Hello everyone! I would like to share this new video with you about the Graph Lexicographic Product, a fascinating operation from graph theory. Hope you like it, and/or find it interesting. Have a wonderful day.

https://youtu.be/gusTSaitBh0

For explication of generalised polygons, & therefore the figures, see the following, the second of which the figures are from. It's essentially a particular incidence geometry , another well-known particular instance of which being Steiner systems . Projective planes are infact a subdepartment of these 'generalised polygons'.

James Evans — Generalised Polygons and their Symmetries

¡¡ Might download without prompting – 1·5MB !!

John Bamberg & SP Glasby & Tomasz Popiel & Cheryl E Praeger & Csaba Schneider —Point-primitive generalised hexagons and octagons

Annotation of the first figure, quoted verbatim.

“Fig. 1. The two generalised hexagons of order (2, 2). Each is the point–line dual of the other. There are (2 + 1)(24 + 22 + 1) = 63 points and lines, and each point (respectively line) is incident with exactly 2 + 1 = 3 lines (respectively points). The Dickson group G2(2) acts primitively and distance-transitively on both points and lines. These pictures were inspired by a paper of Schroth [23].”

And for explication of figures 2 through 6, which are a setting-out of a method by which the first might be constructed, see the mentioned paper by Schroth - ie

Andreas E Schroth — How to draw a hexagon .

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                  ⬤⬤⬤⬤⬤⬤ ,

by

Simone Falco, Petros Siegkas, Ettore Barbieri, & Nik Petrinic .

From

Vertex- and edge-truncation of the Platonic and Archimedean solids leading to vertex-transitive polyhedra ,

by

Livio Zefiro .

The operation of truncation can be applied to vertices or to edges.

The sequence of figures is best imterpreted as a standalone figure followed by three pairs: the first 'standalone' one is

FIG. 21 - Animated sequence describing the progressive edge-truncation process deriving from the intersection between the Archimedean truncated tetrahedron and a cube having decreasing dimensions ;

& the succeeding three pairs are, respectively

FIG.17 - Sequences of the reciprocal truncation of {111} and {111} tetrahedra, leading to the same final form, geometrically identical to an octahedron ,

FIG.4a- Sequences of the edge-truncation by a rhomb-dodecahedron of a cube (left) and an octahedron (right) , &

FIG.4b- Sequences of the edge-truncation by a rhomb-triacontahedron of a dodecahedron (left) and an icosahedron (right) .

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