This video illustrates the Peano space-filling curve,
a mathematical object, a continuous image of a line
segment that covers the unit square. This was a startling
revelation at the time when it was discovered, as it maps
continuously a one-dimensional line onto a two-dimensional
square. It prompted a deeper investigation into the concept
of dimension. The Peano space-filling curve has become an
important example in the part of mathematics called
(point-set) topology.
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also generalizations at
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There is also a book (with emphasis on computing)
on space-filling curves:
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I have followed the original paper by Giuseppe Peano (1858–1932)
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Peano, G. (1890), "Sur une courbe, qui remplit toute une aire plane",
Mathematische Annalen 36 (1): 157–160, doi:10.1007/BF01199438.
available at
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This video shows approximations towards the construction
of the Peano curve (which has become a standard approach),
although the original paper used ternary (base 3)
representation of the numbers between 0 and 1 to define
the continuous map directly. Peano's description (at a time
when computers did not exist) could be considered a recursive
definition of the map, when carried to the limit.
If only finitely many steps are computed, then one
gets approximations of the Peano curve, and I have
literally copied his description to produce the graphs
that constitute the above video.

I used the graphing software Graph, by Ivan Johansen,
available at [ Ссылка ] . (It might be a bit
unorthodox to not use some of the more well-known pieces
of graphing software, but I like Graph, it does what I need.)
Using Graph, I produced avi files, then converted them to
mp4 files using Converter lite [ Ссылка ] .
Finally, I merged the mp4 files and added a soundtrack
using Avidemux [ Ссылка ] .

The soundtrack comes from the Traditional Music Channel
at YouTube, particularly
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(This choppy African drum music seems a good match to
the "mathematically choppy" Peano curve :)

This is version A, I plan to also post versions B and T.
Version T is what one often finds in books, partitioning
the unit square into 9 smaller squares, and traversing
the diagonals in a particular order, repeating recursively.
(This uses the idea of self-similarity too, as in
the description of many fractal objects in mathematics).
Version T might be difficult to watch as each
approximating curve has many self-intersection points.
Version B is based on version A, but skipping some
sample points so as to resemble the version that traverses
the diagonals, but at the same time to obtain
non-self-intersecting approximation curves (which are
easier to follow as they are drawn in the video).
The Peano curve is obtained as the limit, and the limit
is the same for each version A, B, and T.

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