Governance by those who do the work.

Sunday, April 6, 2014

Recurrence for Multidimensional Space-Filling Functions

"Recurrence for Multidimensional Space-Filling Functions" is the culmination of many years' interest in space-filling curves and functions.  The recurrences result from a de novo analysis that was more engineering than mathematical.  Instead of trying to characterize all possible curves or base patterns, I focused on finding a cell algorithm for all ranks and side lengths (greater than 2) and making that work with a self-similar recurrence subject to scaling laws I discovered from previous implementations.

That cell algorithm is "serpentine".  It turns out that both the Peano and Hilbert (multidimensional) space-filling curves are instances of my recurrence working on serpentine patterns.

From there I generalized the alignment function.  In the alignment function there were degrees of freedom for diagonal-corner cells (with odd side lengths) which were easy to employ to improve their isotropy (Peano curves are anisotropic).  There are degrees of freedom for adjacent-corner cells of rank greater than 2, but deeper analysis will be required to understand them.

While the alignment function seems to work for all serpentine cells (I verified thousands of combinations of rank and side-length), this is not yet proven.  And Figure 20 at the end of the paper shows a non-serpentine 3x3x3 adjacent-corner cell whose symmetrical version works with my alignment function, but whose asymmetrical variant does not.  Future explorations should determine if that asymmetrical cell itself is valid, and discover the constraint on cells if that is not the case.

The non-terminating recurrence is elegant but was tricky to discover because its truncated form returns coordinates at the center of sub-cells rather than at their origins as the integer recurrence does.

The centered form was accomplished by generalizing the offsets I earlier discovered for centering the Peano curve.

Hereditary Dislike of Family

The extensive genealogy work my spouse has done has barely penetrated one branch of my family, and it's not because it lacks descendants! Unlike the other relatives, the ones in this branch admit to knowing little or nothing about their parents' and grandparents' generations. Many in this branch have not responded to our inquiries; two have told us to never contact them again.

So why does antipathy towards relatives seem to be concentrated in this line?  An interesting idea is to consider the evolutionary consequences of a heritable dislike-your-family trait.  Such a trait seems present in the life cycles of some social species where only a minority of adults leave their clans.  While this trait would, on average, probably reduce the survivability of the individuals expressing it, that could be outweighed by the increased dispersion of the clan's genes.

Such a trait could have more than one cause.  One possibility is the types of mental illnesses which often result in the breakup of families.  Schizophrenia in particular tends to manifest in late adolescence, the time when individuals become fertile and able to live independently.  Separation from the clan at that time maximizes their chances of mixing their genes with other clans in offspring.

If mental illness benefits the species by increasing genetic diversity (which is evidenced by its global occurrence), then mental illness is an integral and persistant part of the human genome.