Bigram multiplication: Squares and square roots revisited - I
Planar two-dimensional numbers exhibit three out of the four categories of multiplication found in the three-dimensional trigrams. Importantly, that is one more than the one-dimensional number line posesses.  And the two-dimensional plane is significantly the domain in which squares and square roots naturally reside.[1]
The two multiplication categories identical for the trigrams of the cube and the bigrams of the square are those of ortho-multiplication and auto-multiplication.  Meta-multiplication and para-multiplication are, in a sense,  telescoped in two dimensions into a single operation.  This third category shares  with para-multiplication of cubic trigrams the fact that it involves complementary pairs,  but only of two-dimensional bigrams now because the third dimension of cubic space is suppressed.[2]
In the planar context, the identity element of multiplication is OLD YANG, the bigram having two yang lines.  The inversion element is OLD YIN, the bigram having two yin lines. These are complementary bigrams occupying meta-positions to one another.  Two hybrid bigrams, YOUNG YANG  with its yang line below,  in the horizontal dimension, and  YOUNG YIN  with its yang line  above,  in the vertical dimension, are also complementary bigrams occupying meta-positions  with respect to each other.
OLD YIN as operator produces inversion of both Lines of any bigram it is multiplied with,  including itself.  As the  identity element, OLD YANG has no effect at all.  YOUNG YIN as operator  produces inversion of only the first dimensional lower Line of the bigram it acts upon,  while YOUNG YANG  produces  inversion  of only the second dimensional upper Line.  The multiplication operations of planar unit vector numbers described here, functioning in an extended real plane, are the operations that are intended by mandalic geometry to supplant imaginary  and  complex numbers.[3]  Demonstration  of the full power of this method though will not be unleashed until we cover the topic of composite dimension more completely.[4]
In the next post we will consider the important ways in which all of this impacts the matter of square root.
[1] Notwithstanding the insistence of 17th to 21st century mathematics that they somehow inhabit the one-dimensional number line.
[2] The term suppressed is more correctly used here I think  than any term suggesting absence because the two dimensions of the plane still exist within context of the three dimensions of the cube and, indeed, within the context of even higher dimensions. Referring back to the diagram here it should be clear that  the four bigrams  exist still  within the trigrams  of all six planar faces of the cube. They are hidden,  so to speak,  in plain sight.  They play an essential role,  however,  in the transitional dynamics of the geometric structures of any dimension number of which they are a part,  also  of the  transitional energetic dynamics of material particles they are intended to represent.
[3] The eight trigrams perform an analogous though slightly more complex role in the three-dimensional geometry of the cube. In that higher dimensional context, transitions occurring among eight octants of 3-dimensional space are coordinated; here transitions among four quadrants of 2-dimensional space are.
[4] Composite dimension extrapolates the real plane to a four-dimensional spacetime configuration which can, however, be constructed in a manner fully commensurable with the ordinary two-dimensional Cartesian plane.
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