doi:10.1016/S0167-2789(99)00228-6 
Copyright © 2000 Elsevier Science B.V. All rights reserved.
Computing connectedness: disconnectedness and discreteness
V. Robins
, 
, a, J. D. Meissa and E. Bradleyb
a Department of Applied Mathematics, University of Colorado, Boulder, CO 80309, USA
b Department of Computer Science, University of Colorado, Boulder, CO 80309, USA
Received 12 January 1999;
revised 15 October 1999;
accepted 8 November 1999
Communicated by C.K.R.T. Jones
Available online 24 April 2000.
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Abstract
We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its ε-connected components. By extrapolating the limiting behavior of these quantities as ε→0 we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples, including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Hénon attractor, and cantori from four-dimensional symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of ε-components as ε→0 is equivalent to the box-counting dimension.
Author Keywords: Computational topology; Fractal geometry; Minimal spanning tree
PACS classification codes: 07.05.Kf; 02.40.Pc; 47.53.+n
Fig. 1. A finite set of points and its minimal spanning tree. The weight of an edge is the Euclidean distance between the points it joins.
Fig. 2. Template for the iterated function system that generates the Sierpinski triangle relatives.
Fig. 3. (a) 104 points uniformly distributed on the Sierpinski triangle, (b) the corresponding MST, and (c) a close-up of the bottom right corner of the MST.
Fig. 4. C(ε),D(ε) and I(ε) for the Sierpinski triangle. The top row gives results for 104 uniformly distributed points on the fractal and the bottom row for 105 points. All axes are logarithmic. The horizontal axis range is 10−5<ε<1. The solid lines represent C(ε) and D(ε) for ideal data; the dots are the computed values.
Fig. 5. (a) Cutoff resolution, ρ, as a function of the number of points, 103≤N≤105, covering the Sierpinski triangle for two values of p1;(•) marks data for the nonuniform distribution with p1=0.05; (
) marks data for , i.e., a uniform distribution, and (b) cutoff resolution as a function of p1 for 104 data points on the Sierpinski triangle. The error bars are the standard deviation about the mean of 20 calculations of ρ for each value of p1.
Fig. 6. (a) 104 points on the Sierpinski triangle generated by setting p1=0.05 and p2=p3=0.475, and (b) the corresponding MST.
Fig. 7. C(ε),D(ε) and I(ε) for the nonuniformly distributed data set in Fig. 6. All axes are logarithmic. The horizontal axis range is 10−5<ε<1.
Fig. 8. (a) 104 points on the Cantor set generated by (2), and (b) the corresponding MST.
Fig. 9. C(ε),D(ε) and I(ε) for 105 points uniformly distributed over the Cantor set triangle relative. All axes are logarithmic. The horizontal axis range is 10−5<ε<1. The solid lines represent C(ε) and D(ε) for ideal data; the dots are the computed values.
Fig. 10. (a) 104 points on the fractal generated by (3), and (b) the corresponding MST.
Fig. 11. C(ε),D(ε) and I(ε) for a triangle relative with infinitely many components. Again, the data is for 105 points uniformly distributed on the set. All axes are logarithmic. The horizontal axis is 10−5<ε<1. The solid line represents C(ε) and D(ε) for ideal data; the dots are the computed values.
Fig. 12. Cantor sets generated by iterated function systems of four similarity transformations. Both sets have 50 000 points: (a) similarities with contraction ratio , and (b) the upper two similarities have ratio and the lower two have ratio .
Fig. 13. C(ε),D(ε) and I(ε) for the Cantor sets in Fig. 12. The top row is data for Fig. 12a; the second row is for Fig. 12b. All axes are logarithmic. The horizontal axis range is 10−5<ε<1. The solid lines represent C(ε) and D(ε) for ideal data; the dots are the computed values.
Fig. 14. Two Cantor sets with largest gaps of and : (a) a set generated by an IFS of four affine transformations with horizontal contraction of and vertical contraction of , and (b) a fat Cantor set, generated as the cross product of two Cantor sets of positive measure in the real line.
Fig. 15. C(ε),D(ε) and I(ε) for the 2D Cantor sets in Fig. 14. The top row is data for Fig. 14a; the second row for Fig. 14b, the fat Cantor set. All axes are logarithmic. The horizontal axis range is 10−5<ε<1.
Fig. 16. A Cantor set generated by an IFS consisting of four nonlinear affine transformations, each mapping the unit circle into a circle of radius : (a) the data set with circle boundaries, and (b) a close-up of one of the four clusters.
Fig. 17. C(ε),D(ε) and I(ε) for the nonlinear Cantor set of Fig. 16. All axes are logarithmic. The horizontal axis range is 10−5<ε<1.
Fig. 18. An orbit on the Hénon attractor.
Fig. 19. C(ε),D(ε) and I(ε) data for two orbits on the Hénon attractor. The crosses (+) represent calculations for the orbit of 104 iterates and the circles (○) are for an orbit with 5×104 points. All axes are logarithmic. The horizontal axis range is 10−5<ε<1.
Fig. 20. (a) A close-up of the Hénon attractor. The dark spots are points in the three cross-sections considered in the text: slices at x=0.302435,x=0.5 and y=0, and (b) a small part of the slice at x=0.302435, y=0.22 that shows the folding of the attractor. The pairs of vertical lines are the boundaries of the different subslices of widths 2×10−5,2×10−6 and 2×10−7.
Fig. 21. Top row: C(ε), middle: D(ε), and bottom: I(ε) data for three sections of the Hénon attractor. The circles (○) represent calculations for a section of width 2×10−4, the crosses (+) for one of width 2×10−5, the squares (□) of width 2×10−6, and the stars (*) width 2×10−7. All axes are logarithmic. The horizontal axis range is 10−8<ε<1.
Fig. 22. Two examples of cantori generated by symplectic sawtooth maps. Each orbit has 104 points.
Fig. 23. C(ε),D(ε) and I(ε) data for the two cantori: (top row) data for the cantorus in Fig. 22a, and (bottom row) data for the cantorus in Fig. 22b. All axes are logarithmic. The horizontal axis range is 10−15<ε<1.
Fig. 24. C(ε) versus log(ε) for the two cantori of Fig. 22a and 22b, respectively. All axes are logarithmic. The horizontal range is −20<log(ε)<−0.1.
Table 1. This table summarizes values of γ and δ for Cantor subsets of the planea
Table 2. Values of γ and δ for the three sections of the Hénon attractor shown in Fig. 20
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