Fractals & Chaos Recap for 10/28

Report your findings for our final box-counting project here.

After talking so much these past few weeks about fractional dimension—dimension values that fill in the gaps between 1, 2, and 3 dimensions, we turn our perspective in the other direction on the number line: towards the 4th dimension. It stands to reason that the trends we’ve observed could be extended past the 3rd dimension, but considering fractals, or even Euclidean shapes, is immensely challenging for us as 3-dimensional creatures.

What helps is to consider the perspective by analogy: we can better understand the fourth dimension by putting ourselves in the mindset of a 2-dimensional creature considering the third dimension. Fortunately, this is territory that has been well-covered.

In 1884, British teacher and theologian Edwin Abbott Abbott published Flatland: A Romance of Many Dimensions. It is told from the perspective of A. Square, a denizen of the titular 2-dimensional world, and starts off explaining aspects of their universe in great detail. The second part describes his first encounter with the 3-dimensional Spaceland, and the changes to his world-view as a result.

Abbott wrote the book partly as an exercise in geometry, but also partly as a satire on the regimented Victorian-era social hierarchy. As a result, there are some rather uncomfortable characterizations of women as lower-class citizens, among other shocking commentary.

You can read the full book here (I have paper copies if you’d prefer that), but for tomorrow please at least read this excerpt.

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