Elements of subsets of 2^K organized by subset size in columns?
That’s the idea, though I’m not quite sure what you mean by “elements of subsets of 2^K”.
By 2^K i think he means power set of K.
Sorry, I meant elements of 2^K.
Ah, makes sense!
A decade ago I was playing with visualizing your 5th group and discovered a beautiful layout of the connections between them: http://suberic.net/~dmm/mathy.html
Beautiful! Thanks for sharing!
Can you maybe say a bit more about how you discovered this layout, and how it is generated? Is there a nice description of which vertices correspond to which subsets?
I wanted to find a circular layout where each ring had the next layer of sets, the edges were symmetrical, and lines did not have to go across the center or 180 degrees around the diagram. I think that the diagram follows inexorably from those decisions. When I drew it for the first time it seemed instantly familiar, though I believe it is my discovery.
The central point at the origin indicates the empty set. Each of the innermost points is one of the single-element sets – we can call them A, B, C, D, and E. They’re in directions 0, 72, 144, etc from the origin.
The second ring of points corresponds to the two-element sets, and there are two types of those – neighbors such as AB, BC are placed between their component directions at 36, 108, etc. while non-neighbors such as EB, AC, BD are placed at the direction of the skipped element at 0, 72, 144, etc.
The third ring corresponds to the 3-element sets. Neighbors such as ABC are placed based on the central member, while non-neighbors like ACE have two neighboring directions and one loner (EA and C in this case) – those are placed opposite the loner, so ACE will be at 324. This means that there’s a radial line connecting each second-ring element to one third-ring element – non-neighbors get their missing middle and neighbors get their opposing element. This ring is also a reflection of the second ring, naturally.
The fourth ring is for the sets missing one element and they’re placed opposite the corresponding single-element point. The fifth ring is a single point infinitely away in every direction, indicated by the outer circle.
This kind of layout works in two dimensions only for 2, 3, and 5 elements due to symmetry reasons, and 5 is the only interesting-looking one. I think it would only work for 4 elements in three dimensions.
I see, makes sense! But why wouldn’t it work for more elements than 5?
If you have 7 elements and put A at 0 degrees, you have three 2-element sets that need to be on the 0-180 line: BG, CF, and DE. Only two such points are available in the second ring, so you’ll need to arbitrarily break the symmetry on one side or another. (Composite numbers are even worse, as directly opposing points belong in the very center of the diagram.)
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I love it. I wonder if you would publish the diagrams code you used to do it. Diagrams could use some more examples, even/especially if hey are no about certain features but general ideas patterns.
All the code for my blog posts is publicly available at http://hub.darcs.net/byorgey/mathlesstraveled/ , though I suppose it is not necessarily easy/obvious how to find the code corresponding to a particular post. The code for this one can be seen at http://hub.darcs.net/byorgey/mathlesstraveled/browse/pww/Subsets.hs and http://hub.darcs.net/byorgey/mathlesstraveled/browse/pww/subsets.md .
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