# Order-5 truncated pentagonal hexecontahedron

Order-5 truncated pentagonal hexecontahedron
Conway t5gD or wD
Goldberg {5+,3}2,1
Fullerene C140
Faces 72:
60 hexagons
12 pentagons
Edges 210
Vertices 140
Symmetry group Icosahedral (I)
Dual polyhedron Pentakis snub dodecahedron
Properties convex, chiral

The order-5 truncated pentagonal hexecontahedron is a convex polyhedron with 72 faces: 60 hexagons and 12 pentagons triangular, with 210 edges, and 140 vertices. Its dual is the pentakis snub dodecahedron.

It is Goldberg polyhedron {5+,3}2,1 in the icosahedral family, with chiral symmetry. The relationship between pentagons steps into 2 hexagons away, and then a turn with one more step.

It is a Fullerene C140.[1]

## Construction

It is explicitly called a pentatruncated pentagonal hexecontahedron since only the valence-5 vertices of the pentagonal hexecontahedron are truncated.[2]

Its topology can be constructed in Conway polyhedron notation as t5gD and more simply wD as a whirled dodecahedron, reducing original pentagonal faces and adding 5 distorted hexagons around each, in clockwise or counter-clockwise forms. This picture shows its flat construction before the geometry is adjusted into a more spherical form. The snub can create a (5,3) geodesic polyhedron by k5k6.

## Related polyhedra

The whirled dodecahedron creates more polyhedra by basic Conway polyhedron notation. The zip whirled dodecahedron makes a chamfered truncated icosahedron, and Goldberg (4,1). Whirl applied twice produces Goldberg (5,3), and applied twice with reverse orientations produces goldberg (7,0).

Whirled dodecahedron polyhedra
"seed" ambo truncate zip expand bevel snub chamfer whirl whirl-reverse

wD = G(2,1)
wD

awD
awD

twD
twD

zwD = G(4,1)
zwD

ewD
ewD

bwD
bwD

swD
swD

cwD = G(4,2)
cwD

wwD = G(5,3)
wwD

wrwD = G(7,0)
wrwD
dual join needle kis ortho medial gyro dual chamfer dual whirl dual whirl-reverse

dwD
dwD

jwD
jwD

nwD
nwD

kwD
kwD

owD
owD

mwD
mwD

gwD
gwD

dcwD
dcwD

dwwD
dwwD

dwrwD
dwrwD