Ingeometry, ahexagon (fromGreekἕξ,hex, meaning "six", andγωνία,gonía, meaning "corner, angle") is a six-sidedpolygon.[1] The total of the internal angles of anysimple (non-self-intersecting) hexagon is 720°.
A regular hexagon is defined as a hexagon that is bothequilateral andequiangular. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of itsinternal angle is equal to 120°. TheSchläfli symbol denotes this polygon as.[2] However, the regular hexagon can also be considered as thecutting off the vertices of anequilateral triangle, which can also be denoted as.
The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that atriangle with a vertex at the center of the regular hexagon and sharing one side with the hexagon isequilateral, and that the regular hexagon can be partitioned into six equilateral triangles.
The maximaldiameter (which corresponds to the longdiagonal of the hexagon),D, is twice the maximal radius orcircumradius,R, which equals the side length,t. The minimal diameter or the diameter of theinscribed circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base),d, is twice the minimal radius orinradius,r. The maxima and minima are related by the same factor:
The area of a regular hexagon
For any regularpolygon, the area can also be expressed in terms of theapothema and the perimeterp. For the regular hexagon these are given bya =r, andp, so
If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, thenPE + PF = PA + PB + PC + PD.
It follows from the ratio ofcircumradius toinradius that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a longdiagonal of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.
For an arbitrary point in the plane of a regular hexagon with circumradius, whose distances to the centroid of the regular hexagon and its six vertices are and respectively, we have[3]
If are the distances from the vertices of a regular hexagon to any point on its circumcircle, then[3]
A step-by-step animation of the construction of a regular hexagon usingcompass and straightedge, given byEuclid'sElements, Book IV, Proposition 15: this is possible as 6 2 × 3, a product of a power of two and distinctFermat primes.
When the side lengthAB is given, drawing a circular arc from point A and point B gives theintersection M, the center of thecircumscribed circle. Transfer theline segmentAB four times on the circumscribed circle and connect the corner points.
The six lines ofreflection of a regular hexagon, with Dih6 orr12 symmetry, order 12.The dihedral symmetries are divided depending on whether they pass through vertices (d for diagonal) or edges (p for perpendiculars) Cyclic symmetries in the middle column are labeled asg for their central gyration orders. Full symmetry of the regular form isr12 and no symmetry is labeleda1.
A regular hexagon has sixrotational symmetries (rotational symmetry of order six) and sixreflection symmetries (six lines of symmetry), making up thedihedral group D6.[4] There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4cyclic: (Z6, Z3, Z2, Z1) and the trivial (e)
These symmetries express nine distinct symmetries of a regular hexagon.John Conway labels these by a letter and group order.[5]r12 is full symmetry, anda1 is no symmetry.p6, anisogonal hexagon constructed by three mirrors can alternate long and short edges, andd6, anisotoxal hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms areduals of each other and have half the symmetry order of the regular hexagon. Thei4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as anelongatedrhombus, whiled2 andp2 can be seen as horizontally and vertically elongatedkites.g2 hexagons, with opposite sides parallel are also called hexagonalparallelogons.
Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only theg6 subgroup has no degrees of freedom but can be seen asdirected edges.
The 6 roots of thesimple Lie groupA2, represented by aDynkin diagram, are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
The 12 roots of theExceptional Lie groupG2, represented by aDynkin diagram are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
Likesquares andequilateral triangles, regular hexagons fit together without any gaps totile the plane (three hexagons meeting at every vertex), and so are useful for constructingtessellations.[6] The cells of abeehivehoneycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. TheVoronoi diagram of a regular triangular lattice is the honeycomb tessellation of hexagons.
Coxeter states that everyzonogon (a 2m-gon whose opposite sides are parallel and of equal length) can be dissected into1⁄2m(m − 1) parallelograms.[7] In particular this is true forregular polygons with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on aPetrie polygon projection of acube, with 3 of 6 square faces. Otherparallelogons and projective directions of the cube are dissected withinrectangular cuboids.
Dissection of hexagons into three rhombs and parallelograms
A regular hexagon hasSchläfli symbol {6}. A regular hexagon is a part of the regularhexagonal tiling, {6,3}, with three hexagonal faces around each vertex.
A regular hexagon can also be created as atruncatedequilateral triangle, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.
From bees'honeycombs to theGiant's Causeway, hexagonal patterns are prevalent in nature due to their efficiency. In ahexagonal grid each line is as short as it can possibly be if a large area is to be filled with the fewest hexagons. This means that honeycombs require lesswax to construct and gain much strength undercompression.
Irregular hexagons with parallel opposite edges are calledparallelogons and can also tile the plane by translation. In three dimensions,hexagonal prisms with parallel opposite faces are calledparallelohedrons and these can tessellate 3-space by translation.
In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies theConway criterion will tile the plane.
Pascal's theorem (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in anyconic section, and pairs of oppositesides are extended until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
TheLemoine hexagon is acyclic 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 itssymmedian point.
If the successive sides of a cyclic hexagon area,b,c,d,e,f, then the three main diagonals intersect in a single point if and only iface =bdf.[8]
If, for each side of a cyclic hexagon, the adjacent sides are extended to their intersection, forming a triangle exterior to the given side, then the segments connecting the circumcenters of opposite triangles areconcurrent.[9]
If a hexagon has vertices on thecircumcircle of anacute triangle at the six points (including three triangle vertices) where the extended altitudes of the triangle meet the circumcircle, then the area of the hexagon is twice the area of the triangle.[10]: p. 179
Let ABCDEF be a hexagon formed by sixtangent lines of a conic section. ThenBrianchon's theorem states that the three main diagonals AD, BE, and CF intersect at a single point.
Equilateral triangles on the sides of an arbitrary hexagon
If anequilateral triangle is constructed externally on each side of any hexagon, then the midpoints of the segments connecting thecentroids of opposite triangles form another equilateral triangle.[12]: Thm. 1
A regular skew hexagon seen as edges (black) of atriangular antiprism, symmetry D3d, [2+,6], (2*3), order 12.
Askew hexagon is askew polygon with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. Askew zig-zag hexagon has vertices alternating between two parallel planes.
Aregular skew hexagon isvertex-transitive with equal edge lengths. In three dimensions it will be a zig-zag skew hexagon and can be seen in the vertices and side edges of atriangular antiprism with the same D3d, [2+,6] symmetry, order 12.
Thecube andoctahedron (same as triangular antiprism) have regular skew hexagons as petrie polygons.
Aprincipal diagonal of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convexequilateral hexagon (one with all sides equal) with common sidea, there exists[13]: p.184, #286.3 a principal diagonald1 such that
The debate over whether hexagons should be referred to as "sexagons" has its roots in the etymology of the term. The prefix "hex-" originates from the Greek word "hex," meaning six, while "sex-" comes from the Latin "sex," also signifying six. Some linguists and mathematicians argue that since many English mathematical terms derive from Latin, the use of "sexagon" would align with this tradition. Historical discussions date back to the 19th century, when mathematicians began to standardize terminology in geometry. However, the term "hexagon" has prevailed in common usage and academic literature, solidifying its place in mathematical terminology despite the historical argument for "sexagon." The consensus remains that "hexagon" is the appropriate term, reflecting its Greek origins and established usage in mathematics. (seeNumeral_prefix#Occurrences).
^John H. Conway, Heidi Burgiel,Chaim Goodman-Strauss, (2008) The Symmetries of Things,ISBN978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)
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