Category:Concertina cube

Hasse diagrams as skeleton and flattened graph

The concertina cube is an integral polyhedron based on a 4×4×4 cube. It is created by truncating the edges and vertices of the cube that are part of its Petrie hexagon (see here).

Its skeleton is a Hasse diagram. Each of the vertices corresponds to a formula with 3 different variables in first-order logic, and each edge to one of the implications between them.

Each pair of opposite vertices corresponds to one of the weak orderings of 3 elements.

Elements:       (compare row 3 of  A300700)

• 26 vertices       (2 · orderedBell(3))
• 42 edges
• 18 faces: 6 concertina squares, 6 rectangles, 6 rhombi

Surface: hexagon ${\displaystyle 8}$    rectangle ${\displaystyle 2{\sqrt {2}}\approx 2.82}$    rhombus ${\displaystyle {\sqrt {3}}\approx 1.73}$       total ${\displaystyle \approx 75.36}$

Volume:  A300697(3) = 52       (12 cut away from 4×4×4=64)

The coordinates are permutations of these points:       (list)