Hexagon (EntityTopic, 12)
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| - | {{Shape | + | <[#ontology [kind topic] [cats 2D Regular Polytope] [alt [[freebase:0g85j]] [[wikipedia:Hexagon]]]]> |
| - | + | {{STS Shape | |
| name=Hexagon | | name=Hexagon | ||
| dim=2 | | dim=2 | ||
| - | | elements=6, 6 | + | | elements=6 [[digon]]s, 6 [[point]]s |
| genus=0 | | genus=0 | ||
| - | |||
| ssc=G6 | | ssc=G6 | ||
| + | | ssc2=G6 | ||
| pv_circle=<sup>3[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>π</sub> ≈ 0.8270 | | pv_circle=<sup>3[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>π</sub> ≈ 0.8270 | ||
| pv_square=<sup>3[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>4</sub> ≈ 0.6495 | | pv_square=<sup>3[[Sine values|S<sub>4</sub>]]</sup>⁄<sub>4</sub> ≈ 0.6495 | ||
| - | | schlaefli={6} | + | | extra={{STS Matrix| |
| - | | vfigure=[[ | + | 6 0 |
| - | + | 1 1}}{{STS Polytope | |
| - | + | | dual=''Self-dual'' | |
| + | | bowers=Hig | ||
| + | }}{{STS Uniform polytope | ||
| + | | schlaefli={6}, t{3} | ||
| + | | dynkin=x6o, x3x | ||
| + | | vfigure=[[Digon]], length √3 | ||
| + | }}}} | ||
| + | A hexagon is a 6-sided polygon. It is also the truncated [[triangle]]. It is one of the regular polygons that can tile the plain. | ||
| + | ==Coordinates== | ||
| + | The coordinates of a regular hexagon of side 2 are: | ||
| + | <blockquote>(±1, ±√3)<br>(0, ±2)</blockquote> | ||
| - | == | + | == Equations == |
| - | The hexagon of side 1 may be [[ | + | *The area of a regular hexagon with side length ''l'' is equal to six times the area of an equilateral [[triangle]] with side length ''l'', i.e.: |
| + | <blockquote><sup>3√3</sup>∕<sub>2</sub> · ''l''<sup>2</sup></blockquote> | ||
| + | *Because the diameter of a hexagon is twice its side length, the area of a hexagon with diameter ''l'' is a quarter of this, i.e.: | ||
| + | <blockquote><sup>3√3</sup>∕<sub>8</sub> · ''l''<sup>2</sup></blockquote> | ||
| + | |||
| + | == Dissection == | ||
| + | The hexagon of side 1 may be [[dissect]]ed into: | ||
*6× equilateral [[triangle]] with side 1 | *6× equilateral [[triangle]] with side 1 | ||
*3× [[rhombus]] with angles 2×{60°,120°} | *3× [[rhombus]] with angles 2×{60°,120°} | ||
| - | + | <[#polytope [id -6]]> | |
| - | + | ||
| - | + | ||
| - | + | ||
| - | + | ||
{{Dishapes}} | {{Dishapes}} | ||
| - | |||
| - | |||
Latest revision as of 15:56, 26 March 2017
A hexagon is a 6-sided polygon. It is also the truncated triangle. It is one of the regular polygons that can tile the plain.
Coordinates
The coordinates of a regular hexagon of side 2 are:
(±1, ±√3)
(0, ±2)
Equations
- The area of a regular hexagon with side length l is equal to six times the area of an equilateral triangle with side length l, i.e.:
3√3∕2 · l2
- Because the diameter of a hexagon is twice its side length, the area of a hexagon with diameter l is a quarter of this, i.e.:
3√3∕8 · l2
Dissection
The hexagon of side 1 may be dissected into:
Incidence matrix
Usage as facets
- prism: 2× 1-facets of a hexagonal prism
- 20× 1-facets of a truncated icosahedron
- 8× 1-facets of a truncated octahedron
- 4× 1-facets of a truncated tetrahedron
- 20× 1-facets of a rhodopantohedron
- 8× 1-facets of a stauropantohedron
- 1× 1-facets of a triangular cupola
- 1× 1-facets of a hexagonal cupolawedge
- 1× 1-facets of a triangular hebesphenorotunda (named bot)
- 1× 1-facets of a biunpinched cube
- 96× 2-facets of a truncated snub demitesseract
- 288× 2-facets of a truncated snub demitesseract
- 96× 2-facets of a truncated snub demitesseract
- 1× 2-facets of a (dual of bilunabirotunda pseudopyramid)
- 40× 2-facets of a castellated rhodopantohedral prism (named 3 end)
- 4× 2-facets of a bitrigonal diminished pyrocantichoron
- 1× 2-facets of a triangular hebesphenorotunda pseudopyramid (named bot)
- 6× 2-facets of a (dual of triangular hebesphenorotundaeic rhombochoron)
- 2× 2-facets of a triangular hebesphenorotundaeic rhombochoron (named pppppp)
- 6× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 12× 2-facets of a (dual of tetraaugmented triangular hebesphenorotundaeic rhombochoron)
- 2× 2-facets of a tetraaugmented triangular hebesphenorotundaeic rhombochoron (named pppppp - bot)
- 12× 2-facets of a D4.7 (named ooyygg)
- 8× 2-facets of a D4.7 (named oyoyoy)
| Notable Dishapes | |
| Flat: | triangle • square • pentagon • hexagon • octagon • decagon |
| Curved: | circle |
