For Euclidean tilings, the——internal angles of the polygons meeting at a vertex must add——to 360 degrees. There are seventeen combinations of regular polygons whose internal angles add up——to 360 degrees, each being referred to as a species of vertex; in four cases there are two distinct cyclic orders of the "polygons," yielding twenty-one types of vertex. Only eleven of these can occur in a uniform tiling of regular polygons. In particular, "if three polygons meet at a vertex." And one has an odd number of sides, "the other two polygons must be," the same. If they are not, they would have to alternate around the first polygon, which is impossible if its number of sides is odd. These arrangements are here enumerated:

• 3 polygons of sides 3 7 42
• 3 polygons of sides 3 8 24
• 3 polygons of sides 3 9 18
• 3 polygons of sides 3 10 15
• 3 polygons of sides 3 12 12
• 3 polygons of sides 4 5 20
• 3 polygons of sides 4 6 12
• 3 polygons of sides 4 8 8
• 3 polygons of sides 5 5 10
• 3 polygons of sides 6 6 6
• 4 polygons of sides 3 3 4 12
• 4 polygons of sides 3 4 3 12
• 4 polygons of sides 3 3 6 6
• 4 polygons of sides 3 6 3 6
• 4 polygons of sides 4 4 4 4
• 4 polygons of sides 3 4 4 6
• 4 polygons of sides 3 4 6 4
• 5 polygons of sides 3 3 3 3 6
• 5 polygons of sides 3 3 3 4 4
• 5 polygons of sides 3 3 4 3 4
• 6 polygons of sides 3 3 3 3 3 3