Algorithm(s) for counting distinct ways to fold a map (or a strip of stamps)

The function mapFolding.countFolds() counts distinct ways to fold maps and strips of stamps. The function accepts two or more dimensions:

from mapFolding import countFolds
foldsTotal = countFolds( [2,10] )

The directory mapFolding/reference has

• a verbatim transcription of Lunnon's "procedure" published in 1971 by The Computer Journal,
• multiple referential versions of the procedure with explanatory comments including
• hunterNumba.py, a one-size-fits-all, self-contained, reasonably fast, contemporary algorithm that is nevertheless infected by noobaceae ignorancium, and
• miscellaneous notes.

Simple, easy usage based on OEIS IDs

mapFolding directly implements some IDs from The On-Line Encyclopedia of Integer Sequences (BibTex citation).

Usage: command line

After installing (see below), OEIS_for_n will run a computation from the command line.

(mapFolding) C:\apps\mapFolding> OEIS_for_n A001418 5
186086600 distinct folding patterns.
Time elapsed: 1.605 seconds

Use getOEISids to get the most up-to-date list of available OEIS IDs.

(mapFolding) C:\apps\mapFolding> getOEISids

Available OEIS sequences:
  A001415: Number of ways of folding a 2 X n strip of stamps.
  A001416: Number of ways of folding a 3 X n strip of stamps.
  A001417: Number of ways of folding a 2 X 2 X ... X 2 n-dimensional map.
  A001418: Number of ways of folding an n X n sheet of stamps.
  A195646: Number of ways of folding a 3 X 3 X ... X 3 n-dimensional map.

Usage examples:
  Command line:
    OEIS_for_n A001415 8
  Python:
    from mapFolding import oeisIDfor_n
    foldsTotal = oeisIDfor_n('A001415', 8)

Usage: Python module or REPL

Use mapFolding.oeisIDfor_n() to compute a(n) for an OEIS ID.

from mapFolding import oeisIDfor_n
foldsTotal = oeisIDfor_n( 'A001418', 4 )

An "advanced" and likely unnecessary feature: clear mapFolding's cache of OEIS data

Clear The On-Line Encyclopedia of Integer Sequences data from the mapFolding cache:

(mapFolding) C:\apps\mapFolding> clearOEIScache
Cache cleared from C:\apps\mapFolding\mapFolding\.cache

Connections to "Multi-dimensional map-folding" by W. F. Lunnon

The typo-laden algorithm published in 1971

The full paper, W. F. Lunnon, Multi-dimensional map-folding, The Computer Journal, Volume 14, Issue 1, 1971, Pages 75–80, https://doi.org/10.1093/comjnl/14.1.75 (BibTex citation) is available at the DOI link. (As of 3 January 2025, the paper is a PDF of images, not text, and can be accessed without cost or login.)

In foldings.txt, you can find a text transcription of the algorithm as it was printed in 1971. In foldings.AA, I have corrected obvious transcription errors, documented with comments, and I have reformatted line breaks and indentation. For contemporary readers, the result is likely easier to read than the text transcription or the original paper are easy to read. This is especially true if you view the document with semantic highlighting, such as with Algol 60 syntax highlighter.

Java implementation(s) and improvements

archmageirvine (BibTex citation) says about the Java code:

/**
 * A001415 Number of ways of folding a 2 X n strip of stamps.
 * @author Fred Lunnon (ALGOL68, C versions)
 * @author Sean A. Irvine (Java port)
 */
...
  // Implements algorithm as described in "Multi-dimensional map-folding",
  // by W. F. Lunnon, The Computer J, 14, 1, pp. 75--80.  Note the original
  // paper contains a few omissions, so this actual code is based on a C
  // implementation by Fred Lunnon.

Map-folding Video

This caused my neurosis: I enjoyed the following video, which is what introduced me to map folding.

"How Many Ways Can You Fold a Map?" by Physics for the Birds, 2024 November 13 (BibTex citation)

Installation

pip install mapFolding

My recovery