Look-and-Say Sequence

2016-03-05, post № 107

mathematics, programming, Python, #approx., #approximating, #approximation, #John Conway, #lambda, #number sequence, #sequence

An interesting nonmathematical sequence which yet has interesting mathematical properties is the Look-and-say sequence.
As every good sequence it starts with a 𝟣. From there you get each next entry by looking at the previous and saying it. The next entry will thus be “one 𝟣” or 𝟣𝟣. From there “two 𝟣” or 𝟤𝟣, then “one 𝟤; one 𝟣” or 𝟣𝟤𝟣𝟣, “one 𝟣; one 𝟤; two 𝟣” or 𝟣𝟣𝟣𝟤𝟤𝟣 and so on.

Generating the Sequence

The mathematical property, although, does not lie in the entry’s value, but rather in its length.

Calculating the Length of each Entry

The interesting thing about the length is, that it approximately grows by 𝜆 each time. $\big(\lambda=1.303577269034\dots\big)$
This constant 𝜆 can be calculated by getting the real root of a degree 𝟩𝟣 polynomial (see Nathaniel Johnston’s post for further information) or be approximated by dividing entries. $\big(\lambda=\frac{\text{entry}_{n+1}}{\text{entry}_{n}}\text{ for }n\rightarrow\infty\big)$

Approximating 𝜆

Source code: look-and-say-sequence.py

Extra assets: look-and-say-sequence-v2.py, look-and-say-sequence-v3.py, look-and-say-sequence_extra.py, look-and-say-sequence_beginning.pdf, look-and-say-sequence_beginning.tex

Haferman Carpet

2016-02-27, post № 106

mathematics, programming, Pygame, Python, #animated gif, #automation, #black, #black and white, #carpet, #cell, #cell simulation, #cellular, #cellular automation, #cellular simulation, #fractal, #generating, #generation, #gif, #iterations, #weave, #white, #woven

The Haferman Carpet is a fractal, which kind of looks like a woven carpet. To generate it, you start with a single black pixel and apply in each cycle a set of rules.
In each generation every pixel in the carpet will be replaced by nine pixels according to the rules. A black pixel is represented by a 𝟢, a white one by a 𝟣.

The Rules

$0\rightarrow\left(\begin{array}{ccc}1&0&1\\0&1&0\\1&0&1\end{array}\right)\text{and }1\rightarrow\left(\begin{array}{ccc}0&0&0\\0&0&0\\0&0&0\end{array}\right)$

Source code: haferman-carpet.py

J-Filters VI

2016-02-20, post № 105

programming, Pygame, Python, #angle, #around, #filter, #image, #image filter, #image manipulation, #manipulation, #spinning, #spun, #spun around, #stir, #vector

Using a vector and an angle, rotating each pixel’s position by an angle determined by its distance to the image’s center and a constant, this filter gives the image a unique spin.
For demonstration I used — once again — an image from my post Stacking Stones.

Source code: j-filters-vi_spin.py

Extra assets: j-filters-vi_original.jpg, j-filters-vi_spin-0.png, j-filters-vi_spin-1.png, j-filters-vi_spin-2.png, j-filters-vi_spin-3.png, j-filters-vi_spin-4.png, j-filters-vi_spin-5.png, j-filters-vi_spin-6.png, j-filters-vi_spin-7.png, j-filters-vi_spin-8.png, j-filters-vi_spin-9.png, j-filters-vi_spin-12.png, j-filters-vi_spin-13.png, j-filters-vi_spin-15.png, j-filters-vi_spin-16.png, j-filters-vi_spin-17.png

Conway’s Game of Life

2016-02-13, post № 104

games, mathematics, programming, Pygame, Python, #cells, #cell, #cell simulation, #cellular, #cellular automation, #cellular game, #cellular simulation, #John Conway, #simulation

This program is a simulation of the famous cellular automaton created by John Conway. The so-called ‘game’ is rather a simulated evolution of cells in a two-dimensional world. Each cell can either be dead (black) or alive (white) and always has eight neighbours surrounding them. In each cycle of the simulation their state will be changed based on two simple rules, which are listed below.
Despite the rule’s simplicity, the outcomes are often very interesting and complex.
For more details on the game of life, check this Wikipedia entry.

