Augmented truncated tetrahedron

{{Infobox polyhedron

| image = Augmented truncated tetrahedron.png

| type = Johnson
{{math|augmented tridiminished icosahedron – J{{sub|65}} – augmented truncated cube}}

| faces = 8 triangles
3 squares
3 hexagons

| edges = 27

| vertices = 15

| symmetry = {{math|Point groups in three dimensions}}

| vertex_config = {{math|2x3(3.6{{sup|2}})
3(3.4.3.4)
6(3.4.3.6)}}

| properties = convex

| net = Johnson solid 65 net.png

}}

File:J65 augmented truncated tetrahedron.stl

In geometry, the augmented truncated tetrahedron is a polyhedron constructed by attaching a triangular cupola onto a truncated tetrahedron. It is an example of a Johnson solid.

Construction

The augmented truncated tetrahedron is constructed from a truncated tetrahedron by attaching a triangular cupola.{{r|rajwade}} This cupola covers one of the truncated tetrahedron's four hexagonal faces, so that the resulting polyhedron's faces are eight equilateral triangles, three squares, and three regular hexagons.{{r|berman}} Since it has the property of convexity and has regular polygonal faces, the augmented truncated tetrahedron is a Johnson solid, denoted as the sixty-fifth Johnson solid $J_{65}$ .{{r|francis}}

Properties

The surface area of an augmented truncated tetrahedron is:{{r|berman}}

$\frac{6 + 13 \sqrt{3}}{2}a^2 \approx 14.258a^2,$

the sum of the areas of its faces. Its volume can be calculated by slicing it off into both truncated tetrahedron and triangular cupola, and adding their volume:{{r|berman}}

$\frac{11 \sqrt{2}}{4}a^3 \approx 3.889a^3.$

It has the same three-dimensional symmetry group as the triangular cupola, the pyramidal symmetry $C_{3 \mathrm{v}}$ . Its dihedral angles can be obtained by adding the angle of a triangular cupola and an augmented truncated tetrahedron in the following:{{r|johnson}}

its dihedral angle between triangle and hexagon is as in the truncated tetrahedron: 109.47°;
its dihedral angle between adjacent hexagons is as in the truncated tetrahedron: 70.53°;
its dihedral angle between triangle and square is as in the triangular cupola's angle: 125.3°
its dihedral angle between triangle and square, on the edge where the triangular cupola and truncated tetrahedron are attached, is the sum of both triangular cupola's square-hexagon angle and the truncated tetrahedron's triangle-hexagon angle: approximately 164.17°; and
its dihedral angle between triangle and hexagon, on the edge where triangular cupola and truncated tetrahedron are attached, is the sum of the dihedral angle of a triangular cupola and truncated tetrahedron between that: approximately 141.3°;

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

}}

External links

{{Mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid | urlname = AugmentedTruncatedTetrahedron | title = Augmented truncated tetrahedron }}

Category:Johnson solids