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Oppositional geometry gives a mathematical model of oppositional phenomena through "oppositional structures' (logical squares, hexagons, cubes, …). It's so far known formal entities, the backbone of which are the "oppositional bi-simplexes (and poly-simplexes) of dimension m", are distributed into three families (the α-, β- and γ-structures). However, some recent studies by different authors exhibit strange structures, notably strange variations of the notion of "oppositional hexagon" (or "logical hexagon"). In this paper we show that inside the oppositional tetrahexahedron, i.e. the (eta 3)-structure (discovered in 1968 and rediscovered in 2008) – a 3-D solid made of a logical cube and 6 logical 'strong hexagons", containing 14 vertices and 36 implication arrows – there are in fact (mathrm{C}^{6}_{14}=30,030) strange hexagons, which we call "hybrid hexagons". In this paper, through a systematic study of those among them that have as invariant property a regular perimeter made of alternated arrows (henceforth "arrow-hexagons"), we show that they divide into a much smaller number of families, nine, each containing several isomorphic instances of the same oppositional pattern. An interesting result seems to be that when seen from the viewpoint of their mutual transformations (i.e. moving from one to another kind of arrow-hexagon, just by exchanging one of its 6 vertices with one among the remaining (14-6=8) vertices of the tetrahexahedron), these arrow-hexagonal patterns taken as points can be displayed in a new kind of 3-D structure. The latter, by putting into order these points (each representing a family of arrow-hexagons), gives some kind of morphogenetic cartography of the arrow-hexagons of the (eta 3)-structure. As we will argue, since several arrow-hexagons play the role of "attractors", there are reasons for thinking that such a cartography could be very meaningful in the future for modelling "oppositional dynamics", that is the systematic formal study of the situations where a given complex oppositional structure sees its shape change within time.

DOI: 10.1007/978-3-319-15368-1_20

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