Why won’t a regular octagon tessellate the plane by itself? Describe a combination of a regular octagon and another regular polygon that will tessellate the plane.
A regular octagon cannot tessellate the plane by itself because the angles of an octagon sum up to 1080 degrees, while a flat plane requires angles to sum up to 360 degrees for tessellation. To tessellate the plane using an octagon, we need to combine it with another regular polygon that compensates for the excess angles.
One possible combination is to use a square along with an octagon. Both shapes have equal-sized sides and equal angles, but their angle sums do not exceed 360 degrees. The sum of angles in an octagon is 1080 degrees, while the sum of angles in a square is 360 degrees.
To create a tessellation on a plane, you can position an octagon adjacent to a square, aligning their sides. The vertices of the octagon and the square should form a continuous pattern as you repeat it across the plane, without any gaps or overlaps. This combination effectively balances the excess angles of the octagon with the square's angles, allowing for a periodic tiling that fills the entire plane without any voids.
By combining a regular octagon with a regular square, you create a tessellation that repeats indefinitely, covering the entire plane.