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.