Fractals are described by a geometric set (points, lines, areas or volumes) whose measure behaves in such a matter that the larger scale is not proportionately larger, instead it is larger due to more details being visible at the larger scale. This paper focuses particularly on fractals applied in spatial phenomena in four aspects; mathematical, theoretical, cartography and spatial handling data. In a response of measure to scale, the authors describe it as adapting to a real situation, trying to measure accurate lengths of the shore on an island versus the area of a circle.The measurements of the shore depend on cartographic (cartographic scale relays pixel size) generalization, which affects both length and area measurements. The "Steinhaus Paradox" refers to the fact that measured length increases with increasing accuracy. Study in this area about these details that affect measurements aroused from the study of nations and common boundary. With this early study, the scientist behind it discovered that detail becomes apparent at a predictable rate.
The more irregular the line, then the greater the increase between the 2 measurements at different scales. If irregular, D will be greater than 1.
Self-similarity is defined as when any part of a feature is enlarged indistinguishable without vision of the feature as a whole. Irregular features are indistinguishable at all scales. This is used to create topographic scales.
In recursive subdivision of space, a self similar line is regenerated by recursive procedure. This was then used to develop algorithms to generate irregular fractal curves and surfaces. Self similarity is argued as the property of real landscape. Without self similarity, any stimulation would be visually unacceptable because the basic proof of nature is seeing.
The paper stresses that fractals rely on cartography and spatial data handling applications, which is the origin of the math equations.
ESRI definition of a fractal: A geometric pattern that repeats itself, at least roughly, at ever smaller scales to produce self-similar, irregular shapes and surfaces that cannot be represented using classical geometry. If a fractal curve of infinite length serves as the boundary of a plane region, the region itself will be finite. Fractals can be used to model complex natural shapes such as clouds and coastlines