Hexagon

<h2 id="regular-hexagon">Regular hexagon</h2>
<p>A regular hexagon is defined as a hexagon that is both <a href="/facts/Equilateral_polygon/SEUzU3uS">equilateral</a> and <a href="/facts/Equiangular_polygon/Cuo92HxJ">equiangular</a>. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its <a href="/facts/Internal_angle/jd78Qkbt">internal angle</a> is equal to 120°. The <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> denotes this polygon as 
  
    
      
        {
        6
        }
      
    
    {\displaystyle \{6\}}
  
.<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> However, the regular hexagon can also be considered as the <a href="/facts/Truncation_(geometry)/DYb589Je">cutting off the vertices</a> of an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a>, which can also be denoted as 
  
    
      
        
          t
        
        {
        3
        }
      
    
    {\displaystyle \mathrm {t} \{3\}}
  
.
</p><p>A regular hexagon is <a href="/facts/Bicentric_polygon/emVmdWA2">bicentric</a>, meaning that it is both <a href="/facts/Cyclic_polygon/RafYT4am">cyclic</a> (has a circumscribed circle) and <a href="/facts/Tangential_polygon/Lhenc5Vc">tangential</a> (has an inscribed circle). The common length of the sides equals the radius of the <a href="/facts/Circumscribed_circle/RafYT4am">circumscribed circle</a> or <a href="/facts/Circumcircle/RafYT4am">circumcircle</a>, which equals 
  
    
      
        
          
            
              2
              
                3
              
            
          
        
      
    
    {\displaystyle {\tfrac {2}{\sqrt {3}}}}
  
 times the <a href="/facts/Apothem/CHbBbI8t">apothem</a> (radius of the <a href="/facts/Inscribed_figure/ltbko1Eh">inscribed circle</a>).
</p>
<h3>Measurement</h3>
<p>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 a <a href="/facts/Triangle/cRXUwsMn">triangle</a> with a vertex at the center of the regular hexagon and sharing one side with the hexagon is <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral</a>, and that the regular hexagon can be partitioned into six equilateral triangles.
</p>

<p>The maximal <a href="/facts/Diameter/6uNn4VmU">diameter</a> (which corresponds to the long <a href="/facts/Diagonal/39z9sIRb">diagonal</a> of the hexagon), <i>D</i>, is twice the maximal radius or <a href="/facts/Circumradius/RafYT4am">circumradius</a>, <i>R</i>, which equals the side length, <i>t</i>. The minimal diameter or the diameter of the <a href="/facts/Inscribed/ltbko1Eh">inscribed</a> circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), <i>d</i>, is twice the minimal radius or <a href="/facts/Inradius/x9QCn9ML">inradius</a>, <i>r</i>. The maxima and minima are related by the same factor:
</p>

1
            2
          
        
        d
        =
        r
        =
        cos
        ⁡
        (
        
          30
          
            ∘
          
        
        )
        R
        =
        
          
            
              3
            
            2
          
        
        R
        =
        
          
            
              3
            
            2
          
        
        t
      
    
    {\displaystyle {\frac {1}{2}}d=r=\cos(30^{\circ })R={\frac {\sqrt {3}}{2}}R={\frac {\sqrt {3}}{2}}t}
  
   and, similarly, 
  
    
      
        d
        =
        
          
            
              3
            
            2
          
        
        D
        .
      
    
    {\displaystyle d={\frac {\sqrt {3}}{2}}D.}

<p>The area of a regular hexagon
</p>

A
              
              
                
                =
                
                  
                    
                      3
                      
                        
                          3
                        
                      
                    
                    2
                  
                
                
                  R
                  
                    2
                  
                
                =
                3
                R
                r
                =
                2
                
                  
                    3
                  
                
                
                  r
                  
                    2
                  
                
              
            
            
              
              
                
                =
                
                  
                    
                      3
                      
                        
                          3
                        
                      
                    
                    8
                  
                
                
                  D
                  
                    2
                  
                
                =
                
                  
                    3
                    4
                  
                
                D
                d
                =
                
                  
                    
                      3
                    
                    2
                  
                
                
                  d
                  
                    2
                  
                
              
            
            
              
              
                
                ≈
                2.598
                
                  R
                  
                    2
                  
                
                ≈
                3.464
                
                  r
                  
                    2
                  
                
              
            
            
              
              
                
                ≈
                0.6495
                
                  D
                  
                    2
                  
                
                ≈
                0.866
                
                  d
                  
                    2
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A&={\frac {3{\sqrt {3}}}{2}}R^{2}=3Rr=2{\sqrt {3}}r^{2}\\[3pt]&={\frac {3{\sqrt {3}}}{8}}D^{2}={\frac {3}{4}}Dd={\frac {\sqrt {3}}{2}}d^{2}\\[3pt]&\approx 2.598R^{2}\approx 3.464r^{2}\\&\approx 0.6495D^{2}\approx 0.866d^{2}.\end{aligned}}}

<p>For any regular <a href="/facts/Polygon/yAq7RjPz">polygon</a>, the area can also be expressed in terms of the <a href="/facts/Apothem/CHbBbI8t">apothem</a> <i>a</i> and the perimeter <i>p</i>. For the regular hexagon these are given by <i>a</i> = <i>r</i>, and <i>p</i>

=
        6
        R
        =
        4
        r
        
          
            3
          
        
      
    
    {\displaystyle {}=6R=4r{\sqrt {3}}}
  
, so
</p>

A
              
              
                
                =
                
                  
                    
                      a
                      p
                    
                    2
                  
                
              
            
            
              
              
                
                =
                
                  
                    
                      r
                      ⋅
                      4
                      r
                      
                        
                          3
                        
                      
                    
                    2
                  
                
                =
                2
                
                  r
                  
                    2
                  
                
                
                  
                    3
                  
                
              
            
            
              
              
                
                ≈
                3.464
                
                  r
                  
                    2
                  
                
                .
              
