Английская Википедия:Gyrate rhombicosidodecahedron
Шаблон:Short description Шаблон:Infobox polyhedron
In geometry, the gyrate rhombicosidodecahedron is one of the Johnson solids (Шаблон:Math). It is also a canonical polyhedron.
Construction
The gyrate rhombicosidodecahedron can be constructed similarly as rhombicosidodecahedron: it is constructed from parabigyrate rhombicosidodecahedron by attaching two regular pentagonal cupolas onto its decagonal faces. As a result, these pentagonal cupolas cover its dodecagonal faces, so the resulting polyhedron has 20 equilateral triangles, 30 squares, and 10 regular pentagons as its faces. The difference between those two polyhedrons is that one of two pentagonal cupolas from the gyrate rhombicosidodecahedron is rotated through 36°.Шаблон:R A convex polyhedron in which all faces are regular polygons is called the Johnson solid, and the gyrate rhombicosidodecahedron is among them, enumerated as the 72th Johnson solid <math> J_{72} </math>.Шаблон:R Шаблон:Multiple image
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:Шаблон:R <math display="block"> \left(30+5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right)a^2 \approx 59.306a^2. </math> Its volume can be calculated by slicing it into two regular pentagonal cupolas and one parabigyrate rhombicosidodecahedron, and adding their volumes:Шаблон:R <math display="block"> \frac{60+29\sqrt{5}}{3}a^3 \approx 41.615a^3. </math>
The gyrate rhombicosidodecahedron is one of the five Johnson solids that do not have Rupert property, 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, metabigyrate rhombicosidodecahedron, trigyrate rhombicosidodecahedron, and paragyrate diminished rhombicosidodecahedron.Шаблон:R
See also
Alternative Johnson solids, constructed by rotating different cupolae of a rhombicosidodecahedron, are:
- The parabigyrate rhombicosidodecahedron <math> J_{73} </math> where two opposing cupolae are rotated;
- The metabigyrate rhombicosidodecahedron <math> J_{74} </math> where two non-opposing cupolae are rotated;
- And the trigyrate rhombicosidodecahedron <math> J_{75} </math> where three cupolae are rotated.
References
External links
Шаблон:Johnson solids navigator
