Centrality and Hierarchy  from the classical
to the modern. Using the classical for alignment to extend into the
modern settinga 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 theorybasic
triangular lattice.

K=3 hierarchy, marketing principle

K=4 hierarchy, transportation principle

K=7 hierarchy, administrative principle

Fractal central place theory.

Illustration of exact fit of the two approaches, showing that the fractallygenerated
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 mappartially digitized Christaller map.
Draft
map
Hierarchy, Selfsimilarity, 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 Kvalues.

Complete determination of fractal generator shape, that will generate
a complete hierarchy, based solely on numbertheoretic 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 yaxis 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 yaxis; 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 crosscut 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 "hexsteps") simply from the numbertheoretic
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 yaxis, the determination
of generator shape and hierarchy type. Slide
23.

Geometric chart illustrating how to handle offyaxis Kvalues (nonintegral
square roots). Slide
24.

Fractal generators solve the problem (Dacey) of twin Kvalues:
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 spacefilling. 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 