            
          
        
      
    
    {\displaystyle {\begin{aligned}A&={\frac {ap}{2}}\\&={\frac {r\cdot 4r{\sqrt {3}}}{2}}=2r^{2}{\sqrt {3}}\\&\approx 3.464r^{2}.\end{aligned}}}

<p>The regular hexagon fills the fraction 
  
    
      
        
          
            
              
                3
                
                  
                    3
                  
                
              
              
                2
                π
              
            
          
        
        ≈
        0.8270
      
    
    {\displaystyle {\tfrac {3{\sqrt {3}}}{2\pi }}\approx 0.8270}
  
 of its <a href="/facts/Circumscribed_circle/RafYT4am">circumscribed circle</a>.
</p><p>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, then PE + PF = PA + PB + PC + PD.
</p><p>It follows from the ratio of <a href="/facts/Circumradius/RafYT4am">circumradius</a> to <a href="/facts/Inradius/x9QCn9ML">inradius</a> that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long <a href="/facts/Diagonal/39z9sIRb">diagonal</a> of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.
</p>
<h3>Point in plane</h3>
<p>For an arbitrary point in the plane of a regular hexagon with circumradius 
  
    
      
        R
      
    
    {\displaystyle R}
  
, whose distances to the centroid of the regular hexagon and its six vertices are 
  
    
      
        L
      
    
    {\displaystyle L}
  
 and 
  
    
      
        
          d
          
            i
          
        
      
    
    {\displaystyle d_{i}}
  
 
respectively, we have<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a>
</p>

d
          
            1
          
          
            2
          
        
        +
        
          d
          
            4
          
          
            2
          
        
        =
        
          d
          
            2
          
          
            2
          
        
        +
        
          d
          
            5
          
          
            2
          
        
        =
        
          d
          
            3
          
          
            2
          
        
        +
        
          d
          
            6
          
          
            2
          
        
        =
        2
        
          (
          
            
              R
              
                2
              
            
            +
            
              L
              
                2
              
            
          
          )
        
        ,
      
    
    {\displaystyle d_{1}^{2}+d_{4}^{2}=d_{2}^{2}+d_{5}^{2}=d_{3}^{2}+d_{6}^{2}=2\left(R^{2}+L^{2}\right),}

d
          
            1
          
          
            2
          
        
        +
        
          d
          
            3
          
          
            2
          
        
        +
        
          d
          
            5
          
          
            2
          
        
        =
        
          d
          
            2
          
          
            2
          
        
        +
        
          d
          
            4
          
          
            2
          
        
        +
        
          d
          
            6
          
          
            2
          
        
        =
        3
        
          (
          
            
              R
              
                2
              
            
            +
            
              L
              
                2
              
            
          
          )
        
        ,
      
    
    {\displaystyle d_{1}^{2}+d_{3}^{2}+d_{5}^{2}=d_{2}^{2}+d_{4}^{2}+d_{6}^{2}=3\left(R^{2}+L^{2}\right),}

d
          
            1
          
          
            4
          
        
        +
        
          d
          
            3
          
          
            4
          
        
        +
        
          d
          
            5
          
          
            4
          
        
        =
        
          d
          
            2
          
          
            4
          
        
        +
        
          d
          
            4
          
          
            4
          
        
        +
        
          d
          
            6
          
          
            4
          
        
        =
        3
        
          (
          
            
              
                (
                
                  
                    R
                    
                      2
                    
                  
                  +
                  
                    L
                    
                      2
                    
                  
                
                )
              
              
                2
              
            
            +
            2
            
              R
              
                2
              
            
            
              L
              
                2
              
            
          
          )
        
        .
      
    
    {\displaystyle d_{1}^{4}+d_{3}^{4}+d_{5}^{4}=d_{2}^{4}+d_{4}^{4}+d_{6}^{4}=3\left(\left(R^{2}+L^{2}\right)^{2}+2R^{2}L^{2}\right).}

<p>If 
  
    
      
        
          d
          
            i
          
        
      
    
    {\displaystyle d_{i}}
  
 are the distances from the vertices of a regular hexagon to any point on its circumcircle, then <a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>
</p>

(
            
              
                ∑
                
                  i
                  =
                  1
                
                
                  6
                
              
              
                d
                
                  i
                
                
                  2
                
              
            
            )
          
          
            2
          
        
        =
        4
        
          ∑
          
            i
            =
            1
          
          
            6
          
        
        
          d
          
            i
          
          
            4
          
        
        .
      
    
    {\displaystyle \left(\sum _{i=1}^{6}d_{i}^{2}\right)^{2}=4\sum _{i=1}^{6}d_{i}^{4}.}

<h3>Construction</h3>
A step-by-step animation of the construction of a regular hexagon using <a href="/facts/Compass_and_straightedge/lbaHf2m7">compass and straightedge</a>, given by <a href="/facts/Euclid/dHpVyL4R">Euclid</a>'s <i><a href="/facts/Euclid%27s_Elements/nPKvCIxs">Elements</a></i>, Book IV, Proposition 15: this is possible as 6 
  
    
      
        =
      
    
    {\displaystyle =}
  
 2 × 3, a product of a power of two and distinct <a href="/facts/Fermat_prime/SJT8fN9m">Fermat primes</a>.When the side length AB is given, drawing a circular arc from point A and point B gives the <a href="/facts/Intersection/hVW6QFWD">intersection</a> M, the center of the <a href="/facts/Circumscribed_circle/RafYT4am">circumscribed circle</a>. Transfer the <a href="/facts/Line_segment/V8E0M4qk">line segment</a> AB four times on the circumscribed circle and connect the corner points.
<h3>Symmetry</h3>

