In
geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
, the gyrate rhombicosidodecahedron is one of the
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
s (). It is also a
canonical polyhedron
In geometry, the midsphere or intersphere of a convex polyhedron is a sphere which is tangent to every edge of the polyhedron. Not every polyhedron has a midsphere, but the uniform polyhedra, including the regular, quasiregular and semiregul ...
.
Construction
The gyrate rhombicosidodecahedron can be constructed similarly as
rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed of two or more types of regular polygon faces.
It has a total of 62 faces: 20 regular triangular faces, 30 square f ...
: it is constructed from
parabidiminished rhombicosidodecahedron
In geometry, the parabidiminished rhombicosidodecahedron is one of the Johnson solids (). It is also a canonical polyhedron.
It can be constructed as a rhombicosidodecahedron
In geometry, the rhombicosidodecahedron is an Archimedean solid, o ...
by attaching two regular
pentagonal cupola
Properties
The pentagonal cupola (geometry), cupola's faces are five equilateral triangles, five squares, one regular pentagon, and one regular decagon. It has the property of Convex set, convexity and regular polygonal faces, from which it is ...
s onto its
decagonal faces. As a result, these pentagonal cupolas cover its dodecagonal faces, so the resulting polyhedron has 20
equilateral triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...
s, 30
square
In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
s, and 10
regular pentagon
In geometry, a pentagon () is any five-sided polygon or 5-gon. The sum of the internal angles in a simple polygon, simple pentagon is 540°.
A pentagon may be simple or list of self-intersecting polygons, self-intersecting. A self-intersecting ...
s as its faces. The difference between those two polyhedrons is that one of two pentagonal cupolas from the gyrate rhombicosidodecahedron is rotated through 36°. A
convex
Convex or convexity may refer to:
Science and technology
* Convex lens, in optics
Mathematics
* Convex set, containing the whole line segment that joins points
** Convex polygon, a polygon which encloses a convex set of points
** Convex polytop ...
polyhedron in which all faces are
regular polygon
In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...
s is called the
Johnson solid
In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...
, and the gyrate rhombicosidodecahedron is among them, enumerated as the 72th Johnson solid
.
Properties
Because the two aforementioned polyhedrons have similar construction, they have the same surface area and volume. A gyrate rhombicosidodecahedron with edge length has a surface area by adding all of the area of its faces:
Its volume can be calculated by slicing it into two regular pentagonal cupolas and one parabigyrate rhombicosidodecahedron, and adding their volumes:
The gyrate rhombicosidodecahedron is one of the five Johnson solids that do not have
Rupert property
In geometry, Prince Rupert's cube is the largest cube that can pass through a hole cut through a unit cube without splitting it into separate pieces. Its side length is approximately 1.06, 6% larger than the side length 1 of the unit cube through ...
, meaning a polyhedron of the same or larger size and the same shape as it cannot pass through a hole in it. The other Johnson solids with no such property are
parabigyrate rhombicosidodecahedron
In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
Altern ...
,
metabigyrate rhombicosidodecahedron
In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
Alt ...
,
trigyrate rhombicosidodecahedron, and
paragyrate diminished rhombicosidodecahedron.
See also
Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are:
* The
parabigyrate rhombicosidodecahedron
In geometry, the parabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
Altern ...
where two opposing cupolae are rotated;
* The
metabigyrate rhombicosidodecahedron
In geometry, the metabigyrate rhombicosidodecahedron is one of the Johnson solids (). It can be constructed as a rhombicosidodecahedron with two non-opposing pentagonal cupolae rotated through 36 degrees. It is also a canonical polyhedron.
Alt ...
where two non-opposing cupolae are rotated;
* And the
trigyrate rhombicosidodecahedron where three cupolae are rotated.
References
External links
*
Johnson solids
{{Johnson solids navigator