The Rules

A live cell can only survive with exactly 𝟤 or 𝟥 neighbours, or else it dies.
A dead cell with exactly 𝟥 neighbours comes to life.

Controls

Left click turns cell at current mouse position alive,
Right click turns cell at current mouse position dead,
‘r’ turns every cell dead,
‘q’ cycles once,
‘w’ cycles continuously.

Source code: conways-game-of-life.py

Extra assets: conways-game-of-life-2.gif, conways-game-of-life_gif-src.tar.gz, conways-game-of-life_fail-awesome-patterns.py, conways-game-of-life_v1.py, conways-game-of-life_v2.py

J-Filters V

2016-02-06, post № 103

programming, Pygame, Python, #contrast, #detection.sobel, #digital image filter, #edge, #edge detection, #filter, #image, #image filter, #img, #img filter, #J-Filter

This filter runs an edge detection algorithm on a given image.
For demonstration I used — as in my previous filter posts — an image from my post Stacking Stones.

The image with applied edge detection filter:

Source code: j-filters-v_edge-detection.py

Colored Rectangles

2016-01-30, post № 102

programming, Pygame, Python, #color, #mouse, #mouse controlled, #polygons, #rect, #rectangle, #rects

This program lets you spawn in a rectangle with a randomly assigned color. This rectangle will then slowly move towards the screen’s edge, eventually fading away.

Controls

Left click spawns in a new rectangle at current mouse position,
‘Space’ takes a screenshot.

Source code: colored-rectangles.py

Extra assets: colored-rectangles-0.png, colored-rectangles-3.png, colored-rectangles-5.png

Moving Fonts

2016-01-23, post № 101

programming, Pygame, Python, #black, #entities, #entity, #font, #fonts, #get, #get fonts, #hello, #hello world, #moving, #red, #spawn, #world

Experimenting with Pygame’s font functions, I wanted to list every font Pygame could reach with its built-in font.get_fonts() function. This program does so by letting you spawn in the wonderful sentence ’Hello world.’ and assigns it a font based on the previously mentioned function.

Controls

Left click spawns in a new sentence at current mouse position,
‘Space’ saves a screenshot.

Source code: moving-fonts.py

Extra assets: moving-fonts-2.png, moving-fonts-4.png, moving-fonts_font-test.py

The Hundredth Post

2016-01-16, post № 100

programming, Pygame, Python, #100, #100th, #char, #char count, #character, #character distribution, #characters, #chars, #data, #data plot, #distribution, #graph, #graphics, #hundredth, #plot, #plotted, #plotted data, #representating, #Zipf, #Zipf curve

The J-Blog has reached its hundredth post, this one. It being the hundredth post, I want to review my previous posts. But rather than going into the context of my previous 𝟫𝟫 posts, I decided to write a program to do this for me.

The program goes through a list fetched from my Table of contents page, looks up the html code, distills the post and counts how many characters of each type there are.^[1] It then outputs its results in a nice text-based style (can be seen below).

To further visualize the data, I wrote another program to display it graphically. The results look like this. On the left the characters are sorted alphabetically (ASCII) and on the right they are sorted by occurrence (you may click on the images to view them closely).

