elongated triangular gyrobicupola

{{Short description|36th Johnson solid}}

{{Infobox polyhedron

| image = Elongated triangular gyrobicupola.png

| type = Johnson
{{math|elongated triangular orthobicupolaJ{{sub|36}}elongated square gyrobicupola}}

| faces = 8 triangles
12 squares

| edges = 36

| vertices = 18

| symmetry = D_{3h}

| vertex_config = \begin{align}

&6 \times (3 \times 4 \times 3 \times 4) + \\

&12 \times (3 \times 4^3)

\end{align}

| properties = convex

| net = Johnson solid 36 net.png

}}

In geometry, the elongated triangular gyrobicupola is a polyhedron constructed by attaching two regular triangular cupolas to the base of a regular hexagonal prism, with one of them rotated in 60^\circ . It is an example of Johnson solid.

Construction

The elongated triangular gyrobicupola is similarly can be constructed as the elongated triangular orthobicupola, started from a hexagonal prism by attaching two regular triangular cupolae onto its base, covering its hexagonal faces.{{r|rajwade}} This construction process is known as elongation, giving the resulting polyhedron has 8 equilateral triangles and 12 squares.{{r|berman}} The difference between those two polyhedrons is one of two triangular cupolas in the elongated triangular gyrobicupola is rotated in 60^\circ . A convex polyhedron in which all faces are regular is Johnson solid, and the elongated triangular gyrobicupola is one among them, enumerated as 36th Johnson solid J_{36} .{{r|francis}}

Properties

An elongated triangular gyrobicupola with a given edge length a has a surface area by adding the area of all regular faces:{{r|berman}}

\left(12 + 2\sqrt{3}\right)a^2 \approx 15.464a^2.

Its volume can be calculated by cutting it off into two triangular cupolae and a hexagonal prism with regular faces, and then adding their volumes up:{{r|berman}}

\left(\frac{5\sqrt{2}}{3} + \frac{3\sqrt{3}}{2}\right)a^3 \approx 4.955a^3.

Its three-dimensional symmetry groups is the prismatic symmetry, the dihedral group D_{3d} of order 12.{{clarification needed|date=March 2024|reason=Need to explicitly explain the meaning of this symmetry}} Its dihedral angle can be calculated by adding the angle of the triangular cupola and hexagonal prism. The dihedral angle of a hexagonal prism between two adjacent squares is the internal angle of a regular hexagon 120^\circ = 2\pi/3, and that between its base and square face is \pi/2 = 90^\circ . The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately 70.5^\circ , that between each square and the hexagon is 54.7^\circ , and that between square and triangle is 125.3^\circ . The dihedral angle of an elongated triangular orthobicupola between the triangle-to-square and square-to-square, on the edge where the triangular cupola and the prism is attached, is respectively:{{r|johnson}}

\begin{align}

\frac{\pi}{2} + 70.5^\circ &\approx 160.5^\circ, \\

\frac{\pi}{2} + 54.7^\circ &\approx 144.7^\circ.

\end{align}

Related polyhedra and honeycombs

The elongated triangular gyrobicupola forms space-filling honeycombs with tetrahedra and square pyramids.{{Cite web|url=http://woodenpolyhedra.web.fc2.com/J36.html|title = J36 honeycomb}}

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 = Francis | first = Darryl

| title = Johnson solids & their acronyms

| journal = Word Ways

| date = August 2013

| volume = 46 | issue = 3 | page = 177

| url = https://go.gale.com/ps/i.do?id=GALE%7CA340298118

}}

{{cite journal

| last = Johnson | first = Norman W. | authorlink = 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 book

| last = Rajwade | first = A. R.

| title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem

| series = Texts and Readings in Mathematics

| year = 2001

| url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84

| page = 84–89

| publisher = Hindustan Book Agency

| isbn = 978-93-86279-06-4

| doi = 10.1007/978-93-86279-06-4

}}

}}