The Apollonian gasket is a fractal constructed from a triple of circles, where each circle is tangent to the other two. Each level continues this pattern, adding 2·3n more circles on the nth level of the gasket, for a total of 3n+1 + 2 circles after n stages. Repeating this process and taking the limit gives an object like the gasket pictured above on the left.
The Apollonian gasket is also closely related to the undirected graph known as the Apollonian network. The network can be created by first taking three tangent circles, inscribing a circle in the gap created by the three circles, and continuing this process, and then giving each circle a vertex and each pair of tangent circles an edge. This process is seen in the second picture above which shows how it is related to the gasket, and the construction leads to the object pictured on the right.
Pretty interesting relation between the continuous fractal and the discrete graph!