Acronym  ... 
Name  hyperbolic order 4 hexagonal tiling 
©  
Circumradius  1/sqrt(2) = 0.707107 i 
Vertex figure  [6^{4}] 
Dual  x4o6o 
Confer 

External links 
Having the same curvature (resp. circumradius) as o4x∞o, both having even numbered polygons only and both having tetravalent vertex figures with inversional symmetry, these allow for resp. alternating layers, i.e. an corresponding laminat. That one then would have the vertex configuration [4,6,6,∞] and could be described by the generalized Dynkin diagram x3xØx∞xØ*a3*c.
Note that o4o6x further allows for a consistent halving of hexagons (connecting opposite edge midpoints), such as to derive a combinatorical variant of x5o4o. While for that one the edge lengths of the regular pentagons aaaaa can be calculated as given below for comparison, in that decomposition the edge lengths of these nonregular pentagons aabcb would calculate (using the formulas provided here) as:
x6o4o (halving each hexagon gives a topological variant of x5o4o): P0 o+ P1 P1 /  P0 o  P2 \  o+ P1 edges a: hypdist(P0,P0) = 2 arcosh[cos(π/6)/sin(π/4)] = 1.316958 edges b: hypdist(P0,P1) = arcosh[cos(π/6)/sin(π/4)] = 0.658479 edges c: hypdist(P1,P1) = 2 arcosh[cos(π/4)/sin(π/6)] = 1.762747 deformed pentagons aabcb (to be conpared to: x5o4o all edges alike: hypdist(P0,P0) = 2 arcosh[cos(π/5)/sin(π/4)] = 1.061275 regular pentagons aaaaa )
There exists also a series of regular modwraps of this tiling, obtained by identifying every kth vertex on each hole (q.e. straight edge sequence here). Then it allows a representation as (possibly) infinite regular skew polyhedron, which happens to be the facial subset of hexagons of the spherical decachoron (k=3: finite modwrap), of the euclidean bitruncated cubical honeycomb (k=4), of the hyperbolic bitruncated order 5 dodecahedral tesselation (k=5), etc. This series of modwraps generally is being denoted as x6o4ok (with according value of k).
Incidence matrix according to Dynkin symbol
o4o6x (N → ∞) . . .  3N  4  4 +++ . . x  2  6N  2 +++ . o6x  6  6  2N snubbed forms: o4o6s
o6x6o (N → ∞) . . .  3N  4  2 2 +++ . x .  2  6N  1 1 +++ o6x .  6  6  N * . x6o  6  6  * N snubbed forms: o6s6o
© 20042019  top of page 