Wednesday, 29 June 2016

Three cards

One of the drawbacks of using just one or two cards is that when shuffling them it may be possible to keep track of their orientation and position even without looking; it could take quite some time before losing track of which face is which.
Using three cards (considering only one face and two orientations) provides 48 combinations which make the result dependant on the orientation and the position of each card (and it is much more difficult to keep track of three cards at the same time).

I designed the following three cards (these are scanned images as I lost the original jpg!):







where each card can result in being yin or yang depending on both the other two cards.
They are to be used as follow:
  1. Shuffle the three cards at will; remember to rotate one of two of them from time to time while shuffling (it is important!);
  2. Turn the cards and look at the three ideograms on the upper left corner;
  3. If the same sequence appears on the left of one line, draw that line. Otherwise draw the other line. Note that any ideogram can be placed where a red circle appears.
  4. Repeat steps 1-4 other five times drawing the hexagram from bottom to top.
 The following  pictures show some examples.

Result: yin (broken line) as the sequence of the three upper ideograms doesn't appear at the left of any line

Result: moving yin (broken line) as the sequence of three upper ideograms appears at the left of any line (the red circle can be replaced by any ideogram)

Result: yang (solid line) as the sequence of the three upper ideograms appears at the left of any line

Result: moving yang (solid line) as the sequence of the three upper ideograms doesn't appear at the left of any line

Note that if you hold them in your rught hand is even easier to read them as your hand will cover the right side of card which is not relevat for the result.

Probabilities

The lines and the ideograms are set to provide exactly the same probabilities of the sixteen marbles method:

Prob(6) = 1/16
Prob(8) = 7/16
Prob(7) = 5/16
Prob(9) = 3/16
Prob(yin) = Prob(yang) = 1/2


Aleister Crowley's sticks

This method is attributed to the controversial occultist Aleister Crowley. It has been reported that he used six flat sticks prepared as shown in the image (a 3D reconstruction).
Each stick has one side completely flat (representing yang, solid lines) and one side with a circular groove painted in red (representig yin, broken lines).
One of the sticks has one side painted in black, that stick represents a moving line.

The images shows the hexagram 21.5>62.

The method proceeds as follows:
  1. Without looking mix and shuffle the six sticks;
  2. Throw them on the table;
  3. Align them, always without looking, to form the hexagram;
  4. Look at the formed hexagram.

Probabilities

This method, which is identical to the six coins one, always produces exactly one changing line meaning that there are only 276 possible responses (out of the 4096 that are theoretically possible).

The probabilities for each lines are:

Prob(6) = Prob(9) = 1/6 * 1/2 = 8.33%
Prob(8) = Prob(7) = 5/6 * 1/2 = 41.67%
Prob(yin) = Prob(yang) = 1/2


Sunday, 26 June 2016

Creating a casting method

At this point in time, or probably even earlier, you may be asking what this site is all about. Yes, I gave a brief explanation in the About  page but the fundamental problem still remains: is there any real need of so many methods for casting hexagrams? What is needed to create a new method? And, most importantly, why should anyone do it?

We all know (having tested it on the field) that the meaningfulness and the accuracy of I Ching responses do not depend on the method used but only on the ability to relate the question to the response. That's why formulating a good question is so important.

However, I believe, the hexagram casting process plays a key role in setting the right mindset which will lead to a successful interpreation. The problem is that what works for some, may not work for others. The long process of using yarrow stalks puts people in a right medadite state, they say, but for others the long time needed is distracting; their minds keep wandering making them unable to focus on the  question. The instantaneous response of a computer program can be a striking revelation for some, but may look cheap and impersonal to others. As always happens, no size fits all and the many existing methods are there to prove that many people tried to find their own way to connect to the I Ching.

What follows are just ideas that, I hope, could motivate anyone who feels uneasy with their current casting method to create their personal casting method to better fits their needs. Should this ever happen, I would declare full success for my efforts.

Why creating a new method?

I think there at least two key aspects:
  • Practicality: you may want to be able to cast I Ching hexagrams in a small space (e.g. a plane seat) or making no noise, or using as least objects as possible, or making it with as few passages as possible, etc.;
  • Connection: you may want to use a set of objects that have a special meaning to you. They may be something from a beloved person or something that reminds you of an important place or time. Or they just make you feel more inclined to hear what the I Ching has to say to you.
I would add a third one that is important to me: aesthetic. It can be something pleasant to the touch or to the eye; something mathematically elegant or uttely chaothic. Whatever appeals to your own sense of beauty.

