Deltoidal icositetrahedron
The deltoidal icositetrahedron (also deltoidal icositetrahedron is called), a convex Ikositetraeder, so a polyhedron with side faces 24, in which said surfaces mutually congruent deltoids are. It is one of the Catalan bodies . It is dual to the diamond cuboctahedron and has 26 corners and 48 edges.
In crystallography and mineralogy , the deltoidal icositetrahedron is often (shortened) referred to only as an icositetrahedron , and also as a trapezoid or leucitohedron (it is the typical crystal form of leucite ).
Emergence
- If square and triangular pyramids with the flank length and are placed on the 14 boundary surfaces of a cuboctahedron , a general deltoidal icosity tetrahedron is created if and are. The inscribed cuboctahedron has the edge length (i.e. a diagonal of the dragon's square , see below).
- By connecting the centers of four edges that meet in every corner of the room with the rhombic cuboctahedron , a trapezium is created , the circumference of which is also the inscribed circle of the deltoid, the boundary surface of the deltoidal icosity tetrahedron. With this special type, all face angles (≈ 138 ° 7 '5 ") are of the same size and there is a uniform edge spherical radius .
- Let be the edge length of the diamond cuboctahedron, then the resulting side lengths of the deltoid are given by
- The side lengths of the deltoid are therefore in the following relationship:
- This special (regular) deltoidal icosity tetrahedron is the circumscribed body of three regular octagons (with edge length ) standing perpendicular to each other , which intersect at their corners.
- Furthermore, the deltoidalikositetrahedron can be seen as a triple-cut "inflated" cube , which is topologically equivalent with its 24 square boundary surfaces .
Related polyhedra
Dual body: rhombic cuboctahedron
Inscribed octahedron
Inscribed cuboctahedron
Formulas for the regular deltoidal icositetrahedron
For the deltoid
Sizes of the dragon square | |
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Aspect ratio | |
Area | |
Inscribed radius | |
1st diagonal | |
2nd diagonal | |
Pointed angle (3) ≈ 81 ° 34 ′ 44 ″ |
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Obtuse angle (1) ≈ 115 ° 15 ′ 47 ″ |
For the polyhedron
Sizes of a regular Deltoidikositetrahedron with edge length a or b | |
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Volume ≈ 6.9a 3 ≈ 14.91b 3 |
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Surface area ≈ 18.36a 2 ≈ 30.69b 2 |
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Inc sphere radius | |
Edge ball radius | |
Face angle ≈ 138 ° 7 ′ 5 ″ |
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3D edge angle = 135 ° |
Occurrence
In nature z. B. leucite , analcime and spessartine, preferably in the form of deltoidalikositetrahedra. Deltoidalikositetrahedra are also found in crystal form in other minerals of the garnet group or in fluorite . The deltoidalikositetrahedron, that is the form {hll} (with h> l), is either a special form of the crystal class m 3 m, a limiting shape of the pentagonicositetrahedron in the crystal class 432 or a limiting shape of the disdodecahedron in the crystal class m 3 .
Remarks
- ↑ With a the longer of the two sides was called.
Web links
- Eric W. Weisstein : Deltoidalikositetrahedron . In: MathWorld (English).