irrational numbers: No Blame.

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January 13, 2005

No Blame.

Full results (machine-processed, but programmed by me so it still might be wrong ;)) of the yarrow-stalk I Ching, ordered by probability (and secondarily by value*). (Meanwhile please note that my original post had several errors, but the point still stands that the two 'poles' of K'un and Ch'ien are statistically separated by an order of magnitude when using the yarrow stalks, whereas using the coins they would have exactly the same probability of turning up. Hence I would suggest the use of coins has to be regarded as an degraded mode of access to the I Ching that gives the false impression of the absence of internal systemic bias or difference.)
* By "value" I mean the hexagrams' binary value as per Leibniz. The asymmetrical probability of obtaining yin or yang lines has as its consequence a rather nice asymmetrical clustering of the probabilities of these binary values occurring:

{63} hexagrams with only broken lines	117649 / 16777216
{62,61,59,55,47,31} hexagrams with one whole line	84035 / 16777216
{60,58,57,54,53,51,46,45,43,39,30,29,27,23,15} hexagrams with two whole lines	60025 / 16777216
{56,52,50,49,44,42,41,38,37,35,28,26,25,22,21,19,14,13,11,7} hexagrams with three broken, three whole lines	42875 / 16777216
{48,40,36,34,33,24,20,18,17,12,10,9,6,5,3} hexagrams with two broken lines	30625 / 16777216
{32,16,8,4,2,1} hexagrams with one broken line	21875 / 16777216
{0} hexagrams with only whole lines	15625 / 16777216

The same clustering applied to the hexagram's order numbers in the book:
{2}
{24,7,15,16,8,23}
{19,36,46,51,40,62,3,29,39,45,27,4,52,35,20}
{11,54,55,32,60,63,48,17,47,51,41,22,18,21,64,56,42,59,53,12}
{34,5,58,49,28,26,38,30,50,61,37,57,25,28,33}
{43,14,9,10,13,44}
{1}

