We often see floors with patterns as shown in the figure. They are all made of square shaped materials or hexagonal shaped materials. In this way, the materials can be paved into a flat and void free floor (1) If the floor is paved as above, can it be made of pentagonal materials? Why? (2) Can you come up with another plan to use a polygonal (not necessarily regular polygonal) material to lay the floor? Draw a sketch of your plan. (3) please draw a sketch of two different regular polygon materials

We often see floors with patterns as shown in the figure. They are all made of square shaped materials or hexagonal shaped materials. In this way, the materials can be paved into a flat and void free floor (1) If the floor is paved as above, can it be made of pentagonal materials? Why? (2) Can you come up with another plan to use a polygonal (not necessarily regular polygonal) material to lay the floor? Draw a sketch of your plan. (3) please draw a sketch of two different regular polygon materials

(1) The shape of the material used can not be a regular pentagon. Because each inner angle of the regular pentagon is 108 degrees, in order to lay a flat and void free floor, several regular pentagon must be assembled into a circumference (360 degrees), but n that meets the condition of n × 108 degrees = 360 degrees can not be found. Therefore, the floor can not be paved with all regular pentagon materials; (2) draw a sketch according to the requirements; (3)