Will an octagon and an isosceles triangle tessellate a plane?
Yes, under certain conditions.
Each corner of an octagon will leave a 45° isosceles triangle at each corner when attempting to tessellate a plane.
The four isosceles triangles will form a square which can be filled by either 2 or 4 isosceles triangles of 45°.
To determine if an octagon and an isosceles triangle can tessellate a plane, we need to understand the definition of tessellation and the properties of these shapes.
Tessellation refers to the process of covering a plane with shapes without any gaps or overlaps. In other words, the shapes must fit together perfectly, filling the entire plane.
Octagon: An octagon is an eight-sided polygon with equal angles but not necessarily equal side lengths.
Isosceles Triangle: An isosceles triangle is a triangle with two sides of equal length and two equal angles.
For a shape to tessellate a plane, the angles of the shape must fully add up to 360 degrees, and the number of shapes meeting at each vertex must evenly divide into 360 degrees.
Now let's examine the properties of the octagon and isosceles triangle:
Octagon: The sum of the interior angles of an octagon is 1080 degrees ( (n-2) * 180, where n is the number of sides). Dividing 360 degrees (the sum of angles at a vertex of the tessellation) by the interior angle of the octagon (1080 degrees / 8 = 135 degrees) gives us 2.67.
Isosceles Triangle: The sum of the interior angles of an isosceles triangle is always 180 degrees (n-2) * 180, where n is the number of sides). Dividing 360 degrees (the sum of angles at a vertex of the tessellation) by twice the angle of the isosceles triangle (360 degrees / 2 * 180 degrees) gives us 1.
Since the number of shapes meeting at a vertex for an octagon (2.67) is not a whole number, it means that an octagon alone cannot tessellate a plane. On the other hand, for an isosceles triangle, the number (1) is a whole number, so an isosceles triangle alone can tessellate a plane.
However, when we consider combining both shapes, an octagon and an isosceles triangle, it does not change the non-whole number nature of 2.67 for the octagon. Therefore, an octagon and an isosceles triangle together cannot tessellate a plane.