Cast a Hexagram online

As mentioned in a previous post, I've created an application to cast hexagrams online. A compact version of this app is visible in the "Cast a Hexagram" box on each page. The user interface is rather minimal so I thought of giving some explanation here on how to use it.

Quick reference

Cast a hexagram

The basic function is, not surprisingly, casting a hexagram. Each line is generated using random events caused by users moving the mouse, or clicking, on the small image. When the right amount of randomness is collected, a line is generated with the yarrow stalks probabilities. Moving lines are drawn in red, non-moving lines in black.

The line generation is not instantaneous: it requires the user to move the mouse (or click) a few times before a new line can be generated. This is done to avoid bias in the user behaviour (e.g. clicking always the same point in the image) that could destroy the randomness of the casting process.

Hitting on the Reset button (at the bottom left) clears all the enthropy collected up to that point and restart the process.

Record lines

You can also generate the hexagram yourself (e.g. by tossing coins) and record the outcome using this app: use the small buttons at the bottom right (6,7,8,9) to enter the lines one by one. The rightmost button (backspace) allows you to delete the last line in case you made a mistake.
This could be useful if you just want to read the response online but you don't like using software for casting the hexagram.

Get the reading

Once all the lines have been generated, the corresponding hexagram number appears. If there are moving lines, both the response and the relating hexagram numbers are shown (but not the relating hexagram itself).

To get the meaning of the hexagrams you can click on their numbers. You can choose which translation you want to be redirected to by selecting it in the combo box. By default you will get the Wikipedia definition of the hexagram.


Please send me any comment or suggestion you may have to make this little app more useful to you.

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Sixteen beads bracelet

Inspired by the images of buddhist prayer beads, I decided to create an I Ching bracelet with a similar look. Initially, to be faithful to the sixteen marble method, I choose sixteen beads of four different colors, put them on a string and followed the prescribed procedure.

Soon after, however, I realized that one could use the relative position of the beads in the bracelet to simplify the method.
So I reduced the number of colors to two and grouped the beads as shown in the picture on the left.

The procedure is as follows:

1. Pick, without looking, one of the beads
2. Draw the line depending on the selected bead:
   • If it's the single black one, draw  ;
   • If it's one of the seven black ones, draw  ;
   • If it's one of the five wooden ones, draw  ;
   • If it's one of the five wooden ones, draw  ;
3. Repeat steps 1-2 other five times and draw theline from th bottom to the top of the hexagram

And very shortly I understood that even using two colors were unnecessary: just marking the groups was enough:

In the picture above, beads in the same group are directly connected and groups are separated by a short (three links) chain.

The procedure is rather intuitive:

1. Pick, without looking, one of the beads
2. Draw the line depending on the selected bead:
   • If it's the single one, draw  ;
   • If it's one in the group of seven, draw  ;
   • If it's one in the group of five, draw  ;
   • If it's one in the group of three, draw  ;
3. Repeat steps 1-2 other five times and draw theline from th bottom to the top of the hexagram

This is one of my favourite casting devices: you can keep one with you and using it for general meditation (or simply as an anti stress). In the past years I've made and lost many of them, made with every bead I could find.

The only caveat in making one is to ensure to get the right balance between having the groups well seprated and the possibility of recognize them by touching. Otherwise it will be either too difficult to read or too subject to involuntary selection of a specific group.

Probabilities

The arrangement of beads in the bracelets shown in the pictures above are so that:

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

I've never built one but using beads of two colors it's easy to make a bracelets with the three coins probabilities. It's impossible to do the same using just one type of beds.


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

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Paper die (1d8)

Is there an I Ching casting device that one could build out of just a piece of paper? I once asked this question to myself when I wanted to cast an hexagram and no other method was available to me. I was on a plane, so rolling coins on the small table in front of me was not practical and I had no beads nor cards with me.

All that I had was a pen and some sheets of paper. I started cutting sheets to go through the basic origami forms to see if anything could be fit for the purpose.

Very soon, as it is a basic form, I folded the square base, also known as the preliminary base, which has a nice property: it has four different faces each one being a square. This looked promising as I could combine the four faces with the four edges to be able to get sixteen different events (which would be perfect to simulate the yarrow stalks probabilities).

Building it

The diagram below shows how to fold a square base:

Should the diagram above not be clear enough, there are many instructions and videos on the internet on how to fold one.

