File:Nested set V4; circles and trees; parity.svg

From Wikimedia Commons, the free media repository
Jump to navigation Jump to search

Original file ‎(SVG file, nominally 1,645 × 1,002 pixels, file size: 946 KB)

Captions

Captions

Add a one-line explanation of what this file represents

Summary

[edit]
Description
Rings, tree and vectors of the last row
Walsh matrix

The integers between 0 and 15 represented by the elements of the nested set V4 and the corresponding trees

Rings (knots) with an even number of children are blue, those with an odd number are red.
The 16×4 matrices show the respective number of rings (knots) by nesting level. The single columns to the right show their sums.
While the red and blue matrix show the numbers for even and odd rings (knots), the gray one on the right shows their sums.
The second column of the gray matrix is Sloane's A000120, the binary weight of the row number.
The sums of the gray matrix are Sloane's A116549, the total number of rings (knots).

Odd numbers have a darker background. The pattern of each column can be found in a Walsh matrix:

even¬(ABCD), (AD), (BD), (CD)¬()
odd(ABCD), (BC), (CD), ()(AC)
sum¬(), (ABCD), (BC), (CD)¬(AC)

The vectors in the three matrices are not unique:
In the blue matrix rows 7/8 are the same, in the red matrix 4/11 and 5/10 and in the gray matrix 6/9.
But the pair of a blue and red or blue and gray vector uniquely identifies a row. 

 blue_vectors = [
     [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0], [1, 1, 1, 0],
     [0, 0, 0, 1], [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 1],
     [0, 1, 1, 1], [1, 2, 1, 1], [1, 1, 2, 1], [0, 2, 2, 1],
     [1, 1, 1, 2], [0, 2, 1, 2], [0, 1, 2, 2], [1, 2, 2, 2]
 ]
 blue_sums = [1, 1, 1, 3, 1, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 7]
 
 
 red_vectors = [
     [0, 0, 0, 0], [1, 0, 0, 0], [1, 1, 0, 0], [0, 1, 0, 0],
     [1, 1, 1, 0], [0, 1, 1, 0], [0, 2, 1, 0], [1, 2, 1, 0],
     [1, 0, 1, 0], [0, 0, 1, 0], [0, 1, 1, 0], [1, 1, 1, 0],
     [0, 1, 2, 0], [1, 1, 2, 0], [1, 2, 2, 0], [0, 2, 2, 0]
 ]
 red_sums = [0, 1, 2, 1, 3, 2, 3, 4, 2, 1, 2, 3, 3, 4, 5, 4]
 
 gray_vectors = [
     [1, 0, 0, 0], [1, 1, 0, 0], [1, 1, 1, 0], [1, 2, 1, 0],
     [1, 1, 1, 1], [1, 2, 1, 1], [1, 2, 2, 1], [1, 3, 2, 1],
     [1, 1, 2, 1], [1, 2, 2, 1], [1, 2, 3, 1], [1, 3, 3, 1],
     [1, 2, 3, 2], [1, 3, 3, 2], [1, 3, 4, 2], [1, 4, 4, 2]
 ]
 gray_sums = [1, 2, 3, 4, 4, 5, 6, 7, 5, 6, 7, 8, 8, 9, 10, 11]
Source Own work
Author
Watchduck
You can name the author as "T. Piesk", "Tilman Piesk" or "Watchduck".
Other versions
plain black
colored rings only

Licensing

[edit]
I, the copyright holder of this work, hereby publish it under the following licenses:
GNU head Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license is included in the section entitled GNU Free Documentation License.
w:en:Creative Commons
attribution
This file is licensed under the Creative Commons Attribution 3.0 Unported license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
You may select the license of your choice.

File history

Click on a date/time to view the file as it appeared at that time.

Date/TimeThumbnailDimensionsUserComment
current22:57, 17 August 2019Thumbnail for version as of 22:57, 17 August 20191,645 × 1,002 (946 KB)Watchduck (talk | contribs)User created page with UploadWizard

You cannot overwrite this file.

Metadata

This file contains additional information such as Exif metadata which may have been added by the digital camera, scanner, or software program used to create or digitize it. If the file has been modified from its original state, some details such as the timestamp may not fully reflect those of the original file. The timestamp is only as accurate as the clock in the camera, and it may be completely wrong.

Retrieved from "https://commons.wikimedia.org/w/index.php?title=File:Nested_set_V4;_circles_and_trees;_parity.svg&oldid=793305373"