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# 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 ℓ_{1}-norm on R^{n}:

${\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.}$

but see 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

${\LaTeX\kern .15em2_{\textstyle \varepsilon }}$

and MathJax^{TM}

\documentclass{article} \usepackage{tikz} \usetikzlibrary[topaths] % A counter, since TikZ is not clever enough (yet) to handle % arbitrary angle systems. \newcount\mycount \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 \advance\mycount by -1 \multiply\mycount by 45 \advance\mycount by 0 \node[draw,circle,inner sep=0.25cm] (N-\number) at (\the\mycount:5.4cm) {}; } \foreach \number in {9,...,16}{ % Computer angle: \mycount=\number \advance\mycount by -1 \multiply\mycount by 45 \advance\mycount by 22.5 \node[draw,circle,inner sep=0.25cm] (N-\number) at (\the\mycount:5.4cm) {}; } \foreach \number in {1,...,15}{ \mycount=\number \advance\mycount by 1 \foreach \numbera in {\the\mycount,...,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)

##### Arguments

`p`

= dimension of object

##### Value

- points
- location of points
- edges
- edges of the object

see also schloerke.com/geozoo/cube/

**References**

Plain numerical DOI: 10.1021/acs.joc.6b02599

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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.

Plain numerical DOI: 10.1002/bdrc.20196

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Plain numerical DOI: 10.1111/j.1462-2920.2009.01995.x

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