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.
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Material on a geometric model of Central Place Theory (Christaller/Lösch).
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Classical central place theory--basic
triangular lattice.
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K=3 hierarchy, marketing principle
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K=4 hierarchy, transportation principle
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K=7 hierarchy, administrative principle
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Fractal central place theory.
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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
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Use of the Diophantine equation K=x^2+xy+y^2 to generate classes of
higher K values.
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Partition of higher values into mutually exclusive, exhaustive classes
of K-values.
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Complete determination of fractal generator shape, that will generate
a complete hierarchy, based solely on number-theoretic properties of K.
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Solution of sets of unsolved problems.
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Fundamental theorems.
Slides:
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Sample of a higher K value to illustrate the difficulty in figuring
out fractal generators to create the geometrically correct spatial hierarchy...Slide
17.
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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.
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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.
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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.
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Algebraic Table illustrating calculations in detail: Slide
21, Slide
22.
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Geometric Table illustrating, a set of K values on the y-axis, the determination
of generator shape and hierarchy type. Slide
23.
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Geometric chart illustrating how to handle off-y-axis K-values (non-integral
square roots). Slide
24.
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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.
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Fractal dimension formula used to calculate space-filling. Slide
26.
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Some implications: from the urban to the electronic environment.
Slide
27.
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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 |