The problem is that a square base is actually asymmetric: its top (the point marked A) is structurally different from its bottom (the point where C,D and the other two angles meet) wich tends to open up. As it is it would not be usable as a device for casting hexagrams.

To make it symmetric (and also give it a stronger structure) I thought of interlocking two square bases. A possible way to do it is to proceed as shown in the diagram below:

The tricky part is in step 3 where the face Y has to go under the face A while, at the same time, the face Z (on the opposite side) has to go over face B. It is much easier doing it than describing it; with a little practice you'll make one in no time.

The resulting object has four faces: two from one base (A and C) and two from the other base (X and Z, not shown in the picture below). It has two rotational axis: the vertical one with order of rotational symmetry 4 (the four faces) and the horizontal haxis with order of rotational symmetry 2 (the swap between the blue and red dots in the picture below).

To get a rather robust object, it is best to start from a square of 5x5 cm (approx 2x2 in). The easiest way is to cut a sheet of paper (A4 or US Letter) in four strips along the longest side and then cut the squares from them.

Marking the faces (for yarrow stalks probabilities)

What I got in the end was a paper die with four faces, each face has four sides so I marked one side with 6,  seven sides with 8, five sides with 7 and three sides with 9.
Here is how the two squares looked like if I had unfolded them:

Done! ... or so I thought. I soon realized that doing this way, the 6 could only show up in two positions: top right or bottom left; should I develop the habit of picking other sides more frequently, I would lower the chance of getting a 6.

If I were more disciplined, I could assume that I would choose any side with the same frequency and the probabilities would be:

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

Marking the faces (for three coins probabilities)

To avoid bias in this die, I decided to split each face in two so that the line would depend on the die orientation and on the face I would pick. This left me with eight possible outcomes: exactly what is needed for the three coins probabilities.

Here how the new faces looked like:

With this marking the probabilities are:

Prob(6) = Prob(9) = ¹/₈ = 12.5%
Prob(8) = Prob(7) = ³/₈ = 37.5%
Prob(yin) = Prob(yang) = ¹/₂

Gallery

In the end, the plane landed and I didn't ask the question I had in mind. However, I gained a new method for casting hexagrams.  Since that day I have inserted in my pocket copy of the I Ching a couple of pre-printed strips of paper so that I can quickly build one of these dice and cast a hexagram with it.
I've never done it but I like the idea to write the question on the back of the piece of paper before building the die so that the casting is related forever to the question.

If you want to try them, download the PDF files you prefer (yarrow stalks or three coins) and print them. Be careful to set up the option to "keep the orginal size" or you will have trouble when cutting and folding

I built a couple of these dice to give a better sense of what they look like.

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

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Six Coins

I learned about this method on Clarity (a well known site with lots of information and very good community forums)  which reports it to be described  in the book "I Ching Made Easy" by Rod & Amy Sorrell. In line with the book's title, this is an easy way to cast an hexagram and proceeds as follows: Take five identical coins and one coin that is notably different from the others Shake the six coins and throw them on the table Without looking, align them in a vertical line on the table Draw a line for each coin in the order you aligned them: For each of the five identical coins: If it's head, draw  ; If it's tail, draw  ; For the different coin: If it's head, draw  ; If it's tail, draw  ; Probabilites This method 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 Variations  Instead of using a different coin for getting a moving line, you can throw the coins six times generating 6 hexagrams from which you can derive 6 line using the process described for the I Ching Book method. --- I received a good suggestion from Tsukiko for another variation: Take six coins (it doesn't matter if they are all equal or not) Shake the six coins and throw them on the table Without looking, align them in a vertical line on the table Draw a line for each coin in the order you aligned them: If it's head, draw  ; If it's tail, draw  ; Repeat steps 2 and 3 This time for each coin: If it's head, the corresponding line is a moving line; If it's tail, the corresponding line is not a moving line;  This variation gives the same probability to each line: Prob(6) = Prob(9) = Prob(8) = Prob(7) = 1/2 * 1/2 = 1/4             Prob(yin) = Prob(yang) = 1/2                    :

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I Ching Top

Using a top is a great way to generate random events. In 2001, Malford W. Goldberg patented an I Ching Top which he had started selling on his ichingtop.com site since the year before.
It was advertised as a reconstruction of an ancient method for casting hexagrams supposedly in use during the Han dinasty as part of many occult rituals.
The site is no longer on line (the latest archived version on the Internet Archive is dated November 2002) and the product doesn't seem to be available anymore. In his article on I Ching related patents, Joel Biroco provides more infomation on this method.

