Acronym | aze |
Name |
apeirogon, infinitogon, horogon (esp. when used as tile in hyperbolic geometry) |
External links |
Generally {n/d} and {(n/d)'} = {n/(n-d)} denote pairs of pro- and retrograde polygons. That is, when considered without context, their geometric realization is the same. Even so, as abstract polytopes they have to be distinguished, esp. when used as link marks within Dynkin diagrams. – It then is only by transition to infinite values of this quotient of n/d, that {n/d} = x-N/D-o and {2n/d} = x-N/D-x become geometrically identical. But this does not hold true for {n/(n-d)} = x-N/(N-D)-o, were the corresponding limit of the mark value becomes 1 (the retrograde aze), and {2n/(n-d)} = x-2N/(N-D)-o, were the limit of that mark value becomes 2 (the ∞-covered edge). These rather behave like a folding rule, in its unfolded resp. its folded state. That is, x-∞-x is a well-behaved polytope, whereas x-∞'-x rather would be some highly degenerate Grünbaumian.
Incidence matrix according to Dynkin symbol
xNo (N → ∞) . . | N | 2 ----+---+-- x . | 2 | N
xN/(N-1)o (N → ∞) . . | N | 2 ----------+---+-- x . | 2 | N
xNx (N → ∞) . . | 2N | 1 1 ----+----+---- x . | 2 | N * . x | 2 | * N
:o:&##x (N → ∞) → height = 1 o | N | 2 -------+---+-- :o:&#x | 2 | N
:oo:&##x (N → ∞) → height(1,2) = height(2,1') = 1 o. | N * | 1 1 .o | * N | 1 1 --------+-----+---- oo &#x | 1 1 | N * inner :oo:&#x | 1 1 | * N outer
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