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In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and hyperbolic plane. Uniform tilings are related to the finite uniform polyhedra; these can be considered uniform tilings of the sphere.
John Conway called these uniform duals Catalan tilings, in parallel to the Catalan solid polyhedra. Uniform tilings are listed by their vertex configuration, the sequence of faces that exist on each vertex. For example 4.8.8 means one square and two octagons on a vertex. These 11 uniform tilings have 32 different uniform colorings. A uniform ...
For example: 3 6; 3 6; 3 4.6, tells us there are 3 vertices with 2 different vertex types, so this tiling would be classed as a ‘3-uniform (2-vertex types)’ tiling. Broken down, 3 6 ; 3 6 (both of different transitivity class), or (3 6 ) 2 , tells us that there are 2 vertices (denoted by the superscript 2), each with 6 equilateral 3-sided ...
There are 4 symmetry classes of reflection on the sphere, and three in the Euclidean plane. A few of the infinitely many such patterns in the hyperbolic plane are also listed. (Increasing any of the numbers defining a hyperbolic or Euclidean tiling makes another hyperbolic tiling.) Point groups:
An n-dimensional uniform honeycomb can be constructed on the surface of n-spheres, in n-dimensional Euclidean space, and n-dimensional hyperbolic space. A 2-dimensional uniform honeycomb is more often called a uniform tiling or uniform tessellation.
In hyperbolic geometry, a uniform hyperbolic tiling (or regular, quasiregular or semiregular hyperbolic tiling) is an edge-to-edge filling of the hyperbolic plane which has regular polygons as faces and is vertex-transitive (transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other).
Functional theories: Widely used approaches like Kouzes & Posner's Five Leadership Practices model and Adair's Action-Centered Leadership theory assume that once the leader understands – and has been trained in – the required leadership behaviors, he or she will apply them as needed, regardless of their personality. However, as with the ...
Functional leadership theory (Hackman & Walton, 1986; McGrath, 1962) is a theory for addressing specific leader behaviors expected to contribute to organizational or unit effectiveness. This theory argues that the leader's main job is to see that whatever is necessary to group needs is taken care of; thus, a leader can be said to have done ...