In geometry, the elongated triangular cupola is a polyhedron constructed from a hexagonal prism by attaching a triangular cupola. It is an example of a Johnson solid.
| Elongated triangular cupola | |
|---|---|
 | |
| Type | Johnson J17 – J18 – J19 |
| Faces | 4 triangles 9 squares 1 hexagon |
| Edges | 27 |
| Vertices | 15 |
| Vertex configuration | 6(42.6) 3(3.4.3.4) 6(3.43) |
| Symmetry group | C3v |
| Dual polyhedron | - |
| Properties | convex |
| Net | |
 | |
Construction
editThe elongated triangular cupola is constructed from a hexagonal prism by attaching a triangular cupola onto one of its bases, a process known as the elongation.[1] This cupola covers the hexagonal face so that the resulting polyhedron has four equilateral triangles, nine squares, and one regular hexagon.[2] A convex polyhedron in which all of the faces are regular polygons is the Johnson solid. The elongated triangular cupola is one of them, enumerated as the eighteenth Johnson solid .[3]
Properties
editThe surface area of an elongated triangular cupola is the sum of all polygonal face's area. The volume of an elongated triangular cupola can be ascertained by dissecting it into a cupola and a hexagonal prism, after which summing their volume. Given the edge length , its surface and volume can be formulated as:[2]
It has the three-dimensional same symmetry as the triangular cupola, the cyclic group of order 6. Its dihedral angle can be calculated by adding the angle of a triangular cupola and a hexagonal prism:[4]
- the dihedral angle of an elongated triangular cupola between square-to-triangle is that of a triangular cupola between those: 125.3°;
- the dihedral angle of an elongated triangular cupola between two adjacent squares is that of a hexagonal prism, the internal angle of its base 120°;
- the dihedral angle of a hexagonal prism between square-to-hexagon is 90°, that of a triangular cupola between square-to-hexagon is 54.7°, and that of a triangular cupola between triangle-to-hexagonal is an 70.5°. Therefore, the elongated triangular cupola between square-to-square and triangle-to-square, on the edge where a triangular cupola is attached to a hexagonal prism, is 90° 54.7° = 144.7° and 90° 70.5° = 166.5° respectively.
References
edit- ^ Rajwade, A. R. (2001), Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem, Texts and Readings in Mathematics, Hindustan Book Agency, p. 84–89, doi:10.1007/978-93-86279-06-4, ISBN 978-93-86279-06-4.
- ^ a b Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245329-352&rft.date=1971&rft_id=info:doi/10.1016/0016-0032(71)90071-8&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=290245#id-name=MR&rft.aulast=Berman&rft.aufirst=Martin&rfr_id=info:sid/en.wikipedia.org:Elongated triangular cupola" class="Z3988">.
- ^ Francis, Darryl (August 2013), "Johnson solids & their acronyms", Word Ways, 46 (3): 177.
- ^ Johnson, Norman W. (1966), "Convex polyhedra with regular faces", Canadian Journal of Mathematics, 18: 169–200, doi:10.4153/cjm-1966-021-8, MR 0185507, S2CID 122006114, Zbl 0132.14603169-200&rft.date=1966&rft_id=https://zbmath.org/?format=complete&q=an:0132.14603#id-name=Zbl&rft_id=https://mathscinet.ams.org/mathscinet-getitem?mr=0185507#id-name=MR&rft_id=https://api.semanticscholar.org/CorpusID:122006114#id-name=S2CID&rft_id=info:doi/10.4153/cjm-1966-021-8&rft.aulast=Johnson&rft.aufirst=Norman W.&rfr_id=info:sid/en.wikipedia.org:Elongated triangular cupola" class="Z3988">.