to white, around the centre)
Left-facing "container"	Right-facing "container"

The "Avoidance Container" and Swastika can be readily combined together in further experimental animations -- of which the following is one example.

Animation cycle showing emergence of "avoidance container" from merged Swastikas (left- and right-facing)

Swastika and BaGua: The Knight's moves, and the Swastika in both its forms, can be understood in relation to the Chinese BaGua and the 8 trigrams of which it is composed configured around an empty centre. Each trigram is denoted and distinguished by a characteristic configuration of broken and unbroken lines. The relation is represented in the following animation.

• inversion of trigram line pattern for the longer portion of a Knight's move (across the centre)
• single line change in trigram pattern for the shorter portion of a Knight's move (when horizontal)
• double line change in trigram pattern for the shorter portion of a Knight's move (when vertical)

Animation of superposition of Swastika on BaGua "Earlier Heaven" Arrangement (transformation of trigram coding consistent with Knight's move, namely reflection across the centre or change of line pattern for the "sting")
Left-facing Swastika	Right-facing Swastika

Qualitative distinctions: The BaGua pattern is of special interest in that it has been traditionally used to hold and distinguish the eight fundamental meanings associated with them as indicated below. As conditions of nature, these offer cognitive significance through their metaphorical implication.

Chinese interpretation of the eight trigrams -- ?? Baguà
(reproduced from Wikipedia)

? Qián ?	? Duì ?	? Lí ?	? Zhèn ?	? Xùn ?	? Kan ?	? Gèn ?	? Kun ?
Heaven/Sky	Lake/Marsh	Fire	Lightning	Wind	Water	Mountain	Earth
? Tian	?(?) Zé	? Huo	? Léi	?(?) Feng	? Shui	? Shan	? Dì

As suggested here, this makes a qualitative distinction between the different Knight's moves -- each effectively defined by a pair of BaGua conditions (at the start and end of the move) -- in contrast to the purely formal distinction conventionally made between them as in chess. Note that if direction is taken into account there are then 32 Knight's moves. In a Swastika pattern this means that the left- and right-facing Swastika each then holds 16 moves.

Animation of succession of Knight's moves across the BaGua engendering both forms of Swastika: left-facing (green) and right-facing (red) Note the switch in colour and direction -- to the "other" Swastika -- following each "move"

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