This is the second of a set of three videos that illustrate
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.
[ Ссылка ]
also generalizations at
[ Ссылка ]
There is also a book (with emphasis on computing)
on space-filling curves:
[ Ссылка ]

I have followed the original paper by Giuseppe Peano (1858–1932)
[ Ссылка ]

Peano, G. (1890), "Sur une courbe, qui remplit toute une aire plane",
Mathematische Annalen 36 (1): 157–160, doi:10.1007/BF01199438.
available at
[ Ссылка ]
[ Ссылка ]
[ Ссылка ]

This video shows approximating curves towards the construction
of the Peano curve. This 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
for version A. In the present version B (based on A)
some of the sample points were skipped, resulting in curves
consisting of slant line segments. Version B could be
considered intermediate between version A and version T
(to be posted later). Version T, often found in books,
uses partitions of the unit square into smaller squares,
traversing the diagonals in a particular order, repeating
recursively. This uses the idea of self-similarity, as in
the description of many fractal objects in mathematics.
The Peano curve is obtained as the limit of the sequence
of approximating curves, and the limit is the same
for each version A, B, and T.

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 ] .

In version B, I used a Chet Atkins live performance
of Recuerdos de la Alhambra by Francisco Tárrega,
[ Ссылка ] . Many thanks

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