<p>A regular hexagon has six <a href="/facts/Rotational_symmetries/bpvJqcnS">rotational symmetries</a> (<i>rotational symmetry of order six</i>) and six <a href="/facts/Reflection_symmetries/eSeR8nbD">reflection symmetries</a> (<i>six lines of symmetry</i>), making up the <a href="/facts/Dihedral_group/1PUlRh4M">dihedral group</a> D6.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a>. There are 16 subgroups. There are 8 up to isomorphism: itself (D6), 2 dihedral: (D3, D2), 4 <a href="/facts/Cyclic_group/JL5zfFuL">cyclic</a>: (Z6, Z3, Z2, Z1) and the trivial (e)
</p><p>These symmetries express nine distinct symmetries of a regular hexagon. <a href="/facts/John_Horton_Conway/g3PA1zoK">John Conway</a> labels these by a letter and group order.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> r12 is full symmetry, and a1 is no symmetry. p6, an <a href="/facts/Isogonal_figure/5nC8PdBm">isogonal</a> hexagon constructed by three mirrors can alternate long and short edges, and d6, an <a href="/facts/Isotoxal_figure/p5j7GhwX">isotoxal</a> hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are <a href="/facts/Dual_polygon/mbR8QK0n">duals</a> of each other and have half the symmetry order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an <a href="/facts/Elongation_(geometry)/8SR7CyNz">elongated</a> <a href="/facts/Rhombus/T4pU50GF">rhombus</a>, while d2 and p2 can be seen as horizontally and vertically elongated <a href="/facts/Kite_(geometry)/PRcvPntl">kites</a>. g2 hexagons, with opposite sides parallel are also called hexagonal <a href="/facts/Parallelogon/cr60fx16">parallelogons</a>.
</p><p>Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the g6 subgroup has no degrees of freedom but can be seen as <a href="/facts/Directed_edge/YBdfQ0um">directed edges</a>.
</p><p>Hexagons of symmetry g2, i4, and r12, as <a href="/facts/Parallelogon/cr60fx16">parallelogons</a> can tessellate the Euclidean plane by translation. Other <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagon shapes can tile the plane</a> with different orientations.
</p>
<table><tbody><tr><th><i>p</i>6<i>m</i> (*632)</th><th><i>cmm</i> (2*22)</th><th><i>p</i>2 (2222)</th><th><i>p</i>31<i>m</i> (3*3)</th><th colspan="2"><i>pmg</i> (22*)</th><th><i>pg</i> (××)</th></tr><tr><th><a href="/facts/Hexagonal_tiling/HwG5F0YH">r12</a></th><th>i4</th><th>g2</th><th>d2</th><th>d2</th><th>p2</th><th>a1</th></tr><tr><th>Dih6</th><th>Dih2</th><th>Z2</th><th colspan="3">Dih1</th><th>Z1</th></tr></tbody></table>
<table><tbody><tr><td>A2 group roots</td><td>G2 group roots</td></tr></tbody></table>
<p>The 6 roots of the <a href="/facts/Simple_Lie_group/LkHTNYU9">simple Lie group</a> <a href="/facts/Dynkin_diagram/oBhUwRIR">A2</a>, represented by a <a href="/facts/Dynkin_diagram/oBhUwRIR">Dynkin diagram</a> , are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.
</p><p>The 12 roots of the <a href="/facts/Exceptional_Lie_group/LkHTNYU9">Exceptional Lie group</a> <a href="/facts/G2_(mathematics)/Bka4pKjQ">G2</a>, represented by a <a href="/facts/Dynkin_diagram/oBhUwRIR">Dynkin diagram</a>  are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.
</p>
<h3>Tessellations</h3>
<p>Like <a href="/facts/Square_(geometry)/JHJeMvmL">squares</a> and <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a>, regular hexagons fit together without any gaps to <i>tile the plane</i> (three hexagons meeting at every vertex), and so are useful for constructing <a href="/facts/Tessellation/7BWDywXC">tessellations</a>. The cells of a <a href="/facts/Beehive_(beekeeping)/VQMwYFHq">beehive</a> <a href="/facts/Honeycomb/oeAFHaxm">honeycomb</a> are hexagonal for this reason and because the shape makes efficient use of space and building materials. The <a href="/facts/Voronoi_diagram/AhQY5LD8">Voronoi diagram</a> of a regular triangular lattice is the honeycomb tessellation of hexagons.
</p>
<h2 id="dissection">Dissection</h2>
<table><tbody><tr><th><a href="/facts/6-cube/xN74tdD0">6-cube</a> projection</th><th colspan="2">12 rhomb dissection</th></tr><tr><td></td><td></td><td></td></tr></tbody></table>
<p><a href="/facts/Coxeter/D2l5jWGG">Coxeter</a> states that every <a href="/facts/Zonogon/MVJrlMuI">zonogon</a> (a 2<i>m</i>-gon whose opposite sides are parallel and of equal length) can be dissected into 1⁄2<i>m</i>(<i>m</i> − 1) parallelograms.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> In particular this is true for <a href="/facts/Regular_polygon/ws89Nhpt">regular polygons</a> with evenly many sides, in which case the parallelograms are all rhombi. This decomposition of a regular hexagon is based on a <a href="/facts/Petrie_polygon/uDmot1hD">Petrie polygon</a> projection of a <a href="/facts/Cube/65lUaAqN">cube</a>, with 3 of 6 square faces. Other <a href="/facts/Parallelogon/cr60fx16">parallelogons</a> and projective directions of the cube are dissected within <a href="/facts/Rectangular_cuboid/eLLKX4rh">rectangular cuboids</a>.
</p>
<table><tbody><tr><th colspan="12">Dissection of hexagons into three rhombs and parallelograms</th></tr><tr><th rowspan="3">2D</th><th>Rhombs</th><th colspan="3">Parallelograms</th></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td>Regular {6}</td><td colspan="3">Hexagonal <a href="/facts/Parallelogon/cr60fx16">parallelogons</a></td></tr><tr><th rowspan="3">3D</th><th colspan="2">Square faces</th><th colspan="2">Rectangular faces</th></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td colspan="2"><a href="/facts/Cube/65lUaAqN">Cube</a></td><td colspan="2"><a href="/facts/Rectangular_cuboid/eLLKX4rh">Rectangular cuboid</a></td></tr></tbody></table>
<h2 id="related-polygons-and-tilings">Related polygons and tilings</h2>
<p>A regular hexagon has <a href="/facts/Schl%C3%A4fli_symbol/jNQR77UU">Schläfli symbol</a> {6}. A regular hexagon is a part of the regular <a href="/facts/Hexagonal_tiling/HwG5F0YH">hexagonal tiling</a>, {6,3}, with three hexagonal faces around each vertex.
</p><p>A regular hexagon can also be created as a <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a>, with Schläfli symbol t{3}. Seen with two types (colors) of edges, this form only has D3 symmetry.
</p><p>A <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> hexagon, t{6}, is a <a href="/facts/Dodecagon/lKDvfUsm">dodecagon</a>, {12}, alternating two types (colors) of edges. An <a href="/facts/Alternation_(geometry)/H2nVs4xj">alternated</a> hexagon, h{6}, is an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a>, {3}. A regular hexagon can be <a href="/facts/Stellation/T7NY6CWv">stellated</a> with equilateral triangles on its edges, creating a <a href="/facts/Hexagram/IF4PEJbL">hexagram</a>. A regular hexagon can be dissected into six <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a> by adding a center point. This pattern repeats within the regular <a href="/facts/Triangular_tiling/Jx8f8Krh">triangular tiling</a>.
</p><p>A regular hexagon can be extended into a regular <a href="/facts/Dodecagon/lKDvfUsm">dodecagon</a> by adding alternating <a href="/facts/Square/JHJeMvmL">squares</a> and <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangles</a> around it. This pattern repeats within the <a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rhombitrihexagonal tiling</a>.
</p>
<table><tbody><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Regular{6}</th><th>Truncatedt{3} = {6}</th><th colspan="3">Hypertruncated triangles</th><th>Stellated<a href="/facts/Star_figure/I4naM8nS">Star figure</a> <a href="/facts/Hexagram/IF4PEJbL">2{3}</a></th><th>Truncatedt{6} = <a href="/facts/Dodecagon/lKDvfUsm">{12}</a></th><th>Alternatedh{6} = <a href="/facts/Equilateral_triangle/MJomsYDS">{3}</a></th></tr></tbody></table>
<table><tbody><tr><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>Crossedhexagon</th><th>A concave hexagon</th><th>A self-intersecting hexagon (<a href="/facts/Star_polygon/mDfuIe81">star polygon</a>)</th><th>ExtendedCentral {6} in {12}</th><th>A <a href="/facts/Skew_regular_polygon/ws89Nhpt">skew hexagon</a>, within <a href="/facts/Cube/65lUaAqN">cube</a></th><th>Dissected {6}</th><th>projection<a href="/facts/Octahedron/G38q92xW">octahedron</a></th><th><a href="/facts/Complete_graph/AYD6hUqI">Complete graph</a></th></tr></tbody></table>
<h3>Self-crossing hexagons</h3>
<p>There are six <a href="/facts/Star_polygon/mDfuIe81">self-crossing hexagons</a> with the <a href="/facts/Vertex_arrangement/DFgCsP0n">vertex arrangement</a> of the regular hexagon:
</p>
Self-intersecting hexagons with regular vertices<table><tbody><tr><th colspan="3">Dih2</th><th colspan="2">Dih1</th><th>Dih3</th></tr><tr><td>Figure-eight</td><td>Center-flip</td><td><a href="/facts/Unicursal_hexagram/zLJSBnuT">Unicursal</a></td><td>Fish-tail</td><td>Double-tail</td><td>Triple-tail</td></tr></tbody></table>
<h2 id="hexagonal-structures">Hexagonal structures</h2>

