Geometric and cognitive differences between logical diagrams for the Boolean algebra B 4 $\mathbb {B}_{4}$

Abstract

Aristotelian diagrams are used extensively in contemporary research in artificial intelligence. The present paper investigates the geometric and cognitive differences between two types of Aristotelian diagrams for the Boolean algebra B 4. Within the class of 3D visualizations, the main geometric distinction is that between the cube-based diagrams (such as the rhombic dodecahedron) and the tetrahedron-based diagrams. Geometric properties such as collinearity, central symmetry and distance are examined from a cognitive perspective, focusing on diagram design principles such as congruence/isomorphism and apprehension. The cognitive effectiveness of the different visualizations is compared for the representation of implication versus opposition relations, and for subdiagram embeddings.

Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation

“…By viewing the Hasse and Aristotelian rhombic dodecahedra as two vertex-first parallel projections of a tesseract, albeit along different projection axes, we also obtain a unified geometrical explanation of their fundamental diagrammatic differences [56]. In particular, in the tesseract, the logical levels are ordered by means of hyperplanes going from (the vertex corresponding to) 0000 to (the vertex corresponding to) 1111; however, if the projection is precisely along the 1111/0000 axis, then this ordering is completely annihilated, so that the resulting Aristotelian rhombic dodecahedron no longer visually represents this ordering of levels: the L 1 -, L 2 -and L 3 -bitstrings are scattered throughout the polyhedron [55,57].…”

“…The coordinate function c RDH : B 4 → R 3 shown in Table 1 at the end of this section maps the bitstrings of B 4 onto the vertices of the Aristotelian rhombic dodecahedron [55,57]. Note that c RDH (1111) = c RDH (0000) = (0, 0, 0) and c RDH (¬b) = −c RDH (b) for all bitstrings b, i.e., the top and bottom elements of B 4 coincide with each other in the center of the Aristotelian rhombic dodecahedron, and logical negation is visualized by means of central symmetry.…”

“…We have seen in Section 3.1 that the Aristotelian rhombic dodecahedron for B 4 is the vertex-first parallel projection of a tesseract, along the projection axis defined by (the vertices corresponding to) the bitstrings 1111 and 0000. Completely analogously, the Aristotelian nested tetrahedron for B 4 is also the vertex-first projection of a tesseract, along the same projection axis; the only difference is that we now use a perspective projection instead of a parallel projection [55,57]. In the rhombic dodecahedron, the four L 1 -bitstrings and the four L 3 -bitstrings constitute two tetrahedra, which are of equal size (because they are the parallel projections of the L 1 -and L 3 -hyperplanes in the tesseract) and 'interlock' with each other to constitute the cube inside the rhombic dodecahedron (cf.…”

“…One of the crucial insights in this area is that Aristotelian diagrams can also be fruitfully seen as truly geometrical entities and studied by means of tools and techniques such as projection matrices, Euclidean distance, symmetry groups, etc. [54][55][56][57]. (This hybrid perspective on diagrams, treating them simultaneously as diagrammatic visualizations of an underlying abstract structure and as geometrical entities by themselves, can also be found in crystallography [58,59].)…”

“…Similarly, Smessaert et al [63,69] compare the Aristotelian rhombic dodecahedron, tetrakis hexahedron and tetraicosahedron, but these studies again go into far less geometrical detail than the present paper and do not address the Aristotelian nested tetrahedron. Finally, Smessaert et al [55,57] offer an in-depth geometrical comparison of the Aristotelian rhombic dodecahedron and nested tetrahedron for B 4 , but these papers do not deal with the Aristotelian tetrakis hexahedron and tetraicosahedron.…”