square orthobicupola

{{Infobox polyhedron

|image=Square orthobicupola.png

|type=Johnson
{{math|triangular orthobicupola – J{{sub|28}} – square gyrobicupola}}

|faces=8 triangles
2+8 squares

|edges=32

|vertices=16

|symmetry= $D_{4\mathrm{h}}$

|vertex_config= $8 \times (3^2 \times 4^2) + 8 \times (3 \times 4^3)$

|properties=convex

|net=square orthobicupola flat.svg

}}

In geometry, the square orthobicupola is a Johnson solid constructed by two square cupolas base-to-base.

Construction

The square orthobicupola is started by attaching two square cupolae onto their bases.{{r|uehara}} The resulting polyhedron consisted of eight equilateral triangles and ten squares, having eighteen faces in total, as well as thirty-two edges and sixteen vertices. A convex polyhedron in which the faces are all regular polygons is a Johnson solid, and the square orthobicupola is one of them, enumerated as twenty-eighth Johnson solid $J_{28}$ .{{r|berman}} This construction is similar to the next one, the square gyrobicupola, which is twisted one of the cupolae around 45°.{{r|uehara}}

Properties

The square orthobicupola has surface area $A$ of a total sum of its area's faces, eight equilateral triangles and two squares. Its volume $V$ is twice that of the square cupola's volume. With the edge length $a$ , they are:{{r|berman}}

$\begin{align}
A &= \left(2 \cdot \sqrt{3} + 10\right)a^2 \approx 13.464a^2, \\
V &= \left(2+\frac{4\sqrt{2}}{3}\right)a^3 \approx 3.886a^3.
\end{align}$

The square orthobicupola has an axis of symmetry (a line passing through the center of two cupolas at their top) that rotates around one-, two-, and third-fourth of a full turn, and is reflected over the plane so the appearance remains symmetrical. The solid is also symmetrical by reflection over three mutually orthogonal planes.{{r|kovic}}

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

| last = Kovič | first = Junji

| title = Centrally symmetric convex polyhedra with regular polygonal faces

| journal = Mathematical Communications

| volume = 429 | issue = 18 | year = 2013 | pages = 429–440

| url = https://hrcak.srce.hr/file/163337

}}

{{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

}}

External links

{{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| urlname = SquareOrthobicupola | title = Square orthobicupola }}

Category:Johnson solids