![]() ![]() ![]() ![]() If the successive sides of a cyclic hexagon are a, b, c, d, e, f, then the three main diagonals intersect in a single point if and only if ace = bdf. The Lemoine hexagon is a cyclic hexagon (one inscribed in a circle) with vertices given by the six intersections of the edges of a triangle and the three lines that are parallel to the edges that pass through its symmedian point. Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any conic section, and pairs of opposite sides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration. In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the Conway criterion will tile the plane. In three dimensions, hexagonal prisms with parallel opposite faces are called parallelohedrons and these can tessellate 3-space by translation. Irregular hexagons with parallel opposite edges are called parallelogons and can also tile the plane by translation. This means that honeycombs require less wax to construct and gain lots of strength under compression. ![]() In a hexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest number of hexagons. This pattern repeats within the rhombitrihexagonal tiling.Ī self-intersecting hexagon ( star polygon)įrom bees' honeycombs to the Giant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. This pattern repeats within the regular triangular tiling.Ī regular hexagon can be extended into a regular dodecagon by adding alternating squares and equilateral triangles around it. A regular hexagon can be dissected into 6 equilateral triangles by adding a center point. A regular hexagon can be stellated with equilateral triangles on its edges, creating a hexagram. The common length of the sides equals the radius of the circumscribed circle, which equals 2 3. It is bicentric, meaning that it is both cyclic (has a circumscribed circle) and tangential (has an inscribed circle). A regular hexagon is defined as a hexagon that is both equilateral and equiangular. ![]()
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