The pentagonal gyrobicupola is a polyhedron that is constructed by attaching two
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 base-to-base, each of its cupolas is twisted at 36°. It is an example of a
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 a
composite polyhedron
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids.
Definition and examples
A convex polyhedron is said to be ''composite'' if there exists ...
.
Construction
The pentagonal gyrobicupola is a
composite polyhedron
In geometry, a composite polyhedron is a convex polyhedron that produces other polyhedrons when sliced by a plane. Examples can be found in Johnson solids.
Definition and examples
A convex polyhedron is said to be ''composite'' if there exists ...
: it is constructed by attaching two
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 base-to-base. This construction is similar to the
pentagonal orthobicupola; the difference is that one of cupolas in the pentagonal gyrobicupola is twisted at 36°, as suggested by the prefix ''gyro-''. The resulting polyhedron has the same faces as the pentagonal orthobicupola does: those cupolas cover their decagonal bases, replacing it with eight
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, eight
squares
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 ...
, and two
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. 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 of its 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 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 ...
. The pentagonal gyrobicupola has such these, enumerating it as the thirty-first Johnson solid
.
Properties
Because it has a similar construction as the pentagonal orthobicupola, the surface area of a pentagonal gyrobicupola
is the sum of polygonal faces' area, and its volume
is twice the volume of a pentagonal cupola for which slicing it into those:
References
External links
*
{{Johnson solids navigator
Composite polyhedron
Johnson solids