Question

Which of the following shapes is needed to tessellate with octagons? 1)Pentagon 2)Square 3) Decagon 4)Triangle 5)Hexagon

Answer 1

As explained below.

Explanation:

`color(brown)("What does it mean for a shape to tessellate?"`

A tessellation of a flat surface is the tiling of a plane using one or more geometric shapes, called tiles, with no overlaps and no gaps.

In mathematics, tessellations can be generalised to higher dimensions and a variety of geometries.

A tiling that lacks a repeating pattern is called "non-periodic".

`color(brown)("What shapes tessellate and why?"`

Triangles, squares and hexagons are the only regular shapes which tessellate by themselves .

You can have other tessellations of regular shapes if you use more than one type of shape.

There are only three regular tessellations which use a network of equilateral triangles, squares and hexagons.

`color(brown)("What are different types of tessellation?"`

Semi-regular tessellations are made up with two or more types of regular polygon which are fitted together in such a way that the same polygons in the same cyclic order surround every vertex.

Single regular shapes
`color(green)("Triangles" color(crimson)(" Large grid of triangles"`
`color(green)("Squares" color(crimson)(" Large grid of squares"`
`color(green)("Hexagons" color(crimson)("Large grid of hexagons"`

Multiple regular shapes
`color(maroon)("Squares, triangles " color(blue)(" Large grid of squares and triangles"`
`color(maroon)("Hexagons, triangles" color(blue)(" Large grid of hexagons and triangles"`
`color(maroon)("Hexagons, squares, triangles" color(blue)(" Large grid of hexagons, squares and triangles"`
`color(maroon)("Octagons, squares"color(blue)(" Large grid of octagons and squares"`
`color(maroon)("Dodecagons, triangles" color(blue)(" Large grid of dodecagons and triangles"`
`color(maroon)("Dodecagons, hexagons, squares" color(blue)(" Large grid of dodecagons, hexagons and squares"`

Other shapes
`color(purple)("Irregular pentagons" color(magenta)(" Large grid of irregular pentagons"`
`color(purple)("Waffle pattern" color(magenta)(" Large grid of waffle pattern"`
`color(purple)("Fish patterns color(magenta)(" Large grid of fish patterns"`