<p>From bees' <a href="/facts/Honeycomb/oeAFHaxm">honeycombs</a> to the <a href="/facts/Giant%27s_Causeway/QqwrI4lF">Giant's Causeway</a>, hexagonal patterns are prevalent in nature due to their efficiency. In a <a href="/facts/Hexagonal_grid/HwG5F0YH">hexagonal grid</a> 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 less <a href="/facts/Wax/MAXhjzNh">wax</a> to construct and gain much strength under <a href="/facts/Compression_(physics)/2wwcVmM0">compression</a>.
</p><p>Irregular hexagons with parallel opposite edges are called <a href="/facts/Parallelogon/cr60fx16">parallelogons</a> and can also tile the plane by translation. In three dimensions, <a href="/facts/Hexagonal_prism/oxYc0nF2">hexagonal prisms</a> with parallel opposite faces are called <a href="/facts/Parallelohedron/4B0xFmsg">parallelohedrons</a> and these can tessellate 3-space by translation.
</p>
Hexagonal prism tessellations<table><tbody><tr><th>Form</th><th><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a></th><th><a href="/facts/Hexagonal_prismatic_honeycomb/6XVrH9M5">Hexagonal prismatic honeycomb</a></th></tr><tr><th>Regular</th><td></td><td></td></tr><tr><th>Parallelogonal</th><td></td><td></td></tr></tbody></table>
<h2 id="tesselations-by-hexagons">Tesselations by hexagons</h2>
<p class="note">Main article: <a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a></p>
<p>In addition to the regular hexagon, which determines a unique tessellation of the plane, any irregular hexagon which satisfies the <a href="/facts/Conway_criterion/Rd7KcQR5">Conway criterion</a> will tile the plane.
</p>
<h2 id="hexagon-inscribed-in-a-conic-section">Hexagon inscribed in a conic section</h2>
<p><a href="/facts/Pascal%27s_theorem/61dfnDXG">Pascal's theorem</a> (also known as the "Hexagrammum Mysticum Theorem") states that if an arbitrary hexagon is inscribed in any <a href="/facts/Conic_section/teqgLptX">conic section</a>, and pairs of opposite <a href="/facts/Extended_side/wtqKRrBS">sides are extended</a> until they meet, the three intersection points will lie on a straight line, the "Pascal line" of that configuration.
</p>
<h3>Cyclic hexagon</h3>
<p>The <a href="/facts/Lemoine_hexagon/9wur3CNd">Lemoine hexagon</a> is a <a href="/facts/Cyclic_polygon/RafYT4am">cyclic</a> 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 <a href="/facts/Symmedian_point/2MWFKJSg">symmedian point</a>.
</p><p>If the successive sides of a cyclic hexagon are <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>e</i>, <i>f</i>, then the three main diagonals intersect in a single point if and only if <i>ace</i> = <i>bdf</i>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>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 are <a href="/facts/Concurrent_lines/zPT6S4w9">concurrent</a>.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>
</p><p>If a hexagon has vertices on the <a href="/facts/Circumcircle/RafYT4am">circumcircle</a> of an <a href="/facts/Acute_triangle/YrYTg0Ao">acute triangle</a> 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.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: p. 179 
</p>
<h2 id="hexagon-tangential-to-a-conic-section">Hexagon tangential to a conic section</h2>
<p>Let ABCDEF be a hexagon formed by six <a href="/facts/Tangent_line/sF0jH3EI">tangent lines</a> of a conic section. Then <a href="/facts/Brianchon%27s_theorem/pqHflumv">Brianchon's theorem</a> states that the three main diagonals AD, BE, and CF intersect at a single point.
</p><p>In a hexagon that is <a href="/facts/Tangential_polygon/Lhenc5Vc">tangential to a circle</a> and that has consecutive sides <i>a</i>, <i>b</i>, <i>c</i>, <i>d</i>, <i>e</i>, and <i>f</i>,<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>

