Hexagon partitioned into six equilateral triangles. Like squares and equilateral triangles, regular hexagons fit together without any gaps to "tile the plane" (three hexagons meeting at every vertex), and so are useful for constructing tessellations. The cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral. The maximal diameter (which corresponds to the long diagonal of the hexagon), "D", is twice the maximal radius
Sum of angles of a triangle The relation between angular defect and the triangle's area was first proven by Johann Heinrich Lambert. One can easily see how hyperbolic geometry breaks Playfair's axiom, Proclus' axiom (the parallelism, defined as non-intersection, is intransitive in an hyperbolic plane), the equidistance postulate (the points on one side of, and equidistant from, a given line do not form a line), and Pythagoras' theorem. A circle cannot have arbitrarily small curvature, so the three points property also fails. The sum of the angles can be arbitrarily small (but positive). For an ideal triangle, a generalization of hyperbolic triangles, this sum is equal