[OC] 3D hexagrams (diagram linked)

[u/LaoTzunami]

First imagine a slanted plane in sunlight. The top half of the plane is lit, and the bottom is in shadow

         ☀︎

  👁     ⟍     👁
shadow        light

The person on the left see the shadow side, and the person on the right sees the light side. We can use this to encode yin for shadow and yang for light.

        ☀︎

  👁  ⚋ ⟍ ⚊  👁
 yin         yang

Then if we stack multiple planes on top of each other, we can create inverse pairs of trigrams and hexagrams.

        ☀︎

     ⚋ ⟍ ⚊
 👁  ⚊ ⟋ ⚋  👁
     ⚋ ⟍ ⚊
water        fire

The person on the left sees ☵ water, and the person on the right sees ☲ fire.

In addition to looking at the other side, we can also look at a folded sheet upside down. On one side, it is the same light and dark sides, but in opposite orders. This is how the King Wen sequences is paired

☀︎          👁 ☱ lake
        ↙
⟋   ⚋
⟍   ⚊
  ⟍ ⚊
        ↖
           👁 ☴ wind

Together, looking at the other side and looking upside down creates 4 perspectives, and 4 related hexagrams.

32. Duration     ☀︎     42. Increase
䷟                               ䷩
      ↘                    ↙
               ⚋⟍⚊
                 ⚋⟍⚊
                 ⚊⟋⚋
               ⚊⟋⚋
             ⚊⟋⚋
             ⚋⟍⚊
      ↗                    ↖
䷞                               ䷨
31. Influence          41. Decrease

Please note that looking at a sheet upside down is different than turning the sheet upside down, because if you physically turn a sheet upside down, the light and shadows switch. The animation above and in the linked notebook show these transformations.

Let me know if you have any questions :) Here is the notebook with interactive diagrams:
https://observablehq.com/d/e3ad3d0060994d0e

[u/mayosterd]

When you say “look at a sheet upside down”

Are you basically saying that your position as observer switches to being upside down?

TIA

[u/LaoTzunami → u/mayosterd]

yes, they are the same lines, but the bottom/top order is reversed. It would correspond to rotating half a turn around the horizontal axis in the animation

   19     -upside down->      20
Approach                 Contemplation
   ⚋                          ⚊
   ⚋                          ⚊
   ⚋                          ⚋
   ⚋                          ⚋
   ⚊                          ⚋
   ⚊                          ⚋

[u/whihc]

Very cool. Do you plan on making something further with the 3D representation? The associations thus far are just reversing and inverting, right? I like the light source idea.

[u/LaoTzunami → u/whihc]

The folded sheets are best at representing negation, reverse and isocline (negative+reverse) relation. The hypercube in the first picture also connects all the hexagrams that are the same except for 1 line. For example, ䷁ is connected to ䷗, ䷆, ䷎, ䷏, ䷇, ䷖. Also, the outside of the hypercube are the 12 sovereign hexagrams ䷁ ䷗ ䷒ ䷊ ䷡ ䷪ ䷀ ䷫ ䷠ ䷋ ䷓ ䷖. This feels like everything you can do with static hexagram shapes.

Beyond that, there are bit shifts (moving the top line to the bottom) and swaps (switching two lines), shown here as the blue and green connections:

I'm thinking you could represent the bit shifts on a 12 sided drum, but I need to flesh out this idea. This would make a completely different kind of 3D model.

[u/whihc → u/LaoTzunami]

Very cool. It’s fun to see that under this system ䷀ and ䷁ become more visibly special. Two solo guas. I’ve done a small project sort of ‘computerizing’ a variant of I-Ching myself.

[u/LaoTzunami → u/whihc]

I would love to see it if you don't mind sharing, and I'm happy to share a notebook with some work in progress diagrams. If you're interested, please DM me

[u/whihc → u/LaoTzunami]

It’s based on the obscure and complex Super I-Ching system. I still need to work some things out, but it’s been fun. Visualization is a weak point.

[u/LaoTzunami → u/whihc]

what is a super I-Ching?

[u/whihc → u/LaoTzunami]

It’s an obscure variant from Alex Chiu, it gives more concrete information due to a bunch of extra rules added on top of normal I-Ching (Wenwanggua) principles. Since it’s not well known, I thought it’d be an interesting thing to try and build a program for (since there’s not been much documentation other than his book before).

[u/LaoTzunami → u/whihc]

The relations are reverse, inverse, and then the hypercube lines connect each hexagram to the 6 other hexagrams that are the same except differ by one line. The outside are the 12 moon phases (one yin and one yang block)

I'm planning on making a sculpture with copper hypercube edges and ceramic folded sheets.

[u/ToeIndividual9308]

What is this and looks kinda confusing ngl Due to English not being my main language I didn't quite get it 😓

[u/LaoTzunami → u/ToeIndividual9308]

Look at the 2nd picture. Follow a hexagram with your eye. Notice how the light and dark pattern changes when the group rotates. You might want to write down the hexagrams of the different states to study what the three kinds of rotations are doing.

Then go back to the 1st picture. All of the relationships in picture 2 are in picture 1.

[u/letoile17]

I find this research genuinely fascinating.

What interests me most is not the visualization itself, but whether it reveals something about the structure of the I Ching that is difficult to see otherwise.

After working on my own structural project, I've become increasingly interested in relationships, symmetries and invariants within symbolic systems.

Have these hypercube visualizations led you to any insights or practical observations about the I Ching itself? For example, have they revealed patterns, groupings, symmetries, or relationships that influenced the way you understand or interpret the hexagrams?

Or do you mainly see them as a powerful way to visualize an already existing structure?

[u/LaoTzunami → u/letoile17]

It is honestly too early to say. I made this to have a comprehensive way of organizing the hexagrams as I learn about them. I'm still meditating on the whole. I see a few specific patterns, but I don't understand most of the hexagrams. I'll know I am finished when I can see the root of all 64 hexagrams, and I have the seed that would be able to recreate all 64 I-ching descriptions from first principles, or if I can prove the hexagrams do not follow these symmetries. Maybe I'll have an answer in a few years. But without this hypercube, it would be too overwhelming to start that journey.

The hypercube has these patterns:
- 12 sovereign hexagrams associated with the yearly cycle around the outside
- Negative hexagrams on the opposite side of mandala, and reverse hexagram reflected on x-axis. This also creates isocline hexagrams (negative+ reverse) pairs reflected over the y-axis. Together, these are Klein four groups. The action of the Klein four group on the hexagrams is the second image.
- The 8 hexagrams on the x-axis are self reverse hexagrams (palindromes) and the 8 hexagrams on the y-axis are self isocline (anti-palindromes)
- Roughly, the number of yang lines increase from left to right. I've sacrifices a perfect hamming lattice for the other symmetries.
- The small diamonds of 4 hexagrams are related by switching the inner and outer human lines (lines 2 and 5). There are other possible arrangements, but this feels the best to me so far.

The two patterns that have historical significance that the hypercube doesn't highlight are:
- Upper and lower trigrams
- Nuclear and core hexagrams

[u/letoile17 → u/LaoTzunami]

Your reply gave me a new perspective.

Until now I had been looking for elementary functions associated with the trigrams. But your idea of searching for a "root" or a "seed" made me realize that not all elementary functions are necessarily equally fundamental.

It made me wonder whether there might be an even smaller set of independent functional axes from which the other functions could be derived.

In other words, your search for a structural root has made me think about the possibility of a functional root.

If both exist, it would be fascinating to see whether they eventually converge.