geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the small rhombihexacron (or small dipteral disdodecahedron) is the dual of the

small rhombihexahedron In geometry, the small rhombihexahedron (or small rhombicube) is a nonconvex uniform polyhedron, indexed as U18. It has 18 faces (12 squares and 6 octagons), 48 edges, and 24 vertices. Its vertex figure is an antiparallelogram. Related polyhedra ...

. It is visually identical to the

small hexacronic icositetrahedron In geometry, the small hexacronic icositetrahedron is the dual of the small cubicuboctahedron. It is visually identical to the small rhombihexacron. A part of each dart lies inside the solid, hence is invisible in solid models. Proportions Its ...

. Its faces are

antiparallelogram In geometry, an antiparallelogram is a type of self-crossing quadrilateral. Like a parallelogram, an antiparallelogram has two opposite pairs of equal-length sides, but these pairs of sides are not in general parallel. Instead, sides in the lon ...

s formed by pairs of

coplanar In geometry, a set of points in space are coplanar if there exists a geometric plane that contains them all. For example, three points are always coplanar, and if the points are distinct and non-collinear, the plane they determine is unique. How ...

triangle A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- colli ...

Proportions

Each antiparallelogram has two angles of

\arccos(\frac+\frac\sqrt)\approx 16.842\,116\,236\,30^

and two angles of

\arccos(-\frac+\frac\sqrt)\approx 98.421\,058\,118\,15^

. The diagonals of each antiparallelogram intersect at an angle of

\arccos(\frac+\frac\sqrt)\approx 64.736\,825\,645\,55^

. The

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the un ...

equals

\arccos(\frac)\approx 138.117\,959\,055\,51^

. The ratio between the lengths of the long edges and the short ones equals

\sqrt

References

External links

Dual uniform polyhedra {{polyhedron-stub