This is similar to version T which was already posted,
but this one goes directly to the fifth approximation and
draws the smaller filled-in squares in different colors.
Towards the end, this version goes back to approximations
one and two and draws the diagonals of the colored squares.

This video is the fourth (bonus :) 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 electronic 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 version B (intermediate between versions
A and T) some of the sample points were skipped,
resulting in curves consisting of slant line segments.
Version T uses a partitioning of the unit square
into nine smaller squares, traversing the diagonals in
a particular order, repeating recursively with each of the
smaller squares (thus also using the idea of self-similarity,
as in the description of many fractal objects in mathematics).
The present version Tc is like version T, but rearranging
the order of approximating curves and using colors for emphasis.

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, and Tc. (Version T and Tc
show five, rather than four approximations, or levels.)

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 the present version Tc, I used Harpsichord Concerto no. 3 in D,
BWV 1054 by Johann Sebastian Bach,
available at [ Ссылка ]
I followed the links there and downloaded the mp3 files from
[ Ссылка ]
(though they only provide downgraded quality for free).
Other versions (which I liked, but didn't use) of BWV 1054
are available at [ Ссылка ]
and [ Ссылка ]
You might also like Harpsichord Concerto in E major
J.S. Bach BWV 1053 [ Ссылка ]
(but it was a bit longer than what I needed for my video).

When watching the video I ask myself if I used Bach's
BWV 1054 as background music for my video, or whether on
the contrary, the end result was that I made a fancy video
of Peano's curve to serve as the background for this great music.
Either way, both Peano's curve and Bach's concerto
are masterpieces (from different areas :) and they seem
to nicely complement each other here, I take no credit.

Thank you for watching, enjoy!
I made a playlist with these four versions, A, B, T, and Tc, at
[ Ссылка ]

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