Chorand Spheres

The Method of Constructing Chorand Spheres

Henri's Description

Henri described his construction in the following way on the Yixue discussion group in October 2005:

My own stuff - duly copyrighted, of course. It's not a topological sequence, but it's still what I think one can call an interesting topological layout.

I found out that 19 years ago during one of these "a-ha" moments. The idea is to build a couple of topological diagrams, each containing 32 guas, which are "complete" in themselves and do NOT interconnect (the most interesting bit, to me).

I call them hyperspheres, because one can build a 3D model of it, but ideally, it would be great to do that on some synthetic imagery software. And it could well be a 3D "surface" folded on a 4th one (which may remind you about theoretical concepts of the Universe's dimensions, etc., etc.) or a 6D one folded on a 7th one.

I have not tried to pursue my reflexions about the number of dimensions, but okay, no headaches necessary. Here is how to draw a rough sketch on a piece of paper, that may give you an idea.

So, it starts from the way some traditional Chinese diagrams draw one gua at the center of a circle, then draw a hexagon around it where one is to draw, at each of the hexagon's angles, a derived gua where ONE line has changed.

It seems obvious to start building two "stars" (that's how I called them, but they could be snow flakes), starting from guas 1 and 2. That's what I first did. Then I noticed I could "glue" my two drawings of stars together like by putting them as hemispheres (2D surface folded around a 3rd dimention). It worked, in that I had thus grouped together 32 guas. And I could build another pair of "stars" with the remaining 32 hexagrams.

OK, now, here is the building up method.

Choose whatever initial position (line 1 changing) and direction of rotation. You only need to be consistent. For instance:

1st line = 6 o'clock
2nd line = 8 o'clock
3rd line = 10 o'clock
4th line = 12 o'clock
5th line = 2 o'clock
6th line = 4 o'clock

So I thus grouped guas 44, 13, 10, 9, 14, 43 around gua 1 - and guas 24, 7, 15, 16, 8, 23 around gua 2. For readability's sake, draw a straight line between the center gua and each of the "derived" guas.

Now, the "great" idea was to combine each of these "moves" in order to add a second - and larger - circle.

So (with star 1) I "derived" gua 33 from guas 44 and 13. I drew two lines, 33 to 13 and 44 to 13.

Complete the second circle of each star using the same method. For star 1, the second row will include guas 33, 25, 61, 26, 34, 28. For star 2, the second row will include guas 19, 46, 62, 45, 20, 27. Graphically, it seemed obvious to use the following positions:

7 o'clock
9 o'clock
11 o'clock
1 o'clock
3 o'clock
5 o'clock

And the last trick is that you can use "gluon" guas in order to make the two stars glue together, because you will use the same idea in order to make the third row - "derive" 3 contiguous lines. I'll stick to my approximate symmetry and use the same angles (hours) for the third circle as for the first one. For instance, in star 1, at 6 o'clock, I'll put the gua where lines 1, 2, 6 are derived - gua 31, etc.

Here, one can see that the guas of the third circle are the same ones on stars 1 and 2, so at this stage, if you were lucky enough or don't mind using a pair of scissors, you can _join_ stars 1 and 2 with at least one (or two) guas, and draw arrows for the remaining ones in order to get the idea.

OK, and it will work exactly the same with stars made up from guas 63 and 64. So whover wants to explore it all can try to see what there is in these two fully independent sets of guas, beyond the symmetry. Maybe some echoes can appear ...

Anyway, enjoy - and have fun !

Some Comments

One thing comes to mind reading Henri's description. He describes the construction as a "hypersphere" and suggests a number of conjectures as to its geometry. I want to suggest that the simplest of his ideas is the correct one: it is a sphere.  That is, it is a two-dimensional surface, wrapped in the third dimension. The second sphere, which is completely separate from the first, is another two-dimensional surface wrapped in the third dimension. However, because of the six-dimensional nature of the lattice, this second sphere exists in a different three-dimensional space to the first. In that sense, the second sphere is a hypersphere relative to the first. I suggest that the first sphere is external, that is the projection of the six dimensional that is normally visible in three dimensional space. The second sphere is then internal, representing the additional structure. If the Boolean lattice were turned inside out through three dimensional space, then this internal sphere would be the visible surface of the lattice.

We shall now consider some examples, and explore how these spheres are related to the Boolean lattice in more detail.