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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 ℓ1-norm on Rn:
${\displaystyle \{x\in \mathbb {R} ^{n}:\|x\|_{1}\leq 1\}.}$

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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:

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

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

and MathJaxTM

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5 + 1 =

% A counter, since TikZ is not clever enough (yet) to handle
% arbitrary angle systems.
\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:
\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:
\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}{
\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);
% A complete graph
% Author: Quintin Jean-Noël
cross.polytope(p = 16)
p = dimension of object
location of points
edges of the object

see also


Ash, M. G.. (1998). Gestalt psychology in German culture, 1890-1967 : holism and the quest for objectivity. Cambridge Studies in the History of Psychology

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

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.

Rosenberg, E., & Zilber-Rosenberg, I.. (2011). Symbiosis and development: The hologenome concept. Birth Defects Research Part C – Embryo Today: Reviews

Plain numerical DOI: 10.1002/bdrc.20196
directSciHub download

Rosenberg, E., & Zilber-Rosenberg, I.. (2008). Role of microorganisms in the evolution of animals and plants:The hologenome theory of evolution. F. E. M. S. Microbiol. Rev., 32, 723–735.
Rosenberg, E., Sharon, G., & Zilber-Rosenberg, I.. (2009). The hologenome theory of evolution contains Lamarckian aspects within a Darwinian framework. Environmental Microbiology, 11(12), 2959–2962.

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

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