+------+-------+---------+    +------+-------+---------+
| char | count | percent |    | char | count | percent |
+------+-------+---------+    +------+-------+---------+
| all  | 33746 | 100.0   |    | all  | 33746 | 100.0   |
| '!'  |     8 |   0.023 |    | ' '  |  5693 |  16.870 |
| ' '  |  5693 |  16.870 |    | 'e'  |  3323 |   9.847 |
| '%'  |     2 |   0.005 |    | 't'  |  2422 |   7.177 |
| '&'  |     1 |   0.002 |    | 'o'  |  2013 |   5.965 |
| ')'  |   109 |   0.323 |    | 'a'  |  1949 |   5.775 |
| '('  |   109 |   0.323 |    | 'i'  |  1912 |   5.665 |
| '-'  |    45 |   0.133 |    | 'n'  |  1839 |   5.449 |
| ','  |   206 |   0.610 |    | 's'  |  1782 |   5.280 |
| '.'  |   364 |   1.078 |    | 'r'  |  1765 |   5.230 |
| '1'  |    33 |   0.097 |    | 'h'  |  1165 |   3.452 |
| '0'  |    88 |   0.260 |    | 'l'  |  1126 |   3.336 |
| '3'  |    12 |   0.035 |    | 'c'  |  1035 |   3.067 |
| '2'  |    20 |   0.059 |    | 'g'  |   802 |   2.376 |
| '5'  |    11 |   0.032 |    | 'd'  |   799 |   2.367 |
| '4'  |    21 |   0.062 |    | 'm'  |   714 |   2.115 |
| '7'  |     9 |   0.026 |    | 'u'  |   681 |   2.018 |
| '6'  |    14 |   0.041 |    | 'p'  |   656 |   1.943 |
| '9'  |     1 |   0.002 |    | 'y'  |   487 |   1.443 |
| '8'  |     4 |   0.011 |    | 'f'  |   466 |   1.380 |
| ';'  |     1 |   0.002 |    | 'w'  |   437 |   1.294 |
| ':'  |     9 |   0.026 |    | '.'  |   364 |   1.078 |
| '='  |     2 |   0.005 |    | 'b'  |   287 |   0.850 |
| '?'  |     2 |   0.005 |    | 'v'  |   233 |   0.690 |
| 'A'  |    40 |   0.118 |    | 'k'  |   215 |   0.637 |
| 'C'  |    58 |   0.171 |    | ','  |   206 |   0.610 |
| 'B'  |    32 |   0.094 |    | 'I'  |   151 |   0.447 |
| 'E'  |    10 |   0.029 |    | 'T'  |   134 |   0.397 |
| 'D'  |    15 |   0.044 |    | '('  |   109 |   0.323 |
| 'G'  |     8 |   0.023 |    | ')'  |   109 |   0.323 |
| 'F'  |    17 |   0.050 |    | 'x'  |    94 |   0.278 |
| 'I'  |   151 |   0.447 |    | '0'  |    88 |   0.260 |
| 'H'  |    13 |   0.038 |    | 'C'  |    58 |   0.171 |
| 'K'  |     1 |   0.002 |    | 'S'  |    48 |   0.142 |
| 'J'  |     9 |   0.026 |    | '-'  |    45 |   0.133 |
| 'M'  |    20 |   0.059 |    | 'A'  |    40 |   0.118 |
| 'L'  |    27 |   0.080 |    | '1'  |    33 |   0.097 |
| 'O'  |     6 |   0.017 |    | 'U'  |    33 |   0.097 |
| 'N'  |     5 |   0.014 |    | 'B'  |    32 |   0.094 |
| 'Q'  |     1 |   0.002 |    | 'L'  |    27 |   0.080 |
| 'P'  |    19 |   0.056 |    | 'z'  |    26 |   0.077 |
| 'S'  |    48 |   0.142 |    | 'q'  |    24 |   0.071 |
| 'R'  |    19 |   0.056 |    | '4'  |    21 |   0.062 |
| 'U'  |    33 |   0.097 |    | '2'  |    20 |   0.059 |
| 'T'  |   134 |   0.397 |    | 'M'  |    20 |   0.059 |
| 'W'  |    19 |   0.056 |    | 'P'  |    19 |   0.056 |
