Conversion of H&A map (first steps)



Haken and Appel map:

Taken from:
http://www.mathpuzzle.com/4Dec2001.htm (Ed Pegg Jr.)

Referenced by:
http://www.flickr.com/photos/49058045@N00/ (Ibrahim Cahit)

I just started to convert this map into a rectangular map.

To do it manually the full algorithm is explained in the page dedicated to the theorem T2.

Shortly:

  • Consider the map as a jigsaw puzzle
  • Number the faces of the original map starting from an external face
  • Do not make holes while numbering the faces. Last piece will be the ocean (surrounding the entire hexagon)
  • Get a piece from the original map (follow the numbering)
  • Deform it into a rectangle
  • Insert the piece into the rectangular map, respecting the topological properties of the original piece of the original partial map (do not consider the faces not already chosen). For example when you have to work the face number 6, imagine to walk on the border (clockwise) of the original face (the current piece of the puzzle) and write down the sequence of encounters you make. For face number 6 the encounters are: 2 and 5 (face number 6 doesn’t have to be considered because not yet chosen)

For the Haken and Appel map, the encounters are:

  • face 1: no encounters (ocean is the last face to be considered)
  • face 2: 1
  • face 3: 2, 1
  • face 4: 2, 3
  • face 5: 2, 4
  • face 6: 2, 5 (attention not to consider this sequence as a basic rule)
  • face 7: 2. 6
  • face 8: 1, 2, 7 (at this point face number 2 will be completely isolated)
  • face 9: 4, 3
  • face 10: 5, 4, 9 (at this point face number 4 will be completely isolated)

Note:

  • To do it manually after a bit of practice, it takes 10-20 minutes to convert the entire map

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