Regular Hexagons and Equilateral Triangles: Yes, a semiregular uniform tessellation can be created with regular hexagons and equilateral triangles.
Here's why:
Internal angle of hexagon: 120 degrees
Internal angle of equilateral triangle: 60 degrees
Since 120 + 60 = 180, three 60-degree angles (from equilateral triangles) can perfectly fill the gap between two 120-degree angles of a hexagon. This arrangement allows for a seamless and repeating pattern, satisfying the criteria for a tessellation.
Number of Shapes Needed at Each Vertex:
At each vertex of a hexagon, 3 equilateral triangles meet.
At each vertex of an equilateral triangle, 2 hexagons meet.
Therefore, a semiregular uniform tessellation can be created with regular hexagons and equilateral triangles, with the arrangement described above.
Complete question:
Determine whether a semiregular uniform tessellation can be created from the given shapes, assuming that all sides are 1 unit long. If so, determine the number of each shape needed at each vertex to create the tessellation.
Regular hexagons and equilateral triangles