| Revision as of 15:17, 7 April 2012 editDouble sharp (talk | contribs)Autopatrolled, Extended confirmed users, Page movers, File movers, Pending changes reviewers102,062 editsm Double sharp moved page Facetting to Faceting over redirect: it's more commonly spelled this way← Previous edit | Revision as of 15:18, 7 April 2012 edit undoDouble sharp (talk | contribs)Autopatrolled, Extended confirmed users, Page movers, File movers, Pending changes reviewers102,062 editsm change spellingsNext edit → | ||
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| {{otheruses}} | {{otheruses|Facetting (disambiguation)}} | ||
| <div style="float:right; margin:0em 0em 1em 1em; border:1px solid silver; padding:0.5em; text-align:center;"> | <div style="float:right; margin:0em 0em 1em 1em; border:1px solid silver; padding:0.5em; text-align:center;"> | ||
| ]<br> | ]<br> | ||
| Stella octangula as a |
Stella octangula as a faceting of the cube | ||
| </div> | </div> | ||
| In ], ''' |
In ], '''faceting''' (also spelled '''facetting''') is the process of removing parts of a ], ] or ], without creating any new ]. | ||
| Faceting is the reciprocal or ] process to ]. For every stellation of some ], there exists a dual faceting of the ]. | |||
| For example the ''stellated dodecahedron'' and great icosahedron'' are two |
For example the ''stellated dodecahedron'' and great icosahedron'' are two facetings of the icosahedron: | ||
| {| class =wikitable | {| class =wikitable | ||
| !] | !] | ||
| Line 23: | Line 23: | ||
| ==History== | ==History== | ||
| Faceting has not been studied as extensively as ]. | |||
| * In 1619, ] described a ] ] of two ] which fits inside a cube, and which he called the ''']'''. This seems to be the first known example of |
* In 1619, ] described a ] ] of two ] which fits inside a cube, and which he called the ''']'''. This seems to be the first known example of faceting. | ||
| * In 1858, ] derived the regular ] (]) by |
* In 1858, ] derived the regular ] (]) by faceting the ] ] ] and ]. | ||
| * In 1974, Bridge enumerated the more straightforward |
* In 1974, Bridge enumerated the more straightforward facetings of the ] polyhedra, including those of the ]. | ||
| * In 2006, Inchbald described the basic theory of |
* In 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For some vertices, the diagram shows all the possible edges and facets (new faces) which may be used to form facetings of the original hull. It is ] to the ]'s stellation diagram, which shows all the possible edges and vertices for some face plane of the original core. | ||
| ==References== | ==References== | ||
Revision as of 15:18, 7 April 2012
For other uses, see Facetting (disambiguation).
Stella octangula as a faceting of the cube
In geometry, faceting (also spelled facetting) is the process of removing parts of a polygon, polyhedron or polytope, without creating any new vertices.
Faceting is the reciprocal or dual process to stellation. For every stellation of some convex polytope, there exists a dual faceting of the dual polytope.
For example the stellated dodecahedron and great icosahedron are two facetings of the icosahedron:
| icosahedron | small stellated dodecahedron | great icosahedron |
|---|---|---|
|
|
|
History
Faceting has not been studied as extensively as stellation.
- In 1619, Kepler described a regular compound of two tetrahedra which fits inside a cube, and which he called the Stella octangula. This seems to be the first known example of faceting.
- In 1858, Bertrand derived the regular star polyhedra (Kepler-Poinsot polyhedra) by faceting the regular convex icosahedron and dodecahedron.
- In 1974, Bridge enumerated the more straightforward facetings of the regular polyhedra, including those of the dodecahedron.
- In 2006, Inchbald described the basic theory of faceting diagrams for polyhedra. For some vertices, the diagram shows all the possible edges and facets (new faces) which may be used to form facetings of the original hull. It is dual to the dual polyhedron's stellation diagram, which shows all the possible edges and vertices for some face plane of the original core.
References
- Bertrand, J. Note sur la théorie des polyèdres réguliers, Comptes rendus des séances de l'Académie des Sciences, 46 (1858), pp. 79-82.
- Bridge, N.J. Facetting the dodecahedron, Acta crystallographica A30 (1974), pp. 548-552.
- Inchbald, G. Facetting diagrams, The mathematical gazette, 90 (2006), pp. 253-261.
External links
- Weisstein, Eric W. "Faceting". MathWorld.
- Olshevsky, George. "Faceting". Glossary for Hyperspace. Archived from the original on 4 February 2007.
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