Fractals, like the geometry of higher dimensions, has only recently gained its current prominence in the world of mathematics. Both fields were reinvented and revolutionized by computers, and generate compelling, aesthetically interesting images as well. The iterated processes which create fractals are by no means constricted to three dimensions, and some of the more visually compelling fractals exist in more than two dimensions. Two-dimensional fractals can exist on curved surfaces, like the following piece of fractal art: .
Fractals can also be constructed in three dimensions. For example, if we revisit our old sierpinski gasket, (Chapter 2, Page 33 of Banchoff's Beyond the Third Dimension, it's easy to extend this concept into three dimensions. The gasket becomes a pyramid from which one removes a smaller pyramid. The base of this shape is the square which connects the midpoints of four edges which meet at a single vertex. The final vertex lies at the center of the square opposite of the aforementioned vertex. The result reminds one of a sierpinski triangle: there's a hollow center surrounded by filled in space at the vertices (in this case, volume instead of area). There are excellent examples of three-dimensional fractals at http://home.echo-on.net/~push/applets/wireframe/fractal/. Like the famous Hypercube Video, a skeletal illustration allows one to see the insides of the shapes as they rotate in space. However, infinite iterations are only possible in the mind of God, and several iterations are only possible on computers faster than mine. Fortunately, this page gives you the option of changing the number of iterations in addition to the shape of the object. So, although one doesn't have a perfect fractal, it's easy to see how fractals can exist in three-dimensiona space.

However, if one considers dimensionality in terms of the ratio of mass to length as an object as one shrinks or expands an object in scale, then a whole new set of relationships arise which call into question some basic assumptions about dimensionality. Let's start simple. Two dimensions, it turns out, is simpler than one, so we'll start there. When you double the size of a square (ie magnify it to twice its current size), the resulting area is four times bigger. The relationship, familiar to even the most elementary math student, says that the area of a square is one of its sides squared. So, M (in this case, area) = L^2 for this two-dimensional shape.The same proves true for volume in three dimensions, L^3=M. One dimension is slightly more complicated, because both sets of units are length. So, when one doubles the length of a line, it's length doubles. This can be generalized to L^1=M, which in turn suggests that for an d-dimensional shape, the relationship of mass to area (be it length, area, volume, or hyper-volume) is L^d=M.
Armed with this interesting result and perspective, one can consider the strange M-L relationships found in fractals. Consider the Sierpinski Gasket, with its infinite iterations. When you cut the mass in half, the shape has 1/3 the volume it had before. To visualize this, imagine three mini-gaskets, one at each vertex with a mini-gasket-shaped hole in the center. This process can be infinitely repeated. So, when somebody halves the length of a Sierpinski Gasket, they arrive at a smaller sierpinski gasket, within which is a Gasket 1/2 the height of the new one, touching two new gaskets of the same size. Thus, it follows when the length doubles, the area triples. To put this in terms of the relationships discussed above, 2^d = 3, where d is the object's dimension. Have you ever seen an object in the dimension log3/log2?? Can you even imagine it?
Here, not even computers can help, I'm afraid. This new concept of dimensionality requires a whole new mode of thinking about dimensions. We can't conceptualize higher dimensions, but our mind can extrapolate based on past experience. Throughout the course, we have discussed adding and subtracting dimensions with which we are already familiar. However, in order to understand fractional (forgetting about irrational!!) dimensions, we must multiply and divide dimensional values, an operation whose inconcievability dwarfs that of the fourth or fifth dimension. So, we find ourselves in the realm of mathematical abstraction. Some math (dust off your old high school notebooks, and welcome back the logarith function) reveals that the formula for dimensionality of fractals is d= logM/LogL. For a sierpinsky gasket, the dimensionality is somewhere between 1 and 2, which makes sense when you consider its behavior as being somewhere in between linear and planar. The dimensionality of a sierpinski pyramid follows these same rules. When the length is halved, the area becomes one quarter (ie one of the four pyramids which form the larger pyramid becomes the new, half-as-long pyramid). So, surprisingly, the sierpinski pyramid is two-dimensional. Go figure. Other interesting fractals like the Koch curve, also mentioned in Chapter 2, provide fodder for this new technique. Every Kock curve (and for that matter, and section of any koch curve), can be divided into four sections 1/3 as long. So, the relationship here is log 4/log3, another number between one and two.
Considering fractional dimensions frames our own study of higher dimensions in an interesting way, reminding us of the infinite, gaping chasms on the number line which stand between our usual subjects, integral dimensions. In this way, study of fractals lends higher-dimensional mathematics yet another dimension. Like integral spatial dimensions, the fractional dimensions which arise from a study of fractals open up an exponential number of new possibilities.