This is fulfilled when the opposite sides of the hexagon are parallel, and (as a still more special case) when the hexagon is regular.
The town cross is a fine structure standing upon a huge hexagon, surmounted by a stone pillar 12 ft.
Desargues has a special claim to fame on account of his beautiful theorem on the involution of a quadrangle inscribed in a conic. Pascal discovered a striking property of a hexagon inscribed in a conic (the hexagrammum mysticum); from this theorem Pascal is said to have deduced over 400 corollaries, including most of the results obtained by earlier geometers.
Hence if we take two nets of wire with hexagonal meshes, and place one on the other so that the point of concourse of three hexagons of one net coincides with the middle of a hexagon of the other, and if we then, after dipping them in Plateau's liquid, place them horizontally, and gently raise the upper one, we shall develop a system of plane laminae arranged as the walls and floors of the cells are arranged in a honeycomb.
The ringed structure of benzene, C 6 H 61 was first suggested in 1865 by August Kekule, who represented the molecule by six CH groups placed at the six angles of a regular hexagon, the sides of which denoted the valencies saturated by adjacent carbon atoms, the fourth valencies of each carbon atom being represented as saturated along alternate sides.
Consider, for example, a frame whose sides form the six sides of a hexagon ABCDEF and the three diagonals AD, BE, CF; and suppose that it is required to find the stress in CF due to a given system of extraneous forces in equilibrium, acting on the joints.
He numbers the carbon atoms placed at the corners of a hexagon from i to 6, and each side in the same order, so that the carbon atoms i and 2 are connected by the side 1, atoms 2 and 3 by the side 2, and so on.
The octahedral formula discussed by Julius Thomsen (Ber., 1886, 19, p. 2 944) consists of the six carbon atoms placed at the corners of a regular octahedron, and connected together by the full lines as shown in (I); a plane projection gives a hexagon with diagonals (II).
As we see it to-day, it is an open space of ioo acres, set on a hill with a wide prospect east and south and west, in shape an irregular hexagon, enclosed in a circuit of a mile and a half by the massive ruins of a city wall which still stands here and there some 20 ft.
In this way he established the famous theorem that the intersections of the three pairs of opposite sides of a hexagon inscribed in a conic are collinear.