Fractals are self-similar across scale, they look the same, and their details are repeated at different scales or descending levels of magnification. They resemble themselves across categories as well. The classic example of a fractal is a coastline. Coastlines contain bays and peninsulas, which upon closer examination, themselves are made up of bays and peninsulas which are made up of smaller bays and peninsulas on down to boulders and pebbles all the way down to the level of molecules.
The repeating nature of self similarity across scale can also be seen in your cupboard if you happen to have a box of Droste’s Cocoa. The box shows a girl and a boy who is holding a box of cocoa which shows a girl and a boy holding a box of cocoa, which if you look very closely shows the same thing… well you get the picture. A video version of this is found in the Mel Brooks (1987) film Spaceballs. This is known as mise-en-abyme or mise-en abîme (D. L. Miller, personal communication, February 13, 2005). Mise-en-abyme literally means "put in the abyss." The idea is “taken from heraldry via Andre Gide and is used by strategically Derrida and denotes the repetition-in-miniature of a whole within itself, as in the example of a painting within a painting.” (Surfaces, 2005) In the case of heraldry:
a shield/coat-of-arms will have pictured on it an identical shield/coat-of-arms, which has on it an identical . . . etc. . . . to infinity. A more modern example is that of a TV camera taking a picture of a monitor which displays the camera taking a picture of the monitor . . . etc. (Surfaces, 2005)
In such a model, repetition has, as Derrida would say, "always already" taken place: the regression is synchronic, at once originary and teleological. No matter where the regression halts, there will always be the traces of past and future repetitions” (JHU, 2005). This infinite regress can cause a feeling of falling into an abyss. Derrida notes that in mise-en-abyme there is “some kind of fragile radical otherness” within the same (Surfaces, 2005).
This harkens back to the whole notion of the uncanny, and possibly is an unconscious reason why strange attractors are called strange.