Bicupola#Set of gyrobicupolae

{{Short description|Solid made from 2 cupolae joined base-to-base}}

In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. Here, two classes of bicupola are included because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together the result is an orthobicupola; if squares are attached to triangles it is a gyrobicupola.

Forms

=Set of orthobicupolae=

class="wikitable" !Symmetry !Picture !Description
{{math\|D{{sub\|3h}} [2,3] *223}} \|100px \|Triangular orthobicupola ({{math\|J{{sub\|27}}}}): 8 triangles, 6 squares.{{r\|os\|berman}} Its dual is the trapezo-rhombic dodecahedron
{{math\|D{{sub\|4h}} [2,4] *224}} \|100px \|Square orthobicupola ({{math\|J{{sub\|28}}}}): 8 triangles, 10 squares.{{r\|berman}}
{{math\|D{{sub\|5h}} [2,5] *225}} \|100px \|Pentagonal orthobicupola ({{math\|J{{sub\|30}}}}): 10 triangles, 10 squares, 2 pentagons.{{r\|berman}}
{{math\|D{{sub\|nh}} [2,n] *22n}} \| \|{{nowrap\|{{mvar\|n}}-gonal}} orthobicupola: {{math\|2n}} triangles, {{math\|2n}} rectangles, 2 {{nowrap\|{{mvar\|n}}-gons}}

class="wikitable"

!Symmetry

!Picture

!Description

{{math|D{{sub|3h}}
[2,3]
*223}}

|100px

|Triangular orthobicupola ({{math|J{{sub|27}}}}): 8 triangles, 6 squares.{{r|os|berman}} Its dual is the trapezo-rhombic dodecahedron

{{math|D{{sub|4h}}
[2,4]
*224}}

|100px

|Square orthobicupola ({{math|J{{sub|28}}}}): 8 triangles, 10 squares.{{r|berman}}

{{math|D{{sub|5h}}
[2,5]
*225}}

|100px

|Pentagonal orthobicupola ({{math|J{{sub|30}}}}): 10 triangles, 10 squares, 2 pentagons.{{r|berman}}

{{math|D{{sub|nh}}
[2,n]
*22n}}

|{{nowrap|{{mvar|n}}-gonal}} orthobicupola: {{math|2n}} triangles, {{math|2n}} rectangles, 2 {{nowrap|{{mvar|n}}-gons}}

=Set of gyrobicupolae=

A {{mvar|n}}-gonal gyrobicupola has the same topology as a {{mvar|n}}-gonal rectified antiprism, Conway polyhedron notation, {{math|aAn}}.

class="wikitable" !Symmetry !Picture !Description
{{math\|D{{sub\|2d}} [2{{sup\|+}},4] 22}} \|100px \|Gyrobifastigium ({{math\|J{{sub\|26}}}}) or digonal gyrobicupola*: 4 triangles, 4 squares.{{cn\|date=November 2024}}
{{math\|D{{sub\|3d}} [2{{sup\|+}},6] 23}} \|100px \|Triangular gyrobicupola* or cuboctahedron: 8 triangles, 6 squares.{{r\|os\|berman}} Its dual is the rhombic dodecahedron.
{{math\|D{{sub\|4d}} [2{{sup\|+}},8] 2*4}} \|100px \|Square gyrobicupola ({{math\|J{{sub\|29}}}}): 8 triangles, 10 squares.{{r\|berman}} Its dual is the elongated tetragonal trapezohedron
{{math\|D{{sub\|5d}} [2{{sup\|+}},10] 2*5}} \|100px \|Pentagonal gyrobicupola ({{math\|J{{sub\|31}}}}): 10 triangles, 10 squares, 2 pentagons.{{r\|berman}} Its dual is the elongated pentagonal trapezohedron
{{math\|D{{sub\|nd}} [2{{sup\|+}},2n] 2*n}} \| \|{{nowrap\|{{mvar\|n}}-gonal}} gyrobicupola: {{math\|2n}} triangles, {{math\|2n}} rectangles, 2 {{nowrap\|{{mvar\|n}}-gons}}.

class="wikitable"

!Symmetry

!Picture

!Description

{{math|D{{sub|2d}}
[2{{sup|+}},4]
2*2}}

|100px

|Gyrobifastigium ({{math|J{{sub|26}}}}) or digonal gyrobicupola: 4 triangles, 4 squares.{{cn|date=November 2024}}

{{math|D{{sub|3d}}
[2{{sup|+}},6]
2*3}}

|100px

|Triangular gyrobicupola or cuboctahedron: 8 triangles, 6 squares.{{r|os|berman}} Its dual is the rhombic dodecahedron.

{{math|D{{sub|4d}}
[2{{sup|+}},8]
2*4}}

|100px

|Square gyrobicupola ({{math|J{{sub|29}}}}): 8 triangles, 10 squares.{{r|berman}} Its dual is the elongated tetragonal trapezohedron

{{math|D{{sub|5d}}
[2{{sup|+}},10]
2*5}}

|100px

|Pentagonal gyrobicupola ({{math|J{{sub|31}}}}): 10 triangles, 10 squares, 2 pentagons.{{r|berman}} Its dual is the elongated pentagonal trapezohedron

{{math|D{{sub|nd}}
[2{{sup|+}},2n]
2*n}}

|{{nowrap|{{mvar|n}}-gonal}} gyrobicupola: {{math|2n}} triangles, {{math|2n}} rectangles, 2 {{nowrap|{{mvar|n}}-gons}}.

References

{{reflist|refs=

{{cite journal

| last = Berman | first = M.

| doi = 10.1016/0016-0032(71)90071-8

| journal = Journal of the Franklin Institute

| mr = 290245

| pages = 329–352

| title = Regular-faced convex polyhedra

| volume = 291

| year = 1971| issue = 5

}}

{{cite book

| last1 = Ogievetsky | first1 = O.

| last2 = Shlosman | first2 = S.

| editor-last1 = Novikov | editor-first1 = S.

| editor-last2 = Krichever | editor-first2 = I.

| editor-last3 = Ogievetsky | editor-first3 = O.

| editor-last4 = Shlosman | editor-first4 = S.

| year = 2021

| title = Integrability, Quantization, and Geometry: II. Quantum Theories and Algebraic Geometry

| contribution = Platonic compounds and cylinders

| url = https://books.google.com/books?id=UsspEAAAQBAJ&pg=PA477

| page = 477

| publisher = American Mathematical Society

| isbn = 978-1-4704-5592-7

}}

Category:Polyhedra