a
        +
        c
        +
        e
        =
        b
        +
        d
        +
        f
        .
      
    
    {\displaystyle a+c+e=b+d+f.}

<h2 id="equilateral-triangles-on-the-sides-of-an-arbitrary-hexagon">Equilateral triangles on the sides of an arbitrary hexagon</h2>

<p>If an <a href="/facts/Equilateral_triangle/MJomsYDS">equilateral triangle</a> is constructed externally on each side of any hexagon, then the midpoints of the segments connecting the <a href="/facts/Centroid/H6dDbg0h">centroids</a> of opposite triangles form another equilateral triangle.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>: Thm. 1 
</p>

<h2 id="skew-hexagon">Skew hexagon</h2>

<p>A skew hexagon is a <a href="/facts/Skew_polygon/ucW7kB6o">skew polygon</a> with six vertices and edges but not existing on the same plane. The interior of such a hexagon is not generally defined. A <i>skew zig-zag hexagon</i> has vertices alternating between two parallel planes.
</p><p>A regular skew hexagon is <a href="/facts/Vertex-transitive/5nC8PdBm">vertex-transitive</a> 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 a <a href="/facts/Triangular_antiprism/G38q92xW">triangular antiprism</a> with the same D3d, [2+,6] symmetry, order 12.
</p><p>The <a href="/facts/Cube/65lUaAqN">cube</a> and <a href="/facts/Octahedron/G38q92xW">octahedron</a> (same as triangular antiprism) have regular skew hexagons as petrie polygons.
</p>
Skew hexagons on 3-fold axes<table><tbody><tr><td><a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a></td></tr></tbody></table>
<h3>Petrie polygons</h3>
<p>The regular skew hexagon is the <a href="/facts/Petrie_polygon/uDmot1hD">Petrie polygon</a> for these higher dimensional <a href="/facts/Regular_polytope/RYTnEF6Q">regular</a>, uniform and dual polyhedra and polytopes, shown in these skew <a href="/facts/Orthogonal_projection/sElXGkxD">orthogonal projections</a>:
</p>
<table><tbody><tr><th colspan="2">4D</th><th>5D</th></tr><tr><td><a href="/facts/3-3_duoprism/B1dhsgrg">3-3 duoprism</a></td><td><a href="/facts/3-3_duopyramid/B1dhsgrg">3-3 duopyramid</a></td><td><a href="/facts/5-simplex/KIfBFfhl">5-simplex</a></td></tr></tbody></table>
<h2 id="convex-equilateral-hexagon">Convex equilateral hexagon</h2>
<p>A <i>principal diagonal</i> of a hexagon is a diagonal which divides the hexagon into quadrilaterals. In any convex <a href="/facts/Equilateral_polygon/SEUzU3uS">equilateral</a> hexagon (one with all sides equal) with common side <i>a</i>, there exists<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>: p.184, #286.3  a principal diagonal <i>d</i>1 such that
</p>

d
              
                1
              
            
            a
          
        
        ≤
        2
      
    
    {\displaystyle {\frac {d_{1}}{a}}\leq 2}

<p>and a principal diagonal <i>d</i>2 such that
</p>

d
              
                2
              
            
            a
          
        
        >
        
          
            3
          
        
        .
      
