Question

Gary is using a combination of regular polygonal tiles for his kitchen floor. He begins the tessellation at a vertex by placing one triangle and one square.

Which two polygons should he use to fill up the vacant space at the vertex and make it possible to tessellate the floor?

1 square and 1 triangle because 360 – (90 + 120) = 90 + 60

2 squares because 180 = 90 + 90

1 square and 1 hexagon because 360 – (60 + 90) = 90 + 120

1 triangle and 1 hexagon because 180 = 60 + 120

Answer 1

Typical Vertices
Of the regular polygons, only triangles, squares, hexagons, octagons, and dodecagons can be used for tiling around a common vertex - again because of the angle value, and 14 such combinations exist!

This means that each common vertex of any type of regular polygon tessellation must use one of these combinations!

ttt


Combination of Polygons Combination of Angles
6 triangles
6X 60 = 360

4 squares
4 X 90 = 360

3 hexagons
3 X 120 = 360

3 triangles and 2 squares
(3 X 60) + (2 X 90) = 360

3 triangles and 2 squares - another formation
(3 X 60) + (2 X 90) = 360

2 hexagons and 2 triangles
(2 X 120) + (2 X 60) = 360

2 hexagons and 2 triangles - another formation
(2 X 120) + (2 X 60) = 360

1 hexagon, 2 squares, and 1 triangle
(1 X 120) + (2 X 90) + (1 X 60) = 360

1 hexagon, 2 squares, and 1 triangle - another formation
(1 X 120) + (2 X 90) + (1 X 60) = 360

1 hexagon and 4 triangles
(1 x 120) + (4 X 60) = 360

2 octagons and 1 square
(2 X 135) + (1 X 90) = 360

1 dodecagon, 1 square, and 1 hexagon
(1 X 150) + (1 X 90) + (1 X 120) = 360

2 dodecagons and 1 triangle
(2 X 150) + (1 X 60) = 360

1 dodecagon, 2 triangles, and 1 square
(1 X 150) + (2 X 60) + (1 x 90) = 360

http://www.beva.org/math323/asgn5/tess/regpoly.htm