ベン図^2/Edwards' Construction

ベン図と同じ意味だと思うんだけど...

http://www.combinatorics.org/Surveys/ds5/VennWhatEJC.html　より Orz〜
画像：Venn's Construction (集合6個のときのもの)
画像：Edwards' Construction

「For example, below we show the link that arises (after some smooth transformations in 3-space) from Venn's general construction for n = 4. This is clearly the same as the link that arises from Edwards' general construction for n = 4, although the constructions give different links for n > 4.」

画像：これが４個のときの立体視...で...ベン図とエドワード図と同じとわかりますね...^^

画像：5個のときも奇麗なベン図が描けるんだ♪
画像：7個

「A General Construction for Symmetric Venn Diagrams

Recently, Jerrold Griggs, Chip Killian, and Carla Savage [GKS] made a major breakthrough by finding a construction for a symmetric Venn diagram for any prime n. The following page illustrates in detail the ideas used for the general construction:

画像：Symmetric Diagrams, Necklaces, and Chains
画像：11個のときの例

Briefly, the idea is to create the dual graph of one sector of the diagram by building a series of chains that span a set of strings from the Boolean lattice, with the property that when this dual graph is rotated, the dual of the entire Venn diagram is created. The difficult part is specifying which strings to choose to form the appropriate subset (making up 1/nth of the Boolean lattice), using a known rule to build chains out of these strings, and then attaching them together and embedding them to form a planar dual of a sector of the diagram.

The diagrams produced by their construction are monotone and highly non-simple and have the property that there are n vertices through which all n curves pass. Their diagram for n=11 is shown on the page with examples of symmetric diagrams. The diagrams have exactly C(n, floor[ n/2 ] ) vertices, which is minimum for monotone Venn diagrams. In several ways their paper is the culmination of the last decade of research by several authors highlighted in these pages on generating symmetric Venn diagrams. The result has been widely publicised in the scientific press (see, for example, the articles [Ci03] and [Ci04]), and Donald Knuth presented the result in his annual "Christmas Tree" lecture in 2002 [KM].

In further work, Killian, Ruskey, Savage, and Weston [KRSW] showed that more vertices can be added to the construction by applying certain operations; see the bottom of the page on necklace diagrams for more details. The number of vertices can be increased to give asymptotically at least 2n-1 vertices; in practise the numbers are higher than 2n-1, at least for small n, but it appears to be difficult to prove a better bound.

The table below shows the highest known number of vertices in a symmetric diagram, for each n, versus the number of vertices in a simple diagram.

n vertices in simple
　　symmetric diagram best known 　 reference
3 　　6 　　　　　　　6 　　　　　　3-circle diagram
5 　　30 　　　　　　　30 　　　 [Gr75] (the 5-ellipses diagram)
7 　　126 　　　　　　　126 　　　[Ed96], this survey
11 　　2,046 　　　　　　　1,837 　　　 [HPS] (see this page)
13 　　8,190 　　　　　　　5,005 　　　 [KRSW]
17 　　131,070 　　　　 81,787 　　　 [KRSW]
19 　　524,286 　　　　 329,289 [KRSW]

Thus, the search for a simple symmetric Venn diagram of more than 7 curves continues.」

対称な/symmetricなベン図が探されてるってことなんですよね...^^?
わけわからないままに...ただ美しいと思ったもので...^^♪