    
    {\displaystyle {\frac {d_{2}}{a}}>{\sqrt {3}}.}

<h3>Polyhedra with hexagons</h3>
<p>There is no <a href="/facts/Platonic_solid/jraREXFD">Platonic solid</a> made of only regular hexagons, because the hexagons <a href="/facts/Tessellation/7BWDywXC">tessellate</a>, not allowing the result to "fold up". The <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a> with some hexagonal faces are the <a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a>, <a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a>, <a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a> (of <a href="/facts/Soccer_ball/SC2f6csg">soccer ball</a> and <a href="/facts/Fullerene/LPdnl8ox">fullerene</a> fame), <a href="/facts/Truncated_cuboctahedron/nd2ltxcm">truncated cuboctahedron</a> and the <a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">truncated icosidodecahedron</a>. These hexagons can be considered <a href="/facts/Truncation_(geometry)/DYb589Je">truncated</a> triangles, with <a href="/facts/Coxeter_diagram/Ik6LmXJB">Coxeter diagrams</a> of the form  and .
</p>
<table><tbody><tr><th colspan="12">Hexagons in <a href="/facts/Archimedean_solid/wYWeHDYT">Archimedean solids</a></th></tr><tr><th><a href="/facts/Tetrahedral_symmetry/hIVlf9pu">Tetrahedral</a></th><th colspan="2"><a href="/facts/Octahedral_symmetry/qLqhYxgm">Octahedral</a></th><th colspan="2"><a href="/facts/Icosahedral_symmetry/pDlwAy23">Icosahedral</a></th></tr><tr><td></td><td></td><td></td><td></td><td></td></tr><tr><td><a href="/facts/Truncated_tetrahedron/K9ItvmUK">truncated tetrahedron</a></td><td><a href="/facts/Truncated_octahedron/GA98b4TH">truncated octahedron</a></td><td><a href="/facts/Truncated_cuboctahedron/nd2ltxcm">truncated cuboctahedron</a></td><td><a href="/facts/Truncated_icosahedron/LR8ugaY9">truncated icosahedron</a></td><td><a href="/facts/Truncated_icosidodecahedron/d5v9VpFY">truncated icosidodecahedron</a></td></tr></tbody></table>
<p>There are other symmetry polyhedra with stretched or flattened hexagons, like these <a href="/facts/Goldberg_polyhedron/Hp56kpJ1">Goldberg polyhedron</a> G(2,0):
</p>
<table><tbody><tr><th colspan="12">Hexagons in Goldberg polyhedra</th></tr><tr><th><a href="/facts/Tetrahedral_symmetry/hIVlf9pu">Tetrahedral</a></th><th><a href="/facts/Octahedral_symmetry/qLqhYxgm">Octahedral</a></th><th><a href="/facts/Icosahedral_symmetry/pDlwAy23">Icosahedral</a></th></tr><tr><td><a href="/facts/Chamfered_tetrahedron/hQEGYQpl">Chamfered tetrahedron</a></td><td><a href="/facts/Chamfered_cube/hQEGYQpl">Chamfered cube</a></td><td><a href="/facts/Chamfered_dodecahedron/Gqmh0lhd">Chamfered dodecahedron</a></td></tr></tbody></table>
<p>There are also 9 <a href="/facts/Johnson_solid/8SR7CyNz">Johnson solids</a> with regular hexagons:
</p>
<table><tbody><tr><th colspan="12">Johnson solids with hexagons</th></tr><tr><td><a href="/facts/Triangular_cupola/Pyocot8L">triangular cupola</a></td><td><a href="/facts/Elongated_triangular_cupola/tFPPeIEw">elongated triangular cupola</a></td><td><a href="/facts/Gyroelongated_triangular_cupola/MTLrAfcU">gyroelongated triangular cupola</a></td></tr><tr><td><a href="/facts/Augmented_hexagonal_prism/e1C8lUA3">augmented hexagonal prism</a></td><td><a href="/facts/Parabiaugmented_hexagonal_prism/9nKmL9kn">parabiaugmented hexagonal prism</a></td><td><a href="/facts/Metabiaugmented_hexagonal_prism/fFM7FrGD">metabiaugmented hexagonal prism</a></td></tr><tr><td><a href="/facts/Triaugmented_hexagonal_prism/G2QoLwN2">triaugmented hexagonal prism</a></td><td><a href="/facts/Augmented_truncated_tetrahedron/Nxxj4aS2">augmented truncated tetrahedron</a></td><td><a href="/facts/Triangular_hebesphenorotunda/WuMbPpTS">triangular hebesphenorotunda</a></td></tr></tbody></table>
<table><tbody><tr><th colspan="12"><a href="/facts/Prismoid/lPBxnDeG">Prismoids</a> with hexagons</th></tr><tr><td><a href="/facts/Hexagonal_prism/oxYc0nF2">Hexagonal prism</a></td><td><a href="/facts/Hexagonal_antiprism/bt1W4cs6">Hexagonal antiprism</a></td><td><a href="/facts/Hexagonal_pyramid/LVvQdNGd">Hexagonal pyramid</a></td></tr></tbody></table>
<table><tbody><tr><th colspan="12">Tilings with regular hexagons</th></tr><tr><th>Regular</th><th colspan="3">1-uniform</th></tr><tr><td><a href="/facts/Hexagonal_tiling/HwG5F0YH">{6,3}</a></td><td><a href="/facts/Trihexagonal_tiling/spAKhDvd">r{6,3}</a></td><td><a href="/facts/Rhombitrihexagonal_tiling/RJPMacIY">rr{6,3}</a></td><td><a href="/facts/Truncated_trihexagonal_tiling/qMT9oGKC">tr{6,3}</a></td></tr><tr><td></td><td></td><td></td><td></td></tr><tr><td colspan="4"><a href="/facts/2-uniform_tiling/Uuxoua41">2-uniform tilings</a></td></tr><tr><td></td><td></td><td></td><td></td></tr></tbody></table>
<h2 id="hexagon-versus-sexagon">Hexagon versus Sexagon</h2>
<p>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. (see <a href="/facts/Numeral_prefix/b5sechhR">Numeral_prefix#Occurrences</a>).
</p>
<h2 id="gallery-of-natural-and-artificial-hexagons">Gallery of natural and artificial hexagons</h2>

<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/24-cell/tfoGnlAD">24-cell</a>: a <a href="/facts/Four-dimensional_space/vFQd0M7T">four-dimensional</a> figure which, like the hexagon, has <a href="/facts/Orthoplex/erk0byF6">orthoplex</a> facets, is <a href="/facts/Self-dual/8jGuyQIU">self-dual</a> and tessellates <a href="/facts/Euclidean_space/R2UbzmzM">Euclidean space</a></li>
<li><a href="/facts/Hexagonal_crystal_system/sSSok8wo">Hexagonal crystal system</a></li>
<li><a href="/facts/Hexagonal_number/7yXaNygK">Hexagonal number</a></li>
<li><a href="/facts/Hexagonal_tiling/HwG5F0YH">Hexagonal tiling</a>: a <a href="/facts/Regular_tiling/tR5CQACy">regular tiling</a> of hexagons in a plane</li>
<li><a href="/facts/Hexagram/IF4PEJbL">Hexagram</a>: six-sided star within a regular hexagon</li>
<li><a href="/facts/Unicursal_hexagram/zLJSBnuT">Unicursal hexagram</a>: single path, six-sided star, within a hexagon</li>
<li><a href="/facts/Honeycomb_conjecture/g45j09Io">Honeycomb conjecture</a></li>
<li><a href="/facts/Havannah_(board_game)/wmBNpBE3">Havannah</a>: abstract board game played on a six-sided hexagonal grid</li>
<li><a href="/facts/Central_place_theory/w8yEzQjf">Central place theory</a></li></ul>

<h2 id="external-links">External links</h2>

Look up <i>hexagon</i> in Wiktionary, the free dictionary.

