pentagonal cupola

{{short description|5th Johnson solid (12 faces)}}

{{Infobox polyhedron

| image = Pentagonal cupola.png

| type = Johnson
{{math|square cupolaJ{{sub|5}}pentagonal rotunda}}

| faces = 5 triangles
5 squares
1 pentagon
1 decagon

| edges = 25

| symmetry = C_{\mathrm{v}}

| 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 R , and height h , surface area A , and volume V may be applied if all faces are regular with edge length a :{{r|bcp}}

\begin{align}

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}

File:Cupula pentagonal 3D.stl

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 C_{5\mathrm{v}} 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 J_{20} , gyroelongated pentagonal cupola J_{24} , pentagonal orthobicupola J_{30} , pentagonal gyrobicupola J_{31} , pentagonal orthocupolarotunda J_{32} , pentagonal gyrocupolarotunda J_{33} , elongated pentagonal orthobicupola J_{38} , elongated pentagonal gyrobicupola J_{39} , elongated pentagonal orthocupolarotunda J_{40} , gyroelongated pentagonal bicupola J_{46} , gyroelongated pentagonal cupolarotunda J_{47} , augmented truncated dodecahedron J_{68} , parabiaugmented truncated dodecahedron J_{69} , metabiaugmented truncated dodecahedron J_{70} , triaugmented truncated dodecahedron J_{71} , gyrate rhombicosidodecahedron J_{72} , parabigyrate rhombicosidodecahedron J_{73} , metabigyrate rhombicosidodecahedron J_{74} , and trigyrate rhombicosidodecahedron J_{75} . Relatedly, a construction from polyhedra by removing one or more pentagonal cupolas is known as diminishment: diminished rhombicosidodecahedron J_{76} , paragyrate diminished rhombicosidodecahedron J_{77} , metagyrate diminished rhombicosidodecahedron J_{78} , bigyrate diminished rhombicosidodecahedron J_{79} , parabidiminished rhombicosidodecahedron J_{80} , metabidiminished rhombicosidodecahedron J_{81} , gyrate bidiminished rhombicosidodecahedron J_{82} , and tridiminished rhombicosidodecahedron J_{83} .{{r|berman}}

References

{{reflist|refs=

{{cite journal

| 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

}}

{{cite journal

| 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

}}

{{cite journal

| 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

}}

{{cite journal

| 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

}}

{{cite journal

| 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

}}

{{cite book

| 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

}}

}}