triakis truncated tetrahedron

{{Distinguish|truncated triakis tetrahedron}}

{{short description|Space-filling polyhedron with 16 faces}}

{{Infobox polyhedron

| image = Triakis truncated tetrahedron.png

| type = Plesiohedron

| faces = 4 hexagons
12 isosceles triangles

| edges = 30

| vertices = 16

| vertex_config =

| schläfli =

| wythoff =

| conway = {{math|k3tT}}

| coxeter =

| symmetry =

| rotation_group =

| dual = 16{{!}}Order-3 truncated triakis tetrahedron

| properties = convex

| vertex_figure =

| net =

}}

In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb.{{cite book|last1=Conway|first1=John H.|last2=Burgiel|first2=Heidi|last3=Goodman-Strauss|first3=Chaim|title=The Symmetries of Things|page=332|year=2008|isbn=978-1568812205}}{{cite journal|last1=Grünbaum|first1=B|last2=Shephard|first2=G. C.|title=Tilings with Congruent Tiles|journal=Bull. Amer. Math. Soc.|volume=3|issue=3|pages=951–973|year=1980|url=http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183547682|doi=10.1090/s0273-0979-1980-14827-2|doi-access=free}}

The triakis truncated tetrahedron is the shape of the Voronoi cell of the carbon atoms in diamond, which lie on the diamond cubic crystal structure.{{cite journal|first1=L.|last1=Föppl|year=1914|title=Der Fundamentalbereich des Diamantgitters|journal=Phys. Z.|volume=15|pages=191–193}}{{cite web|last=Conway|first=John|title=Voronoi Polyhedron|url=https://groups.google.com/forum/?fromgroups=#!msg/geometry.puzzles/pkL3avbWPoc/ABSaqdQaqu4J|work=geometry.puzzles|accessdate=20 September 2012}} As the Voronoi cell of a symmetric space pattern, it is a plesiohedron.{{citation

| last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum

| last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard

| doi = 10.1090/S0273-0979-1980-14827-2

| issue = 3

| journal = Bulletin of the American Mathematical Society

| mr = 585178

| pages = 951–973

| series = New Series

| title = Tilings with congruent tiles

| volume = 3

| year = 1980| doi-access = free

}}.