The 4-D Case

We obtain the 4-D convergent ratio vector by successive transformations of a base vector by the 4-D triangular matrix:

This leads to the sequence

A comparison of Figure 5 and Figure 9 reveals that while the regular heptagon allows us to form 2 stars, the regular nonagon only forms one star, vertices 3 apart forming only a triangle — well 3∣9 .  In fact a regular polygon with 2n + 1 sides will form n-1 stars if n is prime.  Only when n is prime do we find a matrix field of interest.  In the 4-D case some candidate core matrices have no inverses.

Thus we will not consider the 4-D, 7-D or 10-D cases any further.

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