<table cellspacing=5 width=700><tr> <td>Hexagram;</td><td>Name;</td><td>Binary Value;</td><td>Order;</td><td>Probability;</td></tr> <tr><td> ___    ___ ___    ___ ___    ___ ___    ___ ___    ___ ___    ___ </td> <td>K'un ; (The Receptive)</td><td>63</td><td>2</td><td>117649/16777216 (approx 1/142)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ___    ___ ___    ___ ________ </td> <td>Fu ; (Returning)</td><td>62</td><td>24</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ___    ___ ________ ___    ___ </td> <td>Shih ; (The Army)</td><td>61</td><td>7</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ________ ___    ___ ___    ___ </td> <td>Ch'ien ; (Modesty)</td><td>59</td><td>15</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ___    ___ ___    ___ ___    ___ </td> <td>Yu ; (Harmony)</td><td>55</td><td>16</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ___    ___ ___    ___ ___    ___ </td> <td>Pi ; (Union)</td><td>47</td><td>8</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ___    ___ ___    ___ ___    ___ </td> <td>Po ; (Falling Apart)</td><td>31</td><td>23</td><td>84035/16777216 (approx 1/199)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ___    ___ ________ ________ </td> <td>Lin ; (Approach)</td><td>60</td><td>19</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ________ ___    ___ ________ </td> <td>Ming I ; (Darkening of the Light)</td><td>58</td><td>36</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ________ ________ ___    ___ </td> <td>Sheng ; (Ascending)</td><td>57</td><td>46</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ___    ___ ___    ___ ________ </td> <td>Chen ; (Thunder)</td><td>54</td><td>51</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ___    ___ ________ ___    ___ </td> <td>Hsieh ; (Removing Obstacles)</td><td>53</td><td>40</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ________ ___    ___ ___    ___ </td> <td>Hsiao Kuo ; (Small Excesses)</td><td>51</td><td>62</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ___    ___ ___    ___ ________ </td> <td>Chun ; (Initial Difficulty)</td><td>46</td><td>3</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ___    ___ ________ ___    ___ </td> <td>K'an ; (The Perilous Pit)</td><td>45</td><td>29</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ________ ___    ___ ___    ___ </td> <td>Chien ; (Arresting Movement)</td><td>43</td><td>39</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ___    ___ ___    ___ ___    ___ </td> <td>Ts'ui ; (Gathering Together)</td><td>39</td><td>45</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ___    ___ ___    ___ ________ </td> <td>I ; (Nourishment)</td><td>30</td><td>27</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ___    ___ ________ ___    ___ </td> <td>Meng ; (Youthful Inexperience)</td><td>29</td><td>4</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ________ ___    ___ ___    ___ </td> <td>Ken ; (Mountain)</td><td>27</td><td>52</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ___    ___ ___    ___ ___    ___ </td> <td>Chin ; (Progress)</td><td>23</td><td>35</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ___    ___ ___    ___ ___    ___ </td> <td>Kuan ; (Contempation)</td><td>15</td><td>20</td><td>60025/16777216 (approx 1/279)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ___    ___ ________ ________ ________ </td> <td>T'ai ; (Peace)</td><td>56</td><td>11</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ___    ___ ________ ________ </td> <td>Kuei Mei ; (The Marrying Maiden)</td><td>52</td><td>54</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ________ ___    ___ ________ </td> <td>Feng ; (Abundance)</td><td>50</td><td>55</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ________ ________ ___    ___ </td> <td>Heng ; (Perseverance)</td><td>49</td><td>32</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ___    ___ ________ ________ </td> <td>Chieh ; (Regulation)</td><td>44</td><td>60</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ________ ___    ___ ________ </td> <td>Chi Chi ; (Completion)</td><td>42</td><td>63</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ________ ________ ___    ___ </td> <td>Ching ; (A Well)</td><td>41</td><td>48</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ___    ___ ___    ___ ________ </td> <td>Sui ; (Following)</td><td>38</td><td>17</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ___    ___ ________ ___    ___ </td> <td>K'un ; (Oppression)</td><td>37</td><td>47</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ________ ___    ___ ___    ___ </td> <td>Hsien ; (Influence)</td><td>35</td><td>51</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ___    ___ ________ ________ </td> <td>Sun ; (Decrease)</td><td>28</td><td>41</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ________ ___    ___ ________ </td> <td>Pi ; (Adornment)</td><td>26</td><td>22</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ________ ________ ___    ___ </td> <td>Ku ; (Arresting Decay)</td><td>25</td><td>18</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ___    ___ ___    ___ ________ </td> <td>Shih Ho ; (Biting Through)</td><td>22</td><td>21</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ___    ___ ________ ___    ___ </td> <td>Wei Chi ; (Before Completion)</td><td>21</td><td>64</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ________ ___    ___ ___    ___ </td> <td>Lu ; (Travelling Stranger)</td><td>19</td><td>56</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ___    ___ ___    ___ ________ </td> <td>I ; (Increase)</td><td>14</td><td>42</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ___    ___ ________ ___    ___ </td> <td>Huan ; (Dispersion)</td><td>13</td><td>59</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ________ ___    ___ ___    ___ </td> <td>Chien ; (Gradual Progress)</td><td>11</td><td>53</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ___    ___ ___    ___ ___    ___ </td> <td>P'i ; (Stagnation)</td><td>7</td><td>12</td><td>42875/16777216 (approx 1/391)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ___    ___ ________ ________ ________ ________ </td> <td>Ta Chuang ; (Power of the Great)</td><td>48</td><td>34</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ___    ___ ________ ________ ________ </td> <td>Hsu ; (Waiting)</td><td>40</td><td>5</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ___    ___ ________ ________ </td> <td>Tui ; (Joy)</td><td>36</td><td>58</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ________ ___    ___ ________ </td> <td>Ko ; (Revolution)</td><td>34</td><td>49</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ________ ________ ___    ___ </td> <td>Ta Kuo ; (Excess)</td><td>33</td><td>28</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ___    ___ ________ ________ ________ </td> <td>Ta Ch'u ; (The Great Taming Force)</td><td>24</td><td>26</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ___    ___ ________ ________ </td> <td>K'uei ; (Disunion)</td><td>20</td><td>38</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ________ ___    ___ ________ </td> <td>Li ; (Clinging)</td><td>18</td><td>30</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ________ ________ ___    ___ </td> <td>Ting ; (The Cauldron)</td><td>17</td><td>50</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ___    ___ ________ ________ </td> <td>Chung Fu ; (Inmost Sincerity)</td><td>12</td><td>61</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ________ ___    ___ ________ </td> <td>Chia Jen ; (The Family)</td><td>10</td><td>37</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ________ ________ ___    ___ </td> <td>Sun ; (Gentle Penetration)</td><td>9</td><td>57</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ___    ___ ___    ___ ________ </td> <td>Wu Wang ; (Correctness)</td><td>6</td><td>25</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ___    ___ ________ ___    ___ </td> <td>Sung ; (Conflict)</td><td>5</td><td>28</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ________ ___    ___ ___    ___ </td> <td>Tun ; (Retreat)</td><td>3</td><td>33</td><td>30625/16777216 (approx 1/547)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ___    ___ ________ ________ ________ ________ ________ </td> <td>Kuai ; (Removing Corruption)</td><td>32</td><td>43</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ___    ___ ________ ________ ________ ________ </td> <td>Ta Yu ; (Great Possession)</td><td>16</td><td>14</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ___    ___ ________ ________ ________ </td> <td>Hsiao Ch'u; (The Taming Force)</td><td>8</td><td>9</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ___    ___ ________ ________ </td> <td>Lu ; (Treading Carefully)</td><td>4</td><td>10</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ________ ___    ___ ________ </td> <td>T'ung Jen ; (Fellowship with Men)</td><td>2</td><td>13</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ________ ________ ___    ___ </td> <td>Kou ; (Coming to Meet)</td><td>1</td><td>44</td><td>21875/16777216 (approx 1/766)</td></tr> <tr><td colspan=5>  </td></tr><tr><td> ________ ________ ________ ________ ________ ________ </td> <td>Ch'ien ; (The Creative)</td><td>0</td><td>1</td><td>15625/16777216 (approx 1/1073)</td></tr> <tr><td colspan=5>  </td></tr></table>

Posted by robin at January 13, 2005 09:00 PM