Interdisciplinary matrices

The interactions between disciplines can be geometrically conceptualised as a cross-polytope, viz., each discipline is a node in the polytope.

An n-dimensional cross-polytope (synonymously referred to as “orthoplex”) can be defined as the closed unit ball in the ℓ₁-norm on Rⁿ:
${displaystyle {xin mathbb {R} ^{n}:|x|_{1}leq 1}.}$

but see https://en.wikipedia.org/wiki/Cross-polytope

Generically speaking, each set of k+1 orthogonal vertices corresponds to a distinct k-dimensional component which contains them. The number of k-dimensional components (vertices, edges, faces, …, facets) in an n-dimensional cross-polytope is given by the following equation:

${displaystyle 2^{k+1}{n choose {k+1}}}$

Typesetting in
${LaTeXkern .15em2_{textstyle varepsilon }}$

and MathJax^TM

documentclass{article}
usepackage{tikz}
usetikzlibrary[topaths]
% A counter, since TikZ is not clever enough (yet) to handle
% arbitrary angle systems.
newcountmycount
begin{document}
begin{tikzpicture}[transform shape]
%the multiplication with floats is not possible. Thus I split the loop in two.
foreach number in {1,...,8}{
% Computer angle:
mycount=number
advancemycount by -1
multiplymycount by 45
advancemycount by 0
node[draw,circle,inner sep=0.25cm] (N-number) at (themycount:5.4cm) {};
}
foreach number in {9,...,16}{
% Computer angle:
mycount=number
advancemycount by -1
multiplymycount by 45
advancemycount by 22.5
node[draw,circle,inner sep=0.25cm] (N-number) at (themycount:5.4cm) {};
}
foreach number in {1,...,15}{
mycount=number
advancemycount by 1
foreach numbera in {themycount,...,16}{
path (N-number) edge[->,bend right=3] (N-numbera)  edge[<-,bend
left=3] (N-numbera);
}
}
end{tikzpicture}
end{document}
% A complete graph
% Author: Quintin Jean-Noël
% http://moais.imag.fr/membres/jean-noel.quintin/

#library(devtools)
#install_github("schloerke/geozoo")
#install.packages("geozoo")
cross.polytope(p = 16)

see also http://schloerke.com/geozoo/cube/

References

Knuth, Donald E. (2013), “Two thousand years of combinatorics”, in Wilson, Robin; Watkins, John J., Combinatorics: Ancient and Modern, Oxford University Press, pp. 7–37, ISBN 0191630624.