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Fractals arise in surprising places, in this case, the famous Collatz conjecture in number theory.

A fractal is "a rough or fragmented geometric shape that can be subdivided in parts, each of which is (at least approximately) a reduced-size copy of the whole". The term was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured".

A fractal as a geometric object generally has the following features:

• It has a fine structure at arbitrarily small scales.
• It is too irregular to be easily described in traditional Euclidean geometric language.
• It is self-similar (at least approximately or stochastically).
• It has a Hausdorff dimension which is greater than its topological dimension (although this requirement is not met by space-filling curves such as the Hilbert curve).
• It has a simple and recursive definition.

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.

In his historic work Elements, Euclid assumed the existence of parallel lines with his fifth postulate. The fifth postulate or parallel postulate is equivalent to:

Given a line and a point not on that line, exactly one line can be drawn through that point which does not intersect the original line (see 1).

In the 19th century mathematicians began to seriously question the parallel postulate and found that other forms of geometry are possible. For example elliptical geometry:

Given a line and a point not on that line, all lines drawn through that point will intersect the original line (see 2).

And hyperbolic geometry:

Given a line and a point not on that line, an infinite number of lines can be drawn through the point that do not intersect the original line (see 3).

These other forms of geometry, where the parallel postulate does not hold are called Non-Euclidean geometry.

• ...that there are 6 unsolved mathematics problems whose solutions will earn you one million US dollars each?
• ...that there are different sizes of infinite sets in set theory? More precisely, not all infinite cardinal numbers are equal?
• ...that every natural number can be written as the sum of four squares?
• ...that the largest known prime number is over 12 million digits long?
• ...that the set of rational numbers is equal in size to the subset of integers; that is, they can be put in one-to-one correspondence?
• ...that there are precisely six convex regular polytopes in four dimensions? These are analogs of the five Platonic solids known to the ancient Greeks.
• ...that it is unknown whether π and e are algebraically independent?
• ...that a nonconvex polygon with three convex vertices is called a pseudotriangle?
• ...that it is possible for a three dimensional figure to have a finite volume but infinite surface area? An example of this is Gabriel's Horn.

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