(Sorry about the recent dearth of cp4space postings; I’ve been rather busy in real Euclidean 3-space, .)

Quite a large class of self-similar geometrical objects can be expressed as* iterated function systems*. For instance, the Sierpinski triangle is composed of three copies of itself, each scaled by a factor of ½ and translated by a cube root of unity. More elaborate self-similar sets can be generated by applying other combinations of affine transformations, such as the Barnsley fern:

In most cases, the Hausdorff dimension is not an integer, in which case the object is referred to as a *fractal*. For instance, the aforementioned Sierpinski triangle has a dimension of log(3)/log(2). The Cantor ternary set, on the other hand, has a dimension of log(2)/log(3).

A rather paradoxical fractal is Antoine’s necklace. Firstly, you begin with a plain, bog-standard torus:

Then, place a circular chain of linked tori inside the torus, like so:

We can repeat this process inside each of these tori, and so on *ad infinitum*. Taking the intersection of the (nested) sets obtained at each stage, the result is homeomorphic to the Cantor set. However, the complement is not simply-connected, and therefore not homeomorphic to the complement of the usual embedding of the Cantor set in 3-space. More details are included here.

Antoine’s necklace can be used to construct another unusual object, *Antoine’s horned sphere*. This is obtained by taking a sphere, and extending a tentacle from its surface, which penetrates the initial torus in the construction of Antoine’s necklace. Inside the torus, the tentacle branches into more tentacles, which penetrate each of the sub-tori. This results in a tangled dendritic mess, which is homeomorphic to the 3-ball. However, quite alarmingly, the complement of the horned sphere is not homeomorphic to the complement of the 3-ball. Another such ‘exotic embedding’ of the 3-ball is Alexander’s horned sphere.

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