# Fractal dimensions and geometries of caves

- 117 Downloads
- 24 Citations

## Abstract

Lengths of all caves in a region have been observed previously to be distributed hyperbolically, like self-similar 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 self-similar geometric properties of caves with the following consequences.

Lengths of a cave is defined as the sum of sizes of passage-filling, linked modular elements larger than the cave-defining modulus. If total length of all caves in a region is a self-similar fractal, it has a fractal dimension between 2 and 3; and the total number of linked modular elements in a region is a self-similar 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 self-similarity## Preview

Unable to display preview. Download preview PDF.

### References

- Blumenthal, L. M. and Menger, K., 1970, Studies in geometry: W. H. Freeman, San Francisco, p. 502.Google Scholar
- Chabert, C. and Watson, R. A., 1981, Mapping and measuring caves—a conceptual analysis: Bull. Nat. Speleo. Soc., v. 43, p. 3–11.Google Scholar
- Curl, R. L., 1960, Stochastic models of cavern development: Bull. Nat. Speleo. Soc., v. 22, p. 66–76.Google Scholar
- Curl, R. L., 1964, On the definition of a cave: Bull. Nat. Speleo. Soc., v. 26, p. 1–6.Google Scholar
- Curl, R. L., 1966, Caves as a measure of karst: J. Geol., v. 74, no. 5, part 2, p. 798–830.Google Scholar
- Dubljanski, V. N., Iljuhin, V. V., and Lobanov, J. E., 1980, Some problems relating to the morphometry of karst caves: Nase Jame, v. 21, p. 75–84.Google Scholar
- Fréchet, M., 1941, Sur la loi de répartition de certaines grandeurs géographiques: J. Soc. Stat. Paris, v. 82, p. 114–122.Google Scholar
- Halleck, J. B., 1984, personal communication.Google Scholar
- Korčak, J., 1940, Deux types fondamentaux de distribution statistique: Bull. Inst. Int. Stat., v. 30, p. 295–299.Google Scholar
- Mandelbrot, B. B., 1983, The fractal geometry of nature: W. H. Freeman, San Francisco, 468 p.Google Scholar
- Morrison, D. A. and Clanton, U. S., 1979, Properties of microcraters and cosmic dust of less than 1000 A dimensions: Proceedings of the 10th Lunar Planetary Science Conference, p. 1649–1663.Google Scholar
- Richardson, L. F., 1961, The problem of contiguity: An appendix of statistics of deadly quarrels: Gen. Sys. Yearbook, v. 6, p. 139–187.Google Scholar
- Šušteršič, F., 1978, What is speleometry?: Nase Jame, v. 20, p. 21–29.Google Scholar
- Šušteršič, F., 1980, Some basic dimensions of the speleogenes: Nase Jame, v. 21, p. 61–73.Google Scholar