mr_e_man wrote:Interesting. After seeing this, I considered "almost-uniform" tilings in the Euclidean plane, and found that no such thing exists; any CRF Euclidean tiling with congruent vertices must be uniform. (You probably already knew this.) What about 3D Euclidean space?
Marek14 wrote:Yesterday, we spent some time considering (3,4,5,5) before concluding that this particular combination cannot be made into tiling, not even an aperiodic one.
mr_e_man wrote:Marek14 wrote:Yesterday, we spent some time considering (3,4,5,5) before concluding that this particular combination cannot be made into tiling, not even an aperiodic one.
I can see that easily: both vertices shared by the triangle and the square must be completed by pairs of pentagons; but then the triangle's third vertex has the form 5.3.5.x, which is not 3.4.5.5, regardless of what x is.
Marek14 wrote:As for what "almost uniform" means: there is more than one vertex type, but there is still a finite number of the types and the tiling is overall periodic.
mr_e_man wrote:Marek14 wrote:As for what "almost uniform" means: there is more than one vertex type, but there is still a finite number of the types and the tiling is overall periodic.
Do you allow several vertex configurations?
Do you allow several sets of faces?
Examples: 4.4.3.4.3 and 4.4.4.3.3 are different vertex configurations with the same set of faces. 3^{8} and 3^{2}.8^{3} are different vertex configurations with different sets of faces.
wendy wrote:John Conway described that while they are described by the vertex figure, the cycle of polygons is not enough to describe it.
What you further need is to describe the outbound and inbound edges, where they fall in the cycle, and if they change parity. This is essentially the orbifold.
The first in Mr_e's diagrams is [1,2] [3,5] [4], where [1,2] is the triangle, and [3,5] is the purple squares and [4] are between the purple triangles.
The second is [1,2] [3,5] (4), where the (4) edge is the centre of an order-2 rotation.
It looks like 3% 2% 2% 2% or (1,2) (3) (4) (5). This is an ordinary snub.
In any case, the first two have orbifold nodes of wanders or miracles, which you tell by the presence of non-consecutive numbers in the edges. The third one is a fairly ordinary snub, with squares (as does Miller's mosnster). A pair of consecutive edges in brackets is a cone, or rotation-polygon.
W
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