Centrality and Hierarchy -- from the classical to the modern.  Using the classical for alignment to extend into the modern setting--a strategy useful in a wide range of theoretical and applied research.
• Material on a geometric model of Central Place Theory (Christaller/Lösch).
• Classical central place theory--basic triangular lattice.
• K=3 hierarchy, marketing principle
• K=4 hierarchy, transportation principle
• Fractal central place theory.
• Illustration of exact fit of the two approaches, showing that the fractally-generated tiles fit together precisely to form the classical central place landscapes.

Thus, the result here is that the complex mechanics of classical central place theory come alive as a single dynamic system when viewed using fractal geometry.  The fit is exact.  Now, use the fractal approach to extend the classical into other realms.

Related references

Related map--partially digitized Christaller map.
Draft map

Hierarchy, Self-similarity, and Fractals
• Use of the Diophantine equation K=x^2+xy+y^2 to generate classes of higher K values.
• Partition of higher values into mutually exclusive, exhaustive classes of K-values.
• Complete determination of fractal generator shape, that will generate a complete hierarchy, based solely on number-theoretic properties of K.
• Solution of sets of unsolved problems.
• Fundamental theorems.
Slides:
• Sample of a higher K value to illustrate the difficulty in figuring out fractal generators to create the geometrically correct spatial hierarchy...Slide 17.
• Oblique axes used to separate K values into a number of different subsets:
a.  along the y-axis and elsewhere
b.  along lines parallel to the line y=x
Statements of key theorems...all on Slide 18.
• Procedure for working with K values on the y-axis; note, therefore that the square root of K is always an integer.
a.  equations of horizontal line parallel to y=x
b.  discriminant of the quadratic form
c.  the integral value, j, used to cross-cut the Diophantine equation.
All on Slide 19.
• Chart illustrating how to determine the number of generator sides and the fractal generator shape (in terms of "hex-steps") simply from the number-theoretic properties of K.  That is, the entire central place hierarchy (its geometry) can be generated by understanding the "genetic" code embodied in K.  Slide 20.
• Algebraic Table illustrating calculations in detail:  Slide 21, Slide 22.
• Geometric Table illustrating, a set of K values on the y-axis, the determination of generator shape and hierarchy type.  Slide 23.
• Geometric chart illustrating how to handle off-y-axis K-values (non-integral square roots).  Slide 24.
• Fractal generators solve the problem (Dacey) of twin K-values:  49 can be generated by the pair (0,7) and (5,3).  The procedure separates these geometrically and generates the correct spatial hierarchies for each of them.  Slide 25.
• Fractal dimension formula used to calculate space-filling.  Slide 26.
• Some implications:  from the urban to the electronic environment. Slide 27.
• Geometric suggestion of similar procedure for an environment of squares. K=4; K=7.
 Coordinates Squares: dimension Hexagons: dimension (1,1) 2.0 1.262 (1,2) 1.365 1.129 (0,2) 2.0 1.585 (0,3) 1.465 1.262 (0,4) 1.5 1.161 (0,5) 1.365 1.209 (0,6) 1.387 1.161 (0,7) 1.318 1.129 (0,8) 1.333 1.153 (0,9) 1.290 1.131 (0,10) 1.301 1.114

Ties to one set of basic geographic concepts:
centrality, hierarchy, scale, density, transformation, distance, orientation, geodesic, minimization, connection, adjacency