Actually, I've not seen any other reference to tops used in ancient times for divination in conjuction with the I Ching but I believe it is an interesting idea, worthy to be preserved. The I Ching Top was used as follows:

1. Spin the top;
2. Draw the line which is closer to the axis on the side the top stopped on;
3. Repeat steps 1-2 drawing hexagram lines from bottom to top.

The image below shows the front and the back of the top:
 

The drawings on the faces may suggest other methods for casting hexagrams (for example one trigram at the time).

Probabilities

Looking at the image above, counting the possible outcomes, is clear that the probabilities are the same of the three coins method.

Prob(6) = Prob(9) = ¹/₈ = 12.5%
Prob(8) = Prob(7) = ³/₈ = 37.5%
Prob(yin) = Prob(yang) = ¹/₂

Variations

On Thingiverse I've found another I Ching top which only allows casting the hexagram one trigram at the time. The 3D model is freely downladable.

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Computer casting

There are currently many sites and applications that will generate a hexagram for you. Most of them offer links to commentaries to directly get an interpretation of the hexagram and some also have the possibility to switch between the yarrow stalks and the three coins method probabilites.

While, conceptually, casting hexagrams is an extremely simple task for a computer, there are drawbacks that you should be aware of.

First of all, generating true random number on a computer is extremely difficult. In most cases the program will use the default pseudo-random number generator (PRNG) which, usually, will trade speed for accuracy and generate numbers that are not random at all.
A PRNG will generate a sequence of numbers that is completely fixed once an initial seed is given, however some of them are able to generate sequences that appear to be random to the most common statistical tests. If unbiased randomness is important to you, you should check whether the software you're using has a good PRNG or not.

Second, using a program to cast the hexagram may lack that feeling of a direct connnection between the user and the response: you click a buttom and you instantly get a line (or even an entire hexagram). The experience of counting, shuffling, throwing, etc., is completely taken away, just a click and you're done!
This may be a problem for some (me, for example) and may discourage them from using casting software; others, instead, like exactly the fact that the response is immediate and they see a profound connection of the response with the exact instant in which the question is formulated.
Being a matter of personal preference, it will be up to you to determine if using software is for you or not.

If you come across a softare hexagram generators, to decide if you are going to use it or not, you may want to investigate if it uses one (or more) of these strategies:

1. Use the random number generator available to the language/operating system the software runs on.
2. Use a hardware true random numbers generator (e.g. using white noise radio signals, nuclear decay, ...).
3. Use the date and time to link the hexagram to the time it is cast.
4. Incorporate element from the user (e.g. mouse movement, keyboard tapping).

I wrote a small line hexagrams caster, which uses the two last strategies mentioned above to increase the participation of the user to the casting process. The applet collects random bits from the users mouse movents and clicks and use them to generate the lines (with yarrow stalks probabilites). You can see it in the right bar of this site.
Clicking on the numbers below the response will get you to one of the many sites that offer I Ching commentary text.

Please note that it is not very mobile-friendly (it's somhow bothersome to have to tap many times on the image), I hope I'll have the time to improve it some day.

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Four Coins

Four coins can be used to easily generate hexagram lines with yarrow stalks probabilities.

 This method has been devised by Stuart Anderson and proceeds as follows:

1. Pick three identical coins and one coin that is notably different from the others 
2. Throw the four coins;
3. Look at the single different coin:
• If it's tail, draw  ;
• If it's head, draw  ;
4. Look at the four coins, if among them there are exactly three head, it is a moving line.
5. Repeat steps 1-4 drawing hexagram lines from bottom to top.

Probabilities

The table below summarizes the possible outcomes of this metods, the results that generate a moving line are highlighted in red.

By counting, it's easy to see that the probabilities are:

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

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Liubo die (1d7)

I had heard about the ancient game of Liubo but I didn't know it was related to the I Ching or to divination in general.

Sven Christensen sent me a very interesting article where a parallel between games and divination is drawn and some light on the relationship between Liubo and the I Ching is shed.

Liubo rules are long lost and they were most probably not directly related to divination with I Ching, so the best I can do is to make up a method for casting hexagrams using the same principles that Chris devised for his "One Die Method".

The picture[1] below shows a Liubo dice wich is a 14-sided irregular polyhedron. Since the symbols are repeated twice, it is actually a 7 values die.

