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Thread: The Changes and the Bruijn sequence

  1. #1
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    Default The Changes and the Bruijn sequence

    The problem of sequences of I Cing hexagrams in binary wheels, disks or necklaces was already viewed in the forun:

    64 black+white beads contain all 64 hexagrams:
    http://www.onlineclarity.co.uk/frien...?t=5053&page=2

    I ching on a string:
    http://www.onlineclarity.co.uk/frien...?t=5613&page=3

    *NEW* 64bit Arrangement
    http://www.onlineclarity.co.uk/frien...ead.php?t=5212

    Of course, the arrangement is not new in the sense that there were already objects and descriptions following the same idea.

    The author of the last thread has developped extensively his arrangement in Abrahadabra Forum:

    http://forums.abrahadabra.com/showth...0072#post40072

    http://forums.abrahadabra.com/showth...p-Wheel/page2&

    He built the sequence begining with six yin-0 [000000] and adding sistematically yang-1 each time possible, only adds a yin-0 if otherwise will obtain a repeated hexagram.

    Working with this method he obtains a unique solution, although wondering...

    I'm not sure myself how many unique solutions exist.
    Unique being not reflected or inversed.

    I can't think of another way to generate it myself, just the self referencing method I used.

    Seems the Chinese did know about this arrangement:
    http://www.onlineclarity.co.uk/frien...ead.php?t=5053

    That is the only place I've ever seen it mentioned.
    ...

    I Ching Code Wheel




    From;: http://forums.abrahadabra.com/showth...op-Wheel/page2
    In fact this is one method for building a sequence of bits (yi/yang lines) that generates a wheel with the 64 hexagrams, reading six consecutive bits.

    Another method would be to choose the opposite to the last bit whenever possible, say yin after yang and yang after yin. It will result another sequence.

    There are many possible sequences, as you can see in the following quotes...

    Memory wheels or de Bruijn sequences:
    The problem of constructing memory wheels is known as the rotating drum problem. The circular binary are often called lenght shift register sequences or de Bruijn sequences after the Dutch mathematician N.G. de Bruijn who wrote about them in 1946 (although it turned out that they had been constructed many years before by C. Flie Sainte-Marie). They have been used worldwide in telecomunications, and there have recen applications in biology.

    Fig 4.18 page 84
    Ian Anderson
    A First Curse in Discrete Mathematics
    2048 with 4 bits, imagine with 6! >>> Sorry, I had to say five bits. Ch.
    In his study on memory wheels the Dutch engineer K.Posthumus found that there is exactly one wheel for binari cuplets, two for binary triplets, 16 for binary quadruplets (4 bits) and 2048 forr binary quintuplets (5 bits). <he then conjectured that there are 2^((2^n-1)-n) different memory wheels for binary n-tuples . In 1946 de Bruijn established his conjecture.

    Thomas Koshy
    Discrete mathematics with applications
    Page 703
    Did the chinese know these sequences?

    Ver en Google Books
    Sherman K. Stein:Mathematics: the man-made universe

    Stein traced the first such memory wheel to India of about 1000 AD. The next use of a memory wheel was in France in 1882, where Emile Baudot used it for 32 quintuplet telegraphy. In this past century, memory wheels, also called de Bruijn cycles, have been used in a variety of applications, ranging from probability theory, coding, and communications.
    Stein quotes the Sanskrit sutra, yamatarajabhanasalagam, which describes all possible triplets of short and long syllables, as evidence of the Indian knowledge of a memory wheel of length 3. Since Sanskrit metres are based on the system of short, laghu, and long, guru, syllables, represented traditionally by j and S (by us as 1 and 0), the sutra may be written as:
    1000101110 which represents the sequences 100, 000, 001, 010, 101, 011, 111, 110
    Subhash Kak
    Yamatarajabhanasalagam: An Interesting Combinatoric Sutra
    Indian Journal o history of Science

    At: http://citeseerx.ist.psu.edu/viewdoc...=rep1&type=pdf

    See also:

    http://hitxp.wordpress.com/2007/06/2...toric-formula/

    http://alexandria.tue.nl/repository/books/252901.pdf

    I believe that chinese mathematicians didn´t know the sequences. Persons consulted don´t remember a circular arrangement similar to the posted in Abrahadabra.

    Not strange if we think that the arrangement is not unique and that different diviners, by traying and error might have got different sequences all valid for its use. But for what use? Not for finding an hexagram in the Book of Changes, of course. Only for producing an hexagram by a quick method, like casting coins.

    Maybe for personal use if a necklace or rosary of beads. Maybe for exhibition like a peculiarity, the multiplicity of results, given the lack of precise method maybe caused de low profile of the mechanism.

    Yours,

    Charly
    Last edited by charly; June 29th, 2010 at 02:02 PM.