As a motivation, making money from patenting and selling a casting method may seems appealing but, looking back, many already tried and failed to become rich this way.

How to create a new method? 

These are the three requirements I feel important for any new casting methods:
  1. Each one of the 64 hexagram should be possible in the response.
  2. The 64 hexagrams should all be equiprobable in the response.
  3. Each one of the 4096 possible outcomes (considering the moving linese) should be possible.
Only the requirement 1 is absolutely critical: a method that would rule out a group of hexagrams (say, all those whose second line is yin) would seem just plainly wrong to me.

Requirements 1 and 2 would be fullfilled by any method which assigns to each line the same probability of being yin or yang.

You may note that I refered to the method  probabilities to determine which requirements are more important. This reflects my view that randomness play a key role in the casting process as it determines the relevance of certain aspects (e.g. the number of moving lines) over other.
If you feel that randomness does not play such a key role you can determine which requirements are more important to you.

During the creation steps you will go through the following steps (not necessarily in this order)
  • Choose which objects to use and count the events you can generate with them;
  • Decide on a probability distribution;
  • Define a process to combine/manipulate the objects that will generate the hexagram lines with the desired probability.
As illustrated in the image below, you will probably move from one step to another refiing the method.

Let's go through a full example: the creatione of the one card method.

In this case the need was to have something very portable to tuck into my pocket copy of the I Ching and a single card seemed to be the right choice.
Let's examine our object: a card has two sides (front and back) and two possible orientations (up and down). This means that if we, during the process, rotate and turn the card multiple times we can generate four possible events as the card may end up showing:
  • front side/upward
  • front side/downward
  • back side/upward
  • front side/downward
We could also reduce the number of events, for example we could limit ourselves to only two possible events by having the back of the card to be neutral (like in the regular playing cards) and only considering the orientation. Or by not considering the orientation at all and only considering wether the front or the back face is showing.

Now, if we wanted to mimic the three coins method probabilities ( 1/8 ) we would need to combine at least two operation with the card so to have 16 possible events (4*4).

It is important that the association between the outcomes and the lines are as simple as possible, the users should not be forced to memorize too many things. For example the three coins method requires the user to remember just one things: which side is 2 (the other being 3), the sixteen marbles method requires the user to remember four things: the association between each color and the line types.

That said, for a one card (our object) method with the same probabilities as the three coins method, we need 8 events that we could by combining two operation in the process:
  • The first operation will give yin or yang (e.g by using front/back of the card): 1/2;
  • The second operation will tell if it's a moving line by considering which of the four possible outcomes happened: 1/4.
For the first point we could mark the front of the card with a yang line and the back; for the second point we can mark one of the corner of the front face (e.g. the upper left one) with a dot so we will be able to tell the orientation:
  • front/up → dot in the upper left corner
  • front/down → dot in the lower right corner
  • back/up → no dot visible
  • back/down → no dot visible
we can not tell the two last event apart but we don't really need it. Just being able to identify one of the events (front/up) is enough for our needs.
Here is a possible design for the card:

that you can use with this process:
  1. Without looking, flip, turn and rotate the card.
  2. When you feel it's the right time look at the card
  3. Draw the line you see on the card
  4. Again, without looking, flip, turn and rotate the card.
  5. When you feel it's the right time look at the card
  6. If you see a red circle on the upper left corner of the card, it's a moving line.
And that is done, you can start testing the new method.

Howewer you can start thinking about siimplifying the process considering that many other methods assing the same probabilities to each line. So you can think about modifying the card (your object) to make it possible to generate a line with a single operation:

Now you can define a new process for casting lines:
  1. Without looking, flip, turn and rotate the card.
  2. When you feel it's the right time look at the card
  3. Draw the line you see on the card
  4. If you see a red symbol on the upper left corner of the card, it's a moving line.
Pretty easy, right?

I won't go into the details needed to define a process that would give the yarrow stalks probabilities, you can see how it is done in the one card method that I published already.

There you will also see how I tried to add some image to make it look better. I used some free clipart, a design from a real artist would have made the card 100 times better.

This is just a simple example. Do not hesitate to contact me if you want any more detail.

What about rituals?