A possible method for casting a line (directly inspired to Chris' method) could be the following:

1. Throw the die and look at the face on the top:
2. If it contains lines, draw  ;
3. If it contains a symbol, draw  ;
4. If it is empty, throw the die again and look at the face on the top:
• If it contains lines, draw  ;
• If it contains a symbol, draw  ;
• If it is empty, restart from step 1
5. Repeat steps 1-4 until you have drawn all the lines from the bottom to the top of the hexagram.

Probabilites

Let's begin by noticing that the steps 1-4 above could not produce any line: the probability of such event and the probability of, instead, getting a line are:

Prob(no line) = ¹/₇ * ¹/₇ = ¹/₄₉ 
Prob(line) = 1- ¹/₄₉ = ⁴⁸/₄₉ 

The probability for each possible outcome (conditioned to the event of actually getting a line) is:

Prob(6)  =   Prob(9) = (¹/₇ * ³/₇) * ⁴⁹/₄₈  = ¹/₁₆ = 6.25%
Prob(8)  =  Prob(7) = ³/₇ * ⁴⁹/₄₈             =  ⁷/₁₆ = 43.75%
Prob(yin) = Prob(yang) = ¹/₂                                             

It is interesting to note that this methods has something in common with the Yarrow Stalk method (probabilities as multiple of 1/16, 49 possible outcomes from throwing) and something in common with three coins method (same probability for 9 and 6).

To enahance the fairness of the die, if you decide to build it, I suggest you alternate the symbols so that the the same symbol appears once on a square face and once on an hexagonal face.

3D printed Liubo dice are availble on Shapeways.


[1] http://www.livescience.com/52806-tomb-ancient-board-game-photos.html

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I Ching quipu

For a long time I've thought about a way to cast I Ching hexagrams using knotted strings.

I've tried many different designs and I'm still not 100% satisfied. However this quipu (yes, I know it's not a real quipu) is so simple to build, easy to handle and pleasant to the eye that I decided to present it here.

To see how the two strings are tied togther, just enlarge the picture, it should be pretty straightforward to replicate. In the center there is a triple fisherman's knots, at each end there's a figure 8 knot tightened.

A nice feature of this object is that it can be used to generate lines with different probabilities.

Three Coins Probabilities

1. Pick (without looking) one of the knots at the end of the strings;
2. If it's red, write down 3; if it's black, write down 2;
3. Repeat steps 1-2 other two times and sum up the numbers;
4. Draw the line according the following table:
   

   6	7	8	9

5. Repeat steps 1-4 other five times drawing the resulting lines from the bottom to the top of the hexagram. 

Choosing one of the string ends is equivalent to throwing one of the coin and the probabilities are:

Prob(6) = Prob(9) = ¹/₈ = 12.5%
Prob(8) = Prob(7) = ³/₈ = 37.5%
Prob(yin) = Prob(yang) = ¹/₂

Note how each pick corresponds to one of the coins in the three coins method.

Yarros Stalks Probabilities

1. Pick (without looking) one of the knots at the end of the strings;
2. If it's black, coming out the black knot write down 2, otherwise write 3;
3. Pick (without looking) one of the knots at the end of the strings;
4. If it's red write down 3, if it's black, write down 2;
5. Repeat ste steps 3-4 again and sum up the numbers;
6. Draw the line according the following table:
   

   6	7	8	9

7. Repeat steps 1-6 other five times drawing the resulting lines from the bottom to the top of the hexagram. 

The choice in step 1 will give 2 with probability ¹/₄  and 3 with probability ³/₄ , the other steps will give 2 with probability ²/₄ and 3 with probability ²/₄, meaning that the probabilities are:

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

Note how the first pick corresponds to the first split of the 49 stalks and the others to the subsequent splits.

Equal Probabilities

1. Pick (without looking) one of the knots at the end of the strings;
• If it's black, coming out the red knot, write down 6
• If it's black, coming out the black knot, write down 8
• If it's red, coming out the red knot, write down 7
• If it's red, coming out the black knot, write down 9
2. Draw the line according the following table:
   

   6	7	8	9

3. Repeat steps 1-2 other five times drawing the resulting lines from the bottom to the top of the hexagram. 

Each line has the same probability to be drawn:

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

Variations

Same principle but with a white string instead of a red one. The central knot will help handling the quipu while casting the lines.

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

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