| 'V'  |     7 |   0.020 |    | 'R'  |    19 |   0.056 |
| 'Y'  |    14 |   0.041 |    | 'W'  |    19 |   0.056 |
| 'X'  |     4 |   0.011 |    | 'F'  |    17 |   0.050 |
| '_'  |     2 |   0.005 |    | 'j'  |    17 |   0.050 |
| 'a'  |  1949 |   5.775 |    | 'D'  |    15 |   0.044 |
| 'c'  |  1035 |   3.067 |    | '6'  |    14 |   0.041 |
| 'b'  |   287 |   0.850 |    | 'Y'  |    14 |   0.041 |
| 'e'  |  3323 |   9.847 |    | 'H'  |    13 |   0.038 |
| 'd'  |   799 |   2.367 |    | '3'  |    12 |   0.035 |
| 'g'  |   802 |   2.376 |    | '5'  |    11 |   0.032 |
| 'f'  |   466 |   1.380 |    | 'E'  |    10 |   0.029 |
| 'i'  |  1912 |   5.665 |    | ':'  |     9 |   0.026 |
| 'h'  |  1165 |   3.452 |    | 'J'  |     9 |   0.026 |
| 'k'  |   215 |   0.637 |    | '7'  |     9 |   0.026 |
| 'j'  |    17 |   0.050 |    | '!'  |     8 |   0.023 |
| 'm'  |   714 |   2.115 |    | 'G'  |     8 |   0.023 |
| 'l'  |  1126 |   3.336 |    | 'V'  |     7 |   0.020 |
| 'o'  |  2013 |   5.965 |    | 'O'  |     6 |   0.017 |
| 'n'  |  1839 |   5.449 |    | 'N'  |     5 |   0.014 |
| 'q'  |    24 |   0.071 |    | '8'  |     4 |   0.011 |
| 'p'  |   656 |   1.943 |    | 'X'  |     4 |   0.011 |
| 's'  |  1782 |   5.280 |    | '%'  |     2 |   0.005 |
| 'r'  |  1765 |   5.230 |    | '='  |     2 |   0.005 |
| 'u'  |   681 |   2.018 |    | '?'  |     2 |   0.005 |
| 't'  |  2422 |   7.177 |    | '_'  |     2 |   0.005 |
| 'w'  |   437 |   1.294 |    | '&'  |     1 |   0.002 |
| 'v'  |   233 |   0.690 |    | '~'  |     1 |   0.002 |
| 'y'  |   487 |   1.443 |    | '9'  |     1 |   0.002 |
| 'x'  |    94 |   0.278 |    | ';'  |     1 |   0.002 |
| 'z'  |    26 |   0.077 |    | 'K'  |     1 |   0.002 |
| '~'  |     1 |   0.002 |    | 'Q'  |     1 |   0.002 |
+------+-------+---------+    +------+-------+---------+

The main program (shell-based): the-hundredth-post.py
The visualization program: the-hundredth-post_data-visualization.py

Extra assets: the-hundredth-post_extra-1.png, the-hundredth-post_extra-2.png, the-hundredth-post_extra-3.png, the-hundredth-post_extra-4.png, the-hundredth-post_extra-5.png, the-hundredth-post_extra-6.png, the-hundredth-post_extra-7.png, the-hundredth-post_extra-8.png, the-hundredth-post_extra-9.png, the-hundredth-post_extra-10.png, the-hundredth-post_extra-alphabetically-1.txt, the-hundredth-post_extra-alphabetically-2.txt, the-hundredth-post_extra-highest-count.txt

Posts: 291-284, 283-276, 275-268, 267-260, 259-252, 251-244, 243-236, 235-228, 227-220, 219-212, 211-204, 203-196, 195-188, 187-180, 179-172, 171-164, 163-156, 155-148, 147-140, 139-132, 131-124, 123-116, 115-108, 107-100, 99-92, 91-84, 83-76, 75-68, 67-60, 59-52, 51-44, 43-36, 35-28, 27-20, 19-12, 11-4, 3-1

Jonathan Frech's blog; built 2024/10/27 23:46:30 CET