elongated triangular gyrobicupola

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 $60^\backslash circ$. 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}}

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

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

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}}

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

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 $120^\backslash circ\; =\; 2\backslash pi/3$, and that between its base and square face is $\backslash pi/2\; =\; 90^\backslash circ$. The dihedral angle of a regular triangular cupola between each triangle and the hexagon is approximately $70.5^\backslash circ$, that between each square and the hexagon is $54.7^\backslash circ$, and that between square and triangle is $125.3^\backslash circ$. 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}}

$$\backslash begin\{align\}$$

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

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

\end{align}

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}}

{{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

}}

}}

• {{MathWorld2|title2=Johnson solid|urlname2=JohnsonSolid| urlname=ElongatedTriangularGyrobicupola | title=Elongated triangular gyrobicupola}}

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