  2. The Following 3 Users Say Thank You to charly For This Useful Post:

    bradford (July 1st, 2010), sergio (June 28th, 2010), sparhawk (June 28th, 2010)

  3. #2
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    Thanks Charley!

    Wouldn't it be fun to have this design on a quilt?!!

    Rosada

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    Great stuff, Charly, gracias.

    I'll have to spend some time with it.

    Un abrazo,

    L

  5. #4
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    Quote Originally Posted by rosada View Post
    Thanks Charley!

    Wouldn't it be fun to have this design on a quilt?!!

    Rosada
    Hi, Ros:

    I wonder if it is the same sequence thay you used in your necklace.

    Have you read something in W/B about a MYSTERY SEQUENCE ?

    I have read in Abrahadabra that is similar to the "CODE SEQUENCE", say one of de Bruijn sequences, but does say Wilhelm anything about wheels or necklaces?

    Yours,

    Charly

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    Quote Originally Posted by sparhawk View Post
    ... I'll have to spend some time with it...
    Luis:

    For our interests it's not so difficult as it seems. Do you remember some design like the CODE WHEEL in chinese sources? Do you believe that old chinese diviners have known something about these sequences? Not impossible if one thinks that the Canon of the Supreme Mystery could be written in Han times. Yang Hsiung would have understand de Bruijn sequences.

    Un Abrazo,

    Charly

  7. #6
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    Quote Originally Posted by charly View Post
    ...
    Have you read something in W/B about a MYSTERY SEQUENCE ?
    ...
    This is what Steve Marshall says of the MYSTERY SEQUENCE:
    On the 'I Ching Sequencer' website it is possible to advance the hexagrams one at a time, as well as watch them loop. The three sequences run concurrently, with the 'mystery sequence' displayed largest. I'd never really taken a good look at this particular sequence at the back of Wilhelm-Baynes (pp 730–731, in the third edition), but it appears the animator is right, this is indeed a mystery. I have no idea at present what this sequence is or from where it originates (it doesn't appear in the first edition). But it is certainly a fascinating and aesthetically satisfying sequence when seen as a Flash animation. Notice that through the first 7 hexagrams a single yin line appears to rise up through the all-yang Qian hexagram. Then two yin lines appear to perform the same trick together. Then the trigram Kan rises up through Qian, which suggests that in the first two phases rather than a single yin and two yin rising up it is in fact the trigrams Li and Gen rising up, respectively. But by the 17th hexagram of the sequence this pattern is lost and one suspects that rather than a trigram rising up through a hexagram it is in fact a hexagram rising up through a hexagram, but that would require a more detailed study of the sequence, these are just a few provisional observations, first thoughts. By the 23rd in the sequence we see the familiar pattern again with three yin lines rising up together. By the end of the sequence, in the final seven hexagrams, we see a single yang line rising up through the all-yin hexagram Kun, just as a single yin rose up through all-yang Qian at the start.
    from: http://www.biroco.com/yijing/sequence.htm
    I've not the complet W/B edition and I don't know if some of the Wilhelms said something about the sequence. Does somebody know who was te designer?

    Following the descriptions I'm almost sure that it is a failed de-Bruijn sequence, say not a good solution.

    I've just found a picture and a text, which I go to study, here:

    The source:

    http://iching.egoplex.com/mystery-sequence.html

    Charly

  8. #7
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    For my necklaces I use this pattern:

    Hexagram 1,2,27,5,62,6,48,61,21,29 + 1 yin, 1 yang, 1 yang, 1 yin.

    Ros

  9. #8
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    Quote Originally Posted by charly View Post
    The Mistery Sequence must be read by columns.

    Being 1=yang and 0=yin, the corresponding binary code, from bottom line to top line, for the first column is:


    Row 1: 111111
    Row 2: 011111
    Row 3: 101111
    Row 4: 110111
    Row 5: 111011
    Row 6: 111101
    Row 7: 111110
    Row 8: 001111


    In the first column, rows 1 to 7 depict a de-Bruijn sequence.

    See how taking off a digit from the right (= a line from the top) and adding a digit to the left (a line to the base) in every pair of hexagrams ...

    • line 1 in the precedent corresponds to line 2 in the following
    • line 2 in the precedent corresponds to line 3 in the following
    • line 3 in the precedent corresponds to line 4 in the following
    • line 4 in the precedent corresponds to line 5 in the following
    • line 5 in the precedent corresponds to line 6 in the following
    • line 6 in the precedent disappeared in the following


    It seems that appearing a new line in the following hexagram causes the lines of the precedent to climb one position, overflowing the top line which disappears.

    A movement from bottom to top.

    In words of Steve "Notice that through the first 7 hexagrams a single yin line appears to rise up through the all-yang Qian hexagram."

    But row 8 breaks the sequence.

    Maybe the first digit of row 7 must be a 0 instead of a 1, then adding another 0 row 8 should be correct.