<ul><li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/Hexagon.html">"Hexagon"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul>
<ul><li><a href="http://www.mathopenref.com/hexagon.html">Definition and properties of a hexagon</a> with interactive animation and <a href="http://www.mathopenref.com/consthexagon.html">construction with compass and straightedge</a>.</li>
<li><a href="https://hexnet.org/content/hexagonal-geometry">An Introduction to Hexagonal Geometry</a> on <a href="https://web.archive.org/web/19980204100717/http://www.hexnet.org/">Hexnet</a> a website devoted to hexagon mathematics.</li>
<li><a href="https://www.youtube.com/watch?v=thOifuHs6eY">Hexagons are the Bestagons</a> on <a href="/facts/YouTube_video_(identifier)/db276jst">YouTube</a> – an <a href="/facts/Animation/093f9nUg">animated</a> <a href="/facts/Internet_video/XP3Xxon6">internet video</a> about hexagons by <a href="/facts/CGP_Grey/S1dCuDK5">CGP Grey</a>.</li></ul>

<table><tbody><tr><th colspan="13"><ul><li>v</li><li>t</li><li>e</li></ul>Fundamental convex <a href="/facts/Regular_polytope/RYTnEF6Q">regular</a> and <a href="/facts/Uniform_polytope/nFcNAzLy">uniform polytopes</a> in dimensions 2–10</th></tr><tr><th><a href="/facts/Coxeter_group/bgzHFm1Q">Family</a></th><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>A</i><i>n</i></a></td><td><a href="/facts/Simple_Lie_group/LkHTNYU9"><i>B</i><i>n</i></a></td><td><i>I</i>2(p) / <a href="/facts/Simple_Lie_group/LkHTNYU9"><i>D</i><i>n</i></a></td><td><a href="/facts/E6_(mathematics)/U9YEMtWj"><i>E</i>6</a> / <a href="/facts/E7_(mathematics)/KjFez9PW"><i>E</i>7</a> / <a href="/facts/E8_(mathematics)/NEaKT71H"><i>E</i>8</a> / <a href="/facts/F4_(mathematics)/30H8CQ8v"><i>F</i>4</a> / <a href="/facts/G2_(mathematics)/Bka4pKjQ"><i>G</i>2</a></td><td><a href="/facts/H4_(mathematics)/bgzHFm1Q">Hn</a></td></tr><tr><th><a href="/facts/Regular_polygon/ws89Nhpt">Regular polygon</a></th><td><a href="/facts/Equilateral_triangle/MJomsYDS">Triangle</a></td><td><a href="/facts/Square/JHJeMvmL">Square</a></td><td><a href="/facts/Regular_polygon/ws89Nhpt">p-gon</a></td><td>Hexagon</td><td><a href="/facts/Pentagon/jeHEARHH">Pentagon</a></td></tr><tr><th><a href="/facts/Uniform_polyhedron/PHnprqRC">Uniform polyhedron</a></th><td><a href="/facts/Tetrahedron/7mkSrl6j">Tetrahedron</a></td><td><a href="/facts/Octahedron/G38q92xW">Octahedron</a> • <a href="/facts/Cube/65lUaAqN">Cube</a></td><td><a href="/facts/Tetrahedron/7mkSrl6j">Demicube</a></td><td></td><td><a href="/facts/Regular_dodecahedron/xClYqhcp">Dodecahedron</a> • <a href="/facts/Regular_icosahedron/1U93vmXV">Icosahedron</a></td></tr><tr><th><a href="/facts/Uniform_polychoron/NHxomMxC">Uniform polychoron</a></th><td><a href="/facts/5-cell/Rh6lwDcc">Pentachoron</a></td><td><a href="/facts/16-cell/X0YAfobw">16-cell</a> • <a href="/facts/Tesseract/tlqouBoU">Tesseract</a></td><td><a href="/facts/16-cell/X0YAfobw">Demitesseract</a></td><td><a href="/facts/24-cell/tfoGnlAD">24-cell</a></td><td><a href="/facts/120-cell/W4nroBjJ">120-cell</a> • <a href="/facts/600-cell/tNBtDUe9">600-cell</a></td></tr><tr><th><a href="/facts/Uniform_5-polytope/mqxsoqwY">Uniform 5-polytope</a></th><td><a href="/facts/5-simplex/KIfBFfhl">5-simplex</a></td><td><a href="/facts/5-orthoplex/vU1TvdQ1">5-orthoplex</a> • <a href="/facts/5-cube/XABdhjof">5-cube</a></td><td><a href="/facts/5-demicube/zvCeNbU2">5-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_6-polytope/pxqFTw2F">Uniform 6-polytope</a></th><td><a href="/facts/6-simplex/mInfSIeH">6-simplex</a></td><td><a href="/facts/6-orthoplex/eGV6o0ol">6-orthoplex</a> • <a href="/facts/6-cube/xN74tdD0">6-cube</a></td><td><a href="/facts/6-demicube/1oTufnFp">6-demicube</a></td><td><a href="/facts/1_22_polytope/7XXjWGQ8">122</a> • <a href="/facts/2_21_polytope/nF3nYaQk">221</a></td><td></td></tr><tr><th><a href="/facts/Uniform_7-polytope/suCJSvTN">Uniform 7-polytope</a></th><td><a href="/facts/7-simplex/6DwtPplH">7-simplex</a></td><td><a href="/facts/7-orthoplex/mIJa3rQp">7-orthoplex</a> • <a href="/facts/7-cube/4af1W7tq">7-cube</a></td><td><a href="/facts/7-demicube/fGVUTvTI">7-demicube</a></td><td><a href="/facts/1_32_polytope/um16Pd2V">132</a> • <a href="/facts/2_31_polytope/zkjkKJ6X">231</a> • <a href="/facts/3_21_polytope/AEUzwA1A">321</a></td><td></td></tr><tr><th><a href="/facts/Uniform_8-polytope/M93v3h3c">Uniform 8-polytope</a></th><td><a href="/facts/8-simplex/QUhcnmk2">8-simplex</a></td><td><a href="/facts/8-orthoplex/dXFkXtPn">8-orthoplex</a> • <a href="/facts/8-cube/vEpKH3le">8-cube</a></td><td><a href="/facts/8-demicube/sWAomIhw">8-demicube</a></td><td><a href="/facts/1_42_polytope/DTgYDddB">142</a> • <a href="/facts/2_41_polytope/R9z0wNUz">241</a> • <a href="/facts/4_21_polytope/QFGglCkk">421</a></td><td></td></tr><tr><th><a href="/facts/Uniform_9-polytope/UdFL9ns2">Uniform 9-polytope</a></th><td><a href="/facts/9-simplex/dHV7qhxN">9-simplex</a></td><td><a href="/facts/9-orthoplex/D7LjUcAh">9-orthoplex</a> • <a