is a "turtle" fractal, generated when each of the five segments (represented by arrows) making up the cross below:
is replaced with a copy of the whole thing; and then each of the segments making up those, and so on.
As a fractal, it is neither one-dimensional nor two-dimensional, but has an intermediate dimension, which in this case is:
D = ln (5) / ln (3) ≈ 1.465
because, when the linear dimension increases by a factor of three times, the total length increases by five times (five crosses like the original, before any magnification).
I saw this image for the first time on the back cover of B. Mandelbrot's wonderful (though rather pompous) book "The Fractal Geometry of Nature," where it appeared as a natural feature of a much more complex fractal.
But to me it also has another significance. A deeper one. People sometimes call "crosses" the bad things that happen to them. If they know how to accept them with joy, those "crosses" can become a source of liberation and spiritual renewal. But it is not necessary to wait for something really bad to happen to you. . . .
The fractal cross tells me that my cross can be made of infinitesimal crosses, each one as complete and as beautiful as the whole. And there's an infinite number of them, which you can find at every instant. It's only a matter of looking with a magnifying glass.
Think of this next time you get stuck in traffic.
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