Being a very rational person, I feel rather uneasy discussing this topic. Personally, I do not follow any type of ritual. To me the I Ching is a possibility multiplier, an uncertainty machine, a mirror which reflects my self  I don't feel the need for any ritual in the casting process.
However, I know that rituals are important for many people. They help focusing, they provide additional (sometimes deeper) meaning to the whole process.
Unforunately I have no suggestion about them. I've read many suggestions like: using linen cloths to protect the I Ching book, consacrating (whatever it means) the objects used for casting, point toward East, keeping everything at eye level, etc.
My only advice about this is to experiment: find what it works best for you; everything that helps you focusing better, interpreting the response better, etc. it's, by definition, good.  And you may want to ask the I Ching himself for advice about how good rituals are for you.



Saturday, 25 June 2016

EZ Ching (4d4)

This method, using four 4-sided dice, was devised and made available commercially by Christopher and Avonne Thomson. These dice are no longer available on the internet but there is still a mention of them on Amazon. The original site (www.ezching.com) seems to have disappeard between 2008 and 2010 (according to the Way Back machine).
The four dice were, supposedly, hand made and were engraved so that for each group (yellow or blue) the eight trigrams appeared exactly onc.


They were used as follows:
  1. Throw the four dice;
  2. Pick the blue dice for the lower trigram;
  3. Pick the yellow dice for the upper trigram;
This method generates each hexagram with probability 1/64.

To add changing lines it was suggested to use a regular die (1d6) to identify the moving line.

You can also use these dice to generate an hexagram and then derive a single line as described in the I Ching Book method.




Two Cards

This is just another version of the "Two Aces" method and provides lines with the same probability distribution.
Instead of using two regular playing cards, I've devised a design for two special cards so that you can use them to cast hexagrams.
The pictures below shows the faces of the two cards. I've also made them available as 300dpi images in case you want to print them with your printer or through a custom playing cards service.

 


To cast a hexagram proceed as follows:
  1. Shuffle the two cards rotating one of them while shuffling (this is important);
  2. Turn them and put them one on the other so that two symbols in the upper left corner can be seen;
  3. Draw the same line you see on the top card;
  4. If the two symbols in the upper left corner are the same, it's a moving line.
  5. Repeat steps 1-4 other five times drawing the hexagram from bottom to top.
The image below provides some examples:



Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

Sunday, 19 June 2016

Thirtytwo Tarot Cards

In an online article, Richard T Gault suggested a way to use 32 tarots to cast I Ching hexagrams. The idea is simple:
  • Represent yin lines with Cups and Pentacles (Coins) as they are round and receptive (feminine);
  • Represent yang lines with Swords and Wands as they are aggressive and strong (masculine);
  • Include cards in a proportion that mimics the yarrow stalks probabilities:
    •   2 cards of Cups;
    • 14 cards of Pentacles;
    • 10 cards of Swords;
    •   6 cards of Wands;
Once the association has been memorized, the method is very simple:
  1. Shuffle the 32 cards and pick a card;
  2. Draw a line based on the card suit:
    • If it's Cups,        draw  ;
    • If it's Pentacles, draw  ;
    • If it's Swords,     draw  ;
    • If it's Wands,      draw 
  3. Reinsert the card in the deck;
  4. Repeat steps 1-3 five more times drawing the hexagram from bottom to top

Probabilites

By construction, the method provides the same probabilities as the Method of 16 one.



Cards images: WikiMedia Commons
 

Paper Die (1d4)

In the first post about paper dice, I said I was not satisfied with the version that was marked to generate hexagrams with yarrow stalks probabilities: it could led me to getting too many (or too few) moving yin lines.
I found that the solution to this problem is to consider the paper die as having only four possible outcome (instead of eight) and use the same mechanic of the one card method.
I created another sheet to be printed for your convenience but you can simply fold the die from a blank sheet and draw the dots with a pen.
Each strip (remember: to be split in two squares) would appear like this:


After folding the die, two faces will have 4 dots, one face will have three dots and one will have five dots of which one red. If you are going to draw the dots with a pen, leave the central dot white and fill the other ones (or do the opposite).

