pentagonal cupola
{{short description|5th Johnson solid (12 faces)}}
{{Infobox polyhedron
| image = Pentagonal cupola.png
| type = Johnson
{{math|square cupola – J{{sub|5}} – pentagonal rotunda}}
| faces = 5 triangles
5 squares
1 pentagon
1 decagon
| edges = 25
| symmetry =
| properties = convex, elementary
| net = Pentagonal Cupola.PNG
}}
Properties
The pentagonal cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon.{{r|berman}} It has the property of convexity and regular polygonal faces, from which it is classified as the fifth Johnson solid.{{r|uehara}} This cupola produces two or more regular polyhedrons by slicing it with a plane, an elementary polyhedron's example.{{r|johnson}}
The following formulae for circumradius , and height , surface area , and volume may be applied if all faces are regular with edge length :{{r|bcp}}
h &= \sqrt{\frac{5 - \sqrt{5}}{10}}a &\approx 0.526a, \\
R &= \frac{\sqrt{11+4\sqrt{5}}}{2}a &\approx 2.233a, \\
A &= \frac{20+5\sqrt{3}+\sqrt{5\left(145+62\sqrt{5}\right)}}{4}a^2 &\approx 16.580a^2, \\
V &= \frac{5+4\sqrt{5}}{6}a^3 &\approx 2.324a^3.
\end{align}
It has an axis of symmetry passing through the center of both top and base, which is symmetrical by rotating around it at one-, two-, three-, and four-fifth of a full-turn angle. It is also mirror-symmetric relative to any perpendicular plane passing through a bisector of the hexagonal base. Therefore, it has pyramidal symmetry, the cyclic group of order ten.{{r|johnson}}
Related polyhedron
The pentagonal cupola can be applied to construct a polyhedron. A construction that involves the attachment of its base to another polyhedron is known as augmentation; attaching it to prisms or antiprisms is known as elongation or gyroelongation.{{r|demey|slobodan}} Some of the Johnson solids with such constructions are: elongated pentagonal cupola , gyroelongated pentagonal cupola , pentagonal orthobicupola , pentagonal gyrobicupola , pentagonal orthocupolarotunda , pentagonal gyrocupolarotunda , elongated pentagonal orthobicupola , elongated pentagonal gyrobicupola , elongated pentagonal orthocupolarotunda , gyroelongated pentagonal bicupola , gyroelongated pentagonal cupolarotunda , augmented truncated dodecahedron , parabiaugmented truncated dodecahedron , metabiaugmented truncated dodecahedron , triaugmented truncated dodecahedron , gyrate rhombicosidodecahedron , parabigyrate rhombicosidodecahedron , metabigyrate rhombicosidodecahedron , and trigyrate rhombicosidodecahedron . Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron , paragyrate diminished rhombicosidodecahedron , metagyrate diminished rhombicosidodecahedron , bigyrate diminished rhombicosidodecahedron , parabidiminished rhombicosidodecahedron , metabidiminished rhombicosidodecahedron , gyrate bidiminished rhombicosidodecahedron , and tridiminished rhombicosidodecahedron .{{r|berman}}
References
{{reflist|refs=
| last = Berman | first = Martin
| year = 1971
| title = Regular-faced convex polyhedra
| journal = Journal of the Franklin Institute
| volume = 291
| issue = 5
| pages = 329–352
| doi = 10.1016/0016-0032(71)90071-8
| mr = 290245
}}
| last1 = Braileanu1 | first1 = Patricia I.
| last2 = Cananaul | first2 = Sorin
| last3 = Pasci | first3 = Nicoleta E.
| title = Geometric pattern infill influence on pentagonal cupola mechanical behavior subject to static external loads
| journal = Journal of Research and Innovation for Sustainable Society
| volume = 4 | issue = 2 | year = 2022 | pages = 5–15
| doi = 10.33727/JRISS.2022.2.1:5-15
| issn = 2668-0416
| publisher = Thoth Publishing House
| doi-broken-date = 16 December 2024
| url = https://journals.indexcopernicus.com/search/article?articleId=3753105
| doi-access = free
}}
| last1 = Demey | first1 = Lorenz
| last2 = Smessaert | first2 = Hans
| title = Logical and Geometrical Distance in Polyhedral Aristotelian Diagrams in Knowledge Representation
| journal = Symmetry
| date = 2017
| volume = 9
| issue = 10
| page = 204
| doi = 10.3390/sym9100204
| doi-access = free
| bibcode = 2017Symm....9..204D
}}
| last = Johnson | first = Norman W. | author-link = Norman W. Johnson
| year = 1966
| title = Convex polyhedra with regular faces
| journal = Canadian Journal of Mathematics
| volume = 18
| pages = 169–200
| doi = 10.4153/cjm-1966-021-8
| mr = 0185507
| s2cid = 122006114
| zbl = 0132.14603 | doi-access = free
}}
| last1 = Slobodan | first1 = Mišić
| last2 = Obradović | first2 = Marija
| last3 = Ðukanović | first3 = Gordana
| title = Composite Concave Cupolae as Geometric and Architectural Forms
| year = 2015
| journal = Journal for Geometry and Graphics
| volume = 19
| issue = 1
| pages = 79–91
| url = https://www.heldermann-verlag.de/jgg/jgg19/j19h1misi.pdf
}}
| last = Uehara | first = Ryuhei
| year = 2020
| title = Introduction to Computational Origami: The World of New Computational Geometry
| url = https://books.google.com/books?id=51juDwAAQBAJ&pg=PA62
| page = 62
| publisher = Springer
| isbn = 978-981-15-4470-5
| doi = 10.1007/978-981-15-4470-5
| s2cid = 220150682
}}
}}
External links
- {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname =PentagonalCupola| title =Pentagonal cupola}}
{{Johnson solids navigator}}