Rectified truncated tetrahedron

In geometry, the rectified truncated tetrahedron is a polyhedron, constructed as a rectified, truncated tetrahedron. It has 20 faces: 4 equilateral triangles, 12 isosceles triangles, and 4 regular hexagons.

Rectified truncated tetrahedron
Faces	20: 4 equilateral triangles 12 isosceles triangles 4 hexagons
Edges	48
Vertices	12+18
Schläfli symbol	rt{3,3}
Conway notation	atT
Symmetry group	Td, [3,3], (*332), order 24
Rotation group	T, [3,3]+, (332), order 12
Dual polyhedron	Joined truncated tetrahedron
Properties	convex
Net

Topologically, the triangles corresponding to the tetrahedron's vertices are always equilateral, although the hexagons, while having equal edge lengths, do not have the same edge lengths with the equilateral triangles, having different but alternating angles, causing the other triangles to be isosceles instead.

Related polyhedra

The rectified truncated tetrahedron can be seen in sequence of rectification and truncation operations from the tetrahedron. Further truncation, and alternation operations creates two more polyhedra:

Name	Truncated tetrahedron	Rectified truncated tetrahedron	Truncated rectified truncated tetrahedron	Snub rectified truncated tetrahedron
Coxeter	tT	rtT	trtT	srtT
Conway		atT	btT	stT
Image
Conway	dtT = kT	jtT	mtT	gtT
Dual

See also

• Rectified truncated cube
• Rectified truncated octahedron
• Rectified truncated dodecahedron
• Rectified truncated icosahedron

References

• Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5

External links

• George Hart's Conway interpreter: generates polyhedra in VRML, taking Conway notation as input