To cast a line you need two operations. Proceed as follows:
  1. Without looking, turn and roll the die in your hand;
  2. When you feel the time is right, pick one of the faces and write down the number of black dots (either 3 or 4) you see;
  3. Repeat step 1;
  4. When you feel the time is right, pick one of the faces and write down the number of dots, regardless the color, (either 3, 4 or 5) you see;
  5. Sum up the numbers and draw the line according the following table:
    6 7 8 9
  6. Repeat steps 1-5 other five times and draw theline from th bottom to the top of the hexagram.
Probabilities
The faces are marked so that the first time we can get four possible outcomes: 3,4,4,4 while the second time we can get 3,4,4,5. Summing up as shown in the following table:

gives the same probabilities of the yarrow stalks method:
Prob(6) = 1/16 = 6.25%
Prob(8) = 7/16 = 43.75%
Prob(7) = 5/16 = 31.25%
Prob(9) = 3/16 = 18.75%
Prob(yin) = Prob(yang) = 1/2

Variants

Should you like the method but wanting the three coins pronbabilities you can use exactly the same method but marking the faces as follows:

Note that now there are two red dots in the group of five.
A file ready to be print is available (remember to print it at 100% of its size, with no automatic resize).


Three long dice (3d4)

Lawrence P Kaster, created these three long dice to cast I Ching hexagrams. Two of them are marked (2,2,3,3) the other one is marked (2,3,3,3). Lawrence sells them on Etsy in many variations together with other I Ching related creations.
I don't have them but from the images I feel they are nice to the touch and really pleasant to throw while casting hexagrams.

The method is rather intuitive:
  1. Roll the three dices
  2. Sum the face values and draw the line according the following table:
    6 7 8 9
  3. Repeat steps 1-2 other five times drawing the hexagram from bottom to top.

Probabilities

Assuming the first die is the one marked (2,3,3,3)  we have the following possibilities:
 6 = (2+2+2)                                      
9 = (3+3+3)                                      
8 = (2+3+3) or (3+3+2) or (3+2+3)
7 = (2+2+3) or (2+3+2) or (3+2+2)
 that lead to the following probabilities:
Prob(6) = 1/4 * 2/4 * 2/4 = 4/64 = 1/16
Prob(9) = 3/4 * 2/4 * 2/4 = 12/64 = 3/16
Prob(8) = (1/4 * 2/4 * 2/4) + (3/4 * 2/4 * 2/4) + (3/4 * 2/4 * 2/4) = 28/64 = 7/16
Prob(7) = (1/4 * 2/4 * 2/4) + (1/4 * 2/4 * 2/4) + (3/4 * 2/4 * 2/4) = 20/64 = 5/16
Prob(yin) = Prob(yang) = 1/2
which are the same as the yarrow stalk method.

To get the three coins method probablities, the three dice should have all be marked with (2,2,3,3).

Dutch Sticks (6d4)

This method is based on a pretty simple device: a four faces long die. As shown in the picture, each side is marked with one of the possible lines (the red dot is just to help visualizing the die rotation).
Despite its simplicity (and the fact that long dice were known since ancient times), in 1997 a patent was granted to two Dutch guys for a method of casting I Ching hexagrams using six of these dice.
In his really interesting article on I Ching related patents, Steve Marshall suggests that the sticks  on Francina Pijl's site (where I've originally found this method) and shown in the image below might be the product that originated from that patent. For some time they were also available in US from Amazon.


 
The method is pretty simple:
  1. Roll the six dice;
  2. Stack them in order so to form an hexagram.

Probabilities

This method assign equal probabilities to each possible line:
Prob(6) = Prob(8) = Prob(7) = Prob(9) = 1/4 = 25%
Prob(yin) = Prob(yang) = 1/2

Variations

An alternative use of long dice marked with I Ching lines was devised independently by Lawrence P Kaster who advices not to use them for divination but for keeping track of the lines while casting a hexagram with other methods. The idea is that it would be easier to use them rather than having to stop and draw each line using pen and paper. And I do agree with him.
Lawrence has a shop on Etsy where he sells his I Ching related creations (and some more).

Hanna Moog posted a video on YouTube demonstrating how to use the sticks patented by Dominik Rollè in 2001 that use the same principle of the dutch sticks.
These sticks show, when stacked together, both the primary and the secondary hexagram which makes them very useful as a way to track the lines during the casting process.


Saturday, 18 June 2016

Memory Wheels

Another method taken from Russell Cottrell site: memory wheels.
They are sequences of black and white beads (representing yin and yang lines) that encode the entire set of sixtyfour hexagrams. The picture below, from Russel's site, shows one of those wheels:

Start fom any bead, consider it with the next five beads and draw the hexagram; then move to the neaxt bead and do the same.  After having taken all the 64 beads, you will have drawn all the possible 64 hexagram with no repetition!

