Uniform tilings of the Euclidean plane Uniform tiling

there symmetry groups on euclidean plane constructed fundamental triangles: (4 4 2), (6 3 2), , (3 3 3). each represented set of lines of reflection divide plane fundamental triangles.

these symmetry groups create 3 regular tilings, , 7 semiregular ones. number of semiregular tilings repeated different symmetry constructors.

a prismatic symmetry group represented (2 2 2 2) represents 2 sets of parallel mirrors, in general can have rectangular fundamental domain. generates no new tilings.

a further prismatic symmetry group represented (∞ 2 2) has infinite fundamental domain. constructs 2 uniform tilings, apeirogonal prism , apeirogonal antiprism.

the stacking of finite faces of these 2 prismatic tilings constructs 1 non-wythoffian uniform tiling of plane. called elongated triangular tiling, composed of alternating layers of squares , triangles.

right angle fundamental triangles: (p q 2)

general fundamental triangles: (p q r)

non-simplical fundamental domains

the possible fundamental domain in euclidean 2-space not simplex rectangle (∞ 2 ∞ 2), coxeter diagram: . forms generated become square tiling.