Via the Dimensionism Group on Facebook (a group I run myself):
"If it's exponential, then my guess is that it relates to the square root"---Brian Coppedge [my younger brother by 2 years and 2 days]
Referring to the number of categorical deductions for a given category set, this proved to be true, at least for two dimensions of modular 4.
The simplest equation is to take the square root of the number of categories. Another equation is 2 * ( 2 ^ (M root of C) - 1), where C is the number of categories, and M is the modular value, such as 4 for quadra.
This equation simply gives the number of categorical deductions, not how to tabulate them.
In my formulation, the tabulations are equal to all legal cyclical orders (made complex by modularity), in which the opposite positions are occupied by opposite categories. There are also simpler ways to reach the combinations. The result is essentially binary for every set level, but the exact order counts, and some combinations are superfluous, which is not always obvious. Set-equivalence must be defined in a formalized way.
Link to Facebook for updates: http://www.facebook.com/dimensionism
In my formulation, the tabulations are equal to all legal cyclical orders (made complex by modularity), in which the opposite positions are occupied by opposite categories. There are also simpler ways to reach the combinations. The result is essentially binary for every set level, but the exact order counts, and some combinations are superfluous, which is not always obvious. Set-equivalence must be defined in a formalized way.
Link to Facebook for updates: http://www.facebook.com/dimensionism
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