Fractal geometry

1. Fractal geometry • effective way to represent natural objects • represents phenomenon w/ concepts of • fragmentation • small chunks • self similarity • similar small building blocks • used repeatedly

2. Fractal pattern generation • initiator • the basic element • repetitor • way in which basic element is repeatedly applied • important idea is recursion • I.e. do again • for the definition of recursion - see recursion

3. Fractal patterns can be • deterministic • repetitor is same each time • stochastic • some part of repetitor varies randomly

4. Space filling curves • firstly - why • need ways to refer to specific tiles (cells) in a matrix • one (obvious) way is by x and y coordinates 3 2 1 0 0 1 2 3

5. Problem w/ obvious way • requires storage of two numbers • cannot determine neighbors w/out computation • alternative would permit easy spatial indexing

6. Alternatives to x,y • row order • row prime order • Cantor diagonal order • spiral order • we are generally concerned w/ “space filling curves”

7. Peano ordering • Peano ordering permits use of a key to stand for 2 (or more - if object is volume) dimensions • Peano key is derived using bit interleaving 2 1 0 0 1 2 3

8. Bit interleaving 3 2 X 1 0 0 1 2 3 00 01 10 11 X 0 0 1 1 (3) Y 0 0 1 0 (2) 0 0 0 0 1 1 1 0

9. Topology • relationships in space based on relative positions • absolute position • Thing A is at x=4, y=5 and thing B is at x=3, y=7 • relative position • thing A is near thing B • removal of descriptive geometry

10. Graphs • not x/y graphs • but topological graphs • they reflect relationships • elements • intersections or endpoints of lines (vertices) • lines called edges (but sometimes arcs or chains) • separate links or disconnected sets of lines called subgraphs • vacant spaces (faces) between or outside edges

11. Isomorphic graphs • two graphs are isomorphic if there is a one-to-one correspondence between edges and vertices • shapes can be quite different

12. Graph types • loop, circuit or cyclic graphs • one vertex connected to itself without the need for traversing one edge in both directions • highway systems, electrical systems • tree graph • graphs without loops or cycles • rivers • directed acyclic graph • no circuit but have directed edge(s) • directed edge is simply an edge with direction associated • sewer lines

13. Some special tree types • spanning tree • each vertex must be connected to at least one other vertex • traveling salesman • must be able to return to starting point w/out retracing an edge • radial trees • all peripheral vertices connect to central vertex

14. Properties of graphs • degree of a vertex • number of edges that connect • for directed graphs number of in and out edges • these numbers can be used to assess the overall character of a network • minimum/max degree, average degree etc. • can use these measures to compute Euler number

15. Euler number • V + F = E + S • where • V is total number of vertices • E is total number of edges • F is number of faces • S (or G) is Euler number • if area outside graph is a face S = 2 otherwise 1

16. Computation of Euler numbers S= 1 if “outside” is not counted as a face and S=2 if “outside” is counted as a face V=4, E=4, F=2, S=2 V=6, E=5, F=2, S=1 V=4, F=2, E=5, S=1 V=9, F=3 E=11, S=1