Are there higher-dimensional versions of the concept of rotationally symmetric Venn diagrams, with closed curves replaced by closed surfaces or higher manifolds ?
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The n-simplex as a subset of ${\mathbb R}^{n+2}$ can serve as a stand-in for a Venn diagram. Or if you like you can fatten the vertices until you have over-lapping $n$-balls. More precisely, consider the convex hull of ${e_1, e_2, \ldots , e_{n+1}}$. This is the set $\{ \vec{x} \in R^{n+2}: \sum_i x_i = 1, \ \ \& \ \ 0\le x_i \}$ where $e_i$ represents the $i$th standard unit vector $[0,\ldots, 0,1,0,\ldots, 0]$ that has a $1$ in the $i$th position. Each vertex represents one of your sets. Each edge represents the intersection between two sets. Each triangle represents the intersection among $3$ sets, and each $k$-simplex (which is determined by a subset of size $k+1$ chosen from ${1,2, \ldots , n+2}$ represents the intersection among $(k+1)$ sets. To imagine a model use a protractor and draw the complete graph on $(k+2)$-vertices that are represented by the roots of unity $e^{2\pi i j/(k+2)}$ (The $i$ here is different than the index above). |
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"four intersecting spheres form the highest order Venn diagram that is completely symmetric and can be visually represented" http://en.wikipedia.org/wiki/Venn_diagram#Extensions_to_higher_numbers_of_sets |
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