In
geometry, the rectified truncated icosahedron is a convex
polyhedron. It has 92 faces: 60
isosceles triangle
In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...
s, 12
regular pentagons, and 20
regular hexagon
In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.
Regular hexagon
A '' regular hexagon'' has ...
s. It is constructed as a
rectified,
truncated icosahedron, rectification truncating vertices down to mid-edges.
As a
near-miss Johnson solid
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whos ...
, under
icosahedral symmetry, the pentagons are always regular, although the hexagons, while having equal edge lengths, do not have the same edge lengths with the pentagons, having slightly different but alternating angles, causing the triangles to be
isosceles instead. The shape is a
symmetrohedron
In geometry, a symmetrohedron is a high-symmetry polyhedron containing convex regular polygons on symmetry axes with gaps on the convex hull filled by irregular polygons.
The name was coined by Craig S. Kaplan and George W. Hart.
The trivial ca ...
with notation ''I(1,2,*,
''
Images
Dual
By
Conway polyhedron notation, the dual polyhedron can be called a ''joined truncated icosahedron'', jtI, but it is topologically equivalent to the
rhombic enneacontahedron
In geometry, a rhombic enneacontahedron (plural: rhombic enneacontahedra) is a polyhedron composed of 90 Rhombus, rhombic faces; with three, five, or six rhombi meeting at each vertex. It has 60 broad rhombi and 30 slim. The rhombic enneacontahedr ...
with all rhombic faces.
Related polyhedra
The ''rectified truncated icosahedron'' can be seen in sequence of
rectification
Rectification has the following technical meanings:
Mathematics
* Rectification (geometry), truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points
* Rectifiable curve, in mathematics
* Recti ...
and
truncation
In mathematics and computer science, truncation is limiting the number of digits right of the decimal point.
Truncation and floor function
Truncation of positive real numbers can be done using the floor function. Given a number x \in \mathbb ...
operations from the
truncated icosahedron. Further truncation, and
alternation operations creates two more polyhedra:
See also
*
Near-miss Johnson solid
In geometry, a near-miss Johnson solid is a strictly convex polyhedron whose faces are close to being regular polygons but some or all of which are not precisely regular. Thus, it fails to meet the definition of a Johnson solid, a polyhedron whos ...
*
Rectified truncated tetrahedron
*
Rectified truncated octahedron
*
Rectified truncated cube
*
Rectified truncated dodecahedron
References
*
Coxeter ''
Regular Polytopes'', Third edition, (1973), Dover edition, (pp. 145–154 Chapter 8: Truncation)
*
John H. Conway
John Horton Conway (26 December 1937 – 11 April 2020) was an English people, English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. He also made contributions to ...
, Heidi Burgiel,
Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008,
External links
George Hart's Conway interpreter generates polyhedra in
VRML, taking Conway notation as input
{{Polyhedron-stub
Polyhedra