, Volume 18, Issue 8, pp 765783
First online:
Fractal dimensions and geometries of caves
 Rane L. CurlAffiliated withDepartment of Chemical Engineering, University of Michigan
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Lengths of all caves in a region have been observed previously to be distributed hyperbolically, like selfsimilar geomorphic phenomena identified by Mandelbrot as exhibiting fractal geometry. Proper cave lengths exhibit a fractal dimension of about 1.4. These concepts are extended to other selfsimilar geometric properties of caves with the following consequences.
Lengths of a cave is defined as the sum of sizes of passagefilling, linked modular elements larger than the cavedefining modulus. If total length of all caves in a region is a selfsimilar fractal, it has a fractal dimension between 2 and 3; and the total number of linked modular elements in a region is a selfsimilar fractal of the same dimension. Cave volume in any modular element size range may be calculated from the distribution.
The expected conditional distribution of modular element sizes in a cave, given length and modulus, also is distributed hyperbolically. Data from Little Brush Creek Cave (Utah) agree and yield a fractal dimension of about 2.8 (like the Menger Sponge). The expected number of modular elements in a cave equals approximately the 0.9 power of length of the cave divided by modulus. This result yields an intriguing “parlor trick.” An algorithm for estimating modular element sizes from survey data provides a means for further analysis of cave surveys.
Key words
caves fractals selfsimilarity Title
 Fractal dimensions and geometries of caves
 Journal

Mathematical Geology
Volume 18, Issue 8 , pp 765783
 Cover Date
 198611
 DOI
 10.1007/BF00899743
 Print ISSN
 08828121
 Online ISSN
 15738868
 Publisher
 Kluwer Academic PublishersPlenum Publishers
 Additional Links
 Topics
 Keywords

 caves
 fractals
 selfsimilarity
 Industry Sectors
 Authors

 Rane L. Curl ^{(1)}
 Author Affiliations

 1. Department of Chemical Engineering, University of Michigan, Dow Building, 48109, Ann Arbor, Michigan