"My research idea start with a question that if there is some graphical formation at a particular value of k then what can do to change the graphical formation or to change its position. In his research paper, I have shown that:
How to change any graphical formation into other graphical formation by applying some kind of operation to the value of the k
How can this different graphical formation be used as a frame to create a film/motion picture"
The Core Algorithm (My Original Research)
The technical paper that this project implements is my 2018 preprint, which provides the complete mathematical foundation for these operations.
Paper: "Transformation of the pixels in Tupper's self-referential formula"
Author: Prathamesh Deshmukh (Me)
Publication Date: May 24, 2018
Every image in the $106 \times 17$ window is a 1802-bit number. Numbering the pixels $p_1, p_2, \dots$ from the bottom-left corner, up each column and then on to the next column, gives equation (A): $k = 17 \sum_{i=1}^{1802} p_i \, 2^{i-1}$. Because $k$ is a sum of one term per pixel, arithmetic on $k$ transforms the image:
Pixel manipulation:
$k' = k \pm 17 \times 2^{i-1}$ adds or removes pixel $p_i$. Drawing on the plot does exactly this.
Spatial translation (moving):
Up/Down: $k' = k \times 2^n$ or $k' = k / 2^n$ moves the image $n$ pixels up or down (equations 1 and 2).
Right/Left: $k' = k \times 2^{17n}$ or $k' = k / 2^{17n}$ moves it $n$ columns.
Diagonal: $k' = k \times 2^{17n} \times 2^{n'}$ combines both.
Different transformations for different pixels (B, C):
Split $k$ into parts, transform each part separately and add them back, for example a moving piece $k_s$ over a still background $k_r$.
Films (D):
$k^\circ = f_1 k + f_2 k' + f_3 k'' + \dots$ where exactly one $f_n$ is 1 at a time. Stepping the 1 through $f_1, f_2, \dots$ plays the frames in order.
Pixels are stored column by column, so a pixel moved up past the top row reappears at the bottom of the next column. That is why 17 moves up equal one move right, as the paper derives. Pixels pushed out of the $106 \times 17$ window are dropped.
The $k$ printed in the paper's introduction is the widely quoted value, which assumes reversed axes. In the paper's own pixel layout it draws the formula rotated by 180°, so "Load Tupper's" uses the same image encoded for this layout.
How to Use This Demo
Load an example, draw your own pattern on the plot, or paste a $k$-value into the box.
Move the pattern with the Transform buttons. Each button shows the arithmetic it applies to $k$.
Play the two films from the paper (Eg 3.1 and Eg 3.2), or step through them one frame at a time.
The line under the plot shows the last operation, and the $k$ box always holds the current value.
Citation & Publication History
BibTeX Citation (Recommended)
@article{Deshmukh2018,
author = "P Deshmukh",
title = "{Transformation of the pixels in tupper's self-referential formula}",
year = "2018",
month = "6",
url = "https://figshare.com/articles/preprint/Transformation_of_pixels_pdf/6373046",
doi = "10.6084/m9.figshare.6373046.v2"
}
Indexing & Archives
Preferred Version (Figshare v2)
Deshmukh, P. (2018). Transformation of the pixels in Tupper's self-referential formula. Figshare. (DOI: 10.6084/m9.figshare.6373046)