A regular tessellation is a pattern of repeating shapes that completely covers a plane without any gaps or overlaps. The shapes used in regular tessellations are called regular polygons, which are polygons with equal sides and equal angles. There are only three regular polygons that can be used to create regular tessellations: equilateral triangles, squares, and regular hexagons.

There are only three types of regular tessellations, one for each type of regular polygon:

Triangular Tessellation: In this type, equilateral triangles are used to create the tessellation. Each vertex of the triangle is surrounded by 6 triangles meeting at a point.
Square Tessellation: In this type, squares are used to create the tessellation. Each vertex of the square is surrounded by 4 squares meeting at a point.
Hexagonal Tessellation: In this type, regular hexagons are used to create the tessellation. Each vertex of the hexagon is surrounded by 3 hexagons meeting at a point.

Here are some key points to remember when studying regular tessellations:

Regular tessellations are made using regular polygons.
Only equilateral triangles, squares, and regular hexagons can be used to create regular tessellations.
Regular tessellations cover a plane without any gaps or overlaps.
Each vertex of the regular polygon in the tessellation is surrounded by the same number of polygons.
Regular tessellations are commonly found in art, architecture, and design.

Here are some examples of regular tessellations:

Triangular Tessellation: The pattern of equilateral triangles in a hexagonal grid, often seen in decorative tiling.
Square Tessellation: The pattern of squares commonly found in floor tiling or paving designs.
Hexagonal Tessellation: The pattern of regular hexagons often used in quilting or mosaic designs.