    (to be continued)

    Charly
    Last edited by charly; June 30th, 2010 at 01:06 AM.

  10. #9
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    De Bruijn Sequence:
    All sequences must have 6 consecutive yin [:] lines together with
    6 consecutive yang [|] lines
    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

    This sequence was built with 6 trailing yin lines [::::::]
    The algorithm proceeds adding a yang line whenever it´s possible
    without generating a duplicate hexagram, in which case a yin line is added.
    This method is easy and quick but produces a disbalancement in the
    sequence with more yang lines at the begining and more yin lines at the end.

    Another method trying to add alternative yin/yang lines will produce
    a more balanced result.
    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

    | | | | | | : | | | | : : | | | : | : | | | : : : | | : | | : |

    : : | | : : | : | | : : : : | : | : | : : : | : : | : : : : : :

    __________________________________________________ __________________________

    Generation of hexagrams contained in the cited de-Bruijn Sequence:
    Base line at the left, top line at the right
    One yin [:] or one yang [|] line appears from the left
    The top line overflows by the right
    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _


    __________________________________________________ ________ |:::::
    __________________________________________________ _______ ||::::
    __________________________________________________ ______ |||:::
    __________________________________________________ _____ ||||::
    __________________________________________________ ____ |||||:
    __________________________________________________ ___ |||||| ___ H.1 KWS
    __________________________________________________ __ :|||||
    __________________________________________________ _ |:||||
    __________________________________________________ ||::||
    _________________________________________________ |||::|
    ________________________________________________ :|||::
    _______________________________________________ |:|||:
    ______________________________________________ :|:|||
    _____________________________________________ |:|:||
    ____________________________________________ ||:|:|
    ___________________________________________ |||:|:
    __________________________________________ :|||:|
    _________________________________________ ::|||:
    ________________________________________ :::|||
    _______________________________________ |:::||
    ______________________________________ ||:::|
    _____________________________________ :||:::
    ____________________________________ |:||::
    ___________________________________ ||:||:
    __________________________________ :||:||
    _________________________________ |:||:|
    ________________________________ :|:||:
    _______________________________ ::|:||
    ______________________________ |::|:|
    _____________________________ ||::|:
    ____________________________ :||::|
    ___________________________ ::||::
    __________________________ |::||:
    _________________________ :|::||
    ________________________ |:|::|
    _______________________ ||:|::
    ______________________ :||:|:
    _____________________ ::||:|
    ____________________ :::||:
    ___________________ ::::||
    __________________ |::::|
    _________________ :|::::
    ________________ |:|:::
    _______________ :|:|::
    ______________ |:|:|:
    _____________ :|:|:|
    ____________ ::|:|:
    ___________ :::|:|
    __________ |:::|:
    _________ :|:::|
    ________ ::|:::
    _______ |::|::
    ______ :|::|:
    _____ ::|::|
    ____ :::|::
    ___ ::::|:
    __ :::::|
    _ :::::: ___ H.2 KWS

    __________________________________________________ __________________________

    Binary order of Hexagrams, 0 to 63:
    Maximum binary order (63) belongs to H.1 in King Wen Sequence
    Minimum binary order (00) belongs to H.2 in King Wen Sequence
    _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

    01-03-07-15-31-63-62-61-59-55-47-30-60-57-51-39-14-29-58-53-43-23-46-
    28-56-49-35-06-13-27-54-45-26-52-41-19-38-12-25-50-37-11-22-44-24-48-
    33-02-05-10-21-42-20-40-17-34-04-09-18-36-08-16-32-00
    __________________________________________________ __________________________
    Warning: accuracy was not verified!

    Charly

  11. #10
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    Wilhelm - Baynes said:
    The individual lines enter the hexagram from below and leave it again at
    the top.

    From H.11 Peace
    This is compatible with most commentaries and only possible if the lines are in a circle or loop like a de Bruijn sequence.

    All this sequences have necessarily a segment "000000111111" .

    To compare different sequences we must align it by this segment.

    If we accept that "111111" be the head segment and "000000" the tail of the sequence, the 6 heading hexagrams and the last will be always the same. I believe.


    Head B
    100000
    110000
    111000
    111100
    111110
    111111 = H.1 KWS

    0?????
    00????
    000???
    0000??
    00000?
    000000 = H.2 KWS
    Tail @

    Another form of seeing it is that there are 7 fixed hexagrams, begining wit KUN, and in the middle it is Peace, the hexagrams from which we quote W/B at the begining of this post.

    000000 = H.2 KWS
    100000
    110000
    111000 = H.11 KWS (Peace!)
    111100
    111110
    111111 = H.1 KWS

    Not sure if it happens with all the valid sequences independently of the building method. I have to analyze it more.

    Yours,

    Charly
    Last edited by charly; June 30th, 2010 at 06:34 AM.

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