# Length, Area and Dimension: Measuring Complexity and Scaling Properties

• Heinz-Otto Peitgen
• Hartmut Jürgens
• Dietmar Saupe
Chapter

## Abstract

Geometry has always had two sides, and both together have played very important roles. There has been the analysis of patterns and forms on the one hand; and on the other, the measurement of patterns and forms. The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number π. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus.

## Keywords

Fractal Dimension Hausdorff Dimension Infinite Length Fractal Curve Logarithmic Spiral
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Reference

1. 1.
Benoit B. Mandelbrot, The Fractal Geometry of Nature, Freeman, 1982.Google Scholar
2. 2.
The Encyclopedia AmericanaNew York, 1958 states “Britain has coasts totaling 4650 miles = 7440 km”. Collier’s EncyclopediaLondon, 1986 states “The total mileage of the coastline is slightly under 5000 miles = 8000 km.” Google Scholar
3. 3.
B. B. Mandelbrot, How long is the coast of Britain? Statistical self-similarity and fractional dimension, Science 155 (1967) 636–638.Google Scholar
4. 4.
Here are several methods of getting an answer: (1) Ask all the people in Britain and take the average of their answers. (2) Check encyclopedias. (3) Take a very detailed map of Britain and measure the coast using compasses. (4) Take a very detailed map of Britain and a thin thread, fit it on the coast, and then measure the length of the thread. (5) Walk the coast of Britain.Google Scholar
5. 7.
n: H.-O. Peitgen, H. Jürgens, D. Saupe, C. Zahlten, Fractals - An Animated Discussion, Video film, Freeman, New York, 1990. Also appeared in German as Fraktale in Filmen und Gesprächen, Spektrum der Wissenschaften Videothek, Heidelberg, 1990.Google Scholar
6. 9.
The first measurements of this kind go back to the British scientist R. L. Richardson and his paper The problem of contiguity: an appendix of statistics of deadly quarrels, General Systems Yearbook 6 (1961) 139–187.Google Scholar
7. 10.
We like Utah for many reasons. One of them is that we were introduced to fractals during a sabbatical in Salt Lake City during the 1982/83 academic year. And it was there where we did our first computer graphical experiments on fractals in the Mathematics and Computer Science Departments of the University of Utah.Google Scholar
8. 11.
Two good sources for those who want to pursue the subject are: K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York, 1990 and J. D. Farmer, E. Ott, J. A. Yorke, The dimension of chaotic attractors, Physica 7D (1983) 153–180.
9. 12.
Fractal is derived from the Latin word frangere,which means `to break’.Google Scholar
10. 13.
ausdorff (1868–1942) was a mathematician at the University of Bonn. He was a Jew, and he and his wife committed suicide on January 26 of 1942, after he had learned that his deportation to a concentration camp was only one week away.Google Scholar
11. 15.
See B. B. Mandelbrot, An introduction to multifractal distribution functions, in: Fluctuations and Pattern Formation, H. E. Stanley and N. Ostrowsky (eds.), Kluwer Academic, Dordrecht, 1988. J. Feder, Fractals, Plenum Press, New York, 1988. K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, Wiley, New York, 1990.Google Scholar
12. 16.
F. Hausdorff, Dimension und äußeres Maß, Math. Ann. 79 (1918) 157–179.
13. 17.
G. A. Edgar, Measure, Topology and Fractal Geometry, Springer-Verlag, New York, 1990.
14. 18.
K. Falconer, Fractal Geometry, Mathematical Foundations and Applications, John Wiley und Sons, Chichester, 1990.
15. 20.
P. Bak, The devil’s staircase, Phys. Today 39 (1986) 38–45.Google Scholar

## Authors and Affiliations

• Heinz-Otto Peitgen
• 1
• 2
• Hartmut Jürgens
• 3
• Dietmar Saupe
• 4
1. 1.CeVis and MeVisUniversität BremenBremenGermany
2. 2.Department of MathematicsFlorida Atlantic UniversityBoca RatonUSA
3. 3.CeVis and MeVisUniversität BremenBremenGermany
4. 4.Department of Computer ScienceUniversität FreiburgFreiburgGermany