This seemingly magic property is due to the fact that the beads are arranged in a so called De Bruijn sequence: a way to represent a sequence of all the numbers between 0 and 2n-1 using just 2n bits instead of  n*2n  bits. They have many uses in the field of probability theory, coding and telecommunication.

The compression property they have, makes them a good tool to aid memorization of long sequences of numbers, hence the name of memory wheels. They are reported to be in use since ancient times.

There are many De Bruijn sequence for the sixtyfour I Ching hexagrams; to be precise there are  226 = 67 108 864 of such sequences, each one encoding a different sequence of hexagrams. The following are just two example (0 stands for Black, 1 for white) taken from Russell's site:

0000001111010000100010100100110001101010110010111001110110111111
0000001111010011101111110011011010111000110010110000101010001001

you can find many more there.

De Bruijn sequences have been used to create discs, bracelets, wheels and necklaces related to the I Ching, not all of them, however, for the purpose of divination. There is an interesting summary in the Forum hosted on Clarity on Line.

Sixtyfour beads wheel

A wheel like the one in the picture above has no begininng and no end, it is perfect for casting hexagrams as follows:
  1. Select, without looking, a bead from the wheel;
  2. Draw the line based on the color of the bead:
    • If it's black, draw  ;
    • If it's white, draw  ;
  3. Pick the next bead in the sequence (the direction is not important as long as you are consistent);
  4. Repeat steps 2-3 five more times to get the hexagram.
To get moving lines Russell proposes to use the wheel to draw the hexagram one line at the time and I will describe them in next section.

On Clarity, charly suggested to directly generate the secondary hexagram exactly as done for the primary and determine the moving lines by comparing the two. This approach, which can be used with any method that casts a hexagram at once like the I Ching decks, would provide moving lines with probabilities  1/4.

 

Sixteen beads wheel

A more portable device would use less beads. I've built one with sixteen beads as shown in the picture below.


Probabilites

We can use both the 64 wheel designed by Russell or the 16 wheel shown above, to generate lines with different probability distributions using the following methods.

Three coins probabilities

  1. Select, without looking, a bead from the wheel;
  2. Draw the line based on the color of the bead:
    • If it's black, draw  ;
    • If it's white, draw  ;
  3. Look at next two beads in the sequence, if all the three beads are of the same, it's a moving line;
  4. Repeat steps 1-3 five more times drawing the hexagram from bottom to top;
A De Bruijn sequence with sixteen beads encodes a sequence of 16 numbers between 0 and 8 where each number appears exactly twice. Requiing that moving lines are generated by a sequence of three beads of the same color means we have the following possibilities (B for Black, W for white):

BBB 6    BBW 8    BWB 8    BWW 8
WWW 9    WBW 7    WWB 7    WBB 8

From which we can see that:
Prob(6) = Prob(9) = 1/8 = 12.5%
Prob(8) = Prob(7) = 3/8 = 37.5%
Prob(yin) = Prob(yang) = 1/2
like in the three coins method.

Note that if you just want three coins probabilities and want something smaller, you can build a wheel using eight beads with the following sequence:   B B B W B W W W

Yarrow stalks probabilities

  1. Select, without looking, a bead from the wheel;
  2. Draw the line based on the color of the bead:
    • If it's black, draw  ;
    • If it's white, draw  ;
  3. Look at the next three beads in the sequence, if you can see exactly three white beads among the four beads, it's a moving line;
  4. Repeat steps 1-3 five more times drawing the hexagram from bottom to top;
This is equivalent to the four coins method with white as head and black as tail.

Equal probabilities

  1. Select, without looking, a bead from the wheel;
  2. Draw the line based on the color of the bead:
    • If it's black, draw  ;
    • If it's white, draw  ;
  3. If the next bead in the sequence is of the same color of the selected one, it's a moving line;
  4. Repeat steps 1-3 five more times drawing the hexagram from bottom to top;
Reasoning as we did for the three coins probabilities above, we have (B for Black, W for white):
BB 6    BW 8
WW 9    WB 7

From which we get:
Prob(6) = 1/4 = 25%
Prob(8) = 1/4 = 25%
Prob(7) = 1/4 = 25%
Prob(9) = 1/4 = 25%
Prob(yin) = Prob(yang) = 1/2

Note that if you just want equal probabilities and want something smaller, you can build a wheel using four beads with the following sequence:   B B W W