REGULAR TESSELLATIONS

The three regular tessellations of the plane are closely related to the Platonic solids. Indeed, one can view the Platonic solids as regular tessellations of the sphere. This is done by projecting each solid onto a concentric sphere. The faces project onto regular spherical polygons which exactly cover the sphere. There are three possibilities:

The three regular tilings
 *442

Uniform tiling 44-t0.svg

(3)p6m

Uniform tiling 63-t2.png

*632p4m

Uniform tiling 63-t0.png

{4, 4}

Vertex type 4-4-4-4.svg

44
(t=1, e=1)1-uniform n5.svg

{3, 6}

Vertex type 3-3-3-3-3-3.svg

36
(t=1, e=1)

1-uniform n11.svg

{6, 3)

Vertex type 6-6-6.svg

63
(t=1, e=1)

1-uniform n1.svg

 

 

 

Polygon nets around a vertex
Polyiamond-3-1.svg
{3,3}
Defect 180°
Polyiamond-4-1.svg
{3,4}
Defect 120°
Polyiamond-5-4.svg
{3,5}
Defect 60°
Polyiamond-6-11.svg
{3,6}
Defect 0°
TrominoV.jpg
{4,3}
Defect 90°
Square tiling vertfig.png
{4,4}
Defect 0°
Pentagon net.png
{5,3}
Defect 36°
Hexagonal tiling vertfig.png
{6,3}
Defect 0°
A vertex needs at least 3 faces, and an angle defect.
A 0° angle defect will fill the Euclidean plane with a regular tiling.
By Descartes’ theorem, the number of vertices is 720°/defect.

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