href="/facts/9-cube/RNgCPzoM">9-cube</a></td><td><a href="/facts/9-demicube/dkTpfqss">9-demicube</a></td><td></td><td></td></tr><tr><th><a href="/facts/Uniform_10-polytope/Qc5NglV8">Uniform 10-polytope</a></th><td><a href="/facts/10-simplex/oeft7Qjn">10-simplex</a></td><td><a href="/facts/10-orthoplex/x6ysTC9U">10-orthoplex</a> • <a href="/facts/10-cube/mmmyBfJJ">10-cube</a></td><td><a href="/facts/10-demicube/TskxfLy9">10-demicube</a></td><td></td><td></td></tr><tr><th>Uniform <i>n</i>-<a href="/facts/Polytope/a76nXrL0">polytope</a></th><td><i>n</i>-<a href="/facts/Simplex/0FCfNRN8">simplex</a></td><td><i>n</i>-<a href="/facts/Cross-polytope/erk0byF6">orthoplex</a> • <i>n</i>-<a href="/facts/Hypercube/q2mRPbIz">cube</a></td><td><i>n</i>-<a href="/facts/Demihypercube/ackHKsEa">demicube</a></td><td><a href="/facts/Uniform_1_k2_polytope/eqTstjgl">1k2</a> • <a href="/facts/Uniform_2_k1_polytope/pQv1I7AC">2k1</a> • <a href="/facts/Uniform_k_21_polytope/vv3lWMYd">k21</a></td><td><i>n</i>-<a href="/facts/Pentagonal_polytope/4uxePdKt">pentagonal polytope</a></td></tr><tr><th colspan="13">Topics: <a href="/facts/Polytope_families/Gun0RgmR">Polytope families</a> • <a href="/facts/Regular_polytope/RYTnEF6Q">Regular polytope</a> • <a href="/facts/List_of_regular_polytopes_and_compounds/Kng25Ymq">List of regular polytopes and compounds</a></th></tr></tbody></table>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Cube picture <a href="https://deimel.org/images/plain_cube.gif" target="_blank">https://deimel.org/images/plain_cube.gif</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Wenninger, Magnus J. (1974), Polyhedron Models, Cambridge University Press, p. 9, ISBN 9780521098595, archived from the original on 2016-01-02, retrieved 2015-11-06. <a href="9780521098595" target="_blank">9780521098595</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340. doi:10.26713/cma.v11i3.1420 (inactive 1 November 2024).{{cite journal}}:  CS1 maint: DOI inactive as of November 2024 (link) <a href="https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065" target="_blank">https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Meskhishvili, Mamuka (2020). "Cyclic Averages of Regular Polygons and Platonic Solids". Communications in Mathematics and Applications. 11: 335–355. arXiv:2010.12340. doi:10.26713/cma.v11i3.1420 (inactive 1 November 2024).{{cite journal}}:  CS1 maint: DOI inactive as of November 2024 (link) <a href="https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065" target="_blank">https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Johnston, Bernard L.; Richman, Fred (1997), Numbers and Symmetry: An Introduction to Algebra, CRC Press, p. 92. <a href="http://books.google.com/books?id=koUfrlgsmUcC&pg=PA92" target="_blank">http://books.google.com/books?id=koUfrlgsmUcC&pg=PA92</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, ISBN 978-1-56881-220-5 (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278) <a href="/wiki/Chaim_Goodman-Strauss" target="_blank">/wiki/Chaim_Goodman-Strauss</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Coxeter, Mathematical recreations and Essays, Thirteenth edition, p.141 <a href="/wiki/Coxeter" target="_blank">/wiki/Coxeter</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Cartensen, Jens, "About hexagons", Mathematical Spectrum 33(2) (2000–2001), 37–40. <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Dergiades, Nikolaos (2014). "Dao's theorem on six circumcenters associated with a cyclic hexagon". Forum Geometricorum. 14: 243–246. Archived from the original on 2014-12-05. Retrieved 2014-11-17. <a href="http://forumgeom.fau.edu/FG2014volume14/FG201424index.html" target="_blank">http://forumgeom.fau.edu/FG2014volume14/FG201424index.html</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Johnson, Roger A., Advanced Euclidean Geometry, Dover Publications, 2007 (orig. 1960). <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Gutierrez, Antonio, "Hexagon, Inscribed Circle, Tangent, Semiperimeter", [1] Archived 2012-05-11 at the Wayback Machine, Accessed 2012-04-17. <a href="http://gogeometry.com/problem/p343_circumscribed_hexagon_tangent_semiperimeter.htm" target="_blank">http://gogeometry.com/problem/p343_circumscribed_hexagon_tangent_semiperimeter.htm</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Dao Thanh Oai (2015). "Equilateral triangles and Kiepert perspectors in complex numbers". Forum Geometricorum. 15: 105–114. Archived from the original on 2015-07-05. Retrieved 2015-04-12. <a href="http://forumgeom.fau.edu/FG2015volume15/FG201509index.html" target="_blank">http://forumgeom.fau.edu/FG2015volume15/FG201509index.html</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Inequalities proposed in "Crux Mathematicorum", [2] Archived 2017-08-30 at the Wayback Machine. <a href="/wiki/Crux_Mathematicorum" target="_blank">/wiki/Crux_Mathematicorum</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
</ol>

Hexagon open-in-new

Hexagon