A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured."
Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex (in informal terms). Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, and snow flakes. However, not all self-similar objects are fractals—for example, the real line (a straight Euclidean line) is formally self-similar but fails to have other fractal characteristics; for instance, it is regular enough to be described in Euclidean terms.
From a Unification perspective, the following points can be made:
- The study and construction of fractals are useful in understanding and drawing certain shapes. Fractals show how the principles of mathematics are relevant to and valuable for our daily lives. In a sense, fractals are related to the internal, invisible principles that underlie the external shape of an object.
- By learning the principles of mathematics, people have been able to produce a variety of systems with many valuable functions. The internal lesson for humans is that if they use the principles of nature to work in harmony with the natural world, they can reap tremendous benefits.
- The concept and applications of fractals are outward manifestations of internal human attributes, such as intelligence and rational thought—attributes that are connected to the human spiritual dimension, setting humans apart from all other known species. These attributes are a reflection of God's intelligent faculties.
- By expressing this type of understanding in harmony with the natural world, people can exert stewardship of true love over the created world. The Unification principle calls this the fulfillment of God's Third Great Blessing to humanity.
- If people use their knowledge wisely, to serve the needs of others as well as themselves, they extend the cause of peace in their society. Thus, knowledge of fractals and the proper application of this knowledge provide humans with an internal guiding principle: To live for the sake of others. If people live according to this principle, humankind can establish a world of peace and happiness.