Gyroelongated pentagonal birotunda

{{Short description|48th Johnson solid}}

{{Infobox polyhedron

|image=gyroelongated_pentagonal_birotunda.png

|type=Johnson
{{math|gyroelongated pentagonal cupolarotunda – J{{sub|48}} – augmented triangular prism}}

|faces=4x10 triangles
2+10 pentagons

|edges=90

|vertices=40

|symmetry={{math|Dihedral symmetry in three dimensions}}

|vertex_config={{math|2x10(3.5.3.5)
2.10(3{{sup|4}}.5)}}

|dual=-

|properties=convex, chiral

|net=Johnson solid 48 net.png

}}

In geometry, the gyroelongated pentagonal birotunda is one of the Johnson solids ({{math|J{{sub|48}}}}). As the name suggests, it can be constructed by gyroelongating a pentagonal birotunda (either {{math|pentagonal orthobirotunda}} or the icosidodecahedron) by inserting a decagonal antiprism between its two halves.

{{Johnson solid}}

The gyroelongated pentagonal birotunda is one of five Johnson solids which are chiral, meaning that they have a "left-handed" and a "right-handed" form. In the illustration to the right, each pentagonal face on the bottom half of the figure is connected by a path of two triangular faces to a pentagonal face above it and to the left. In the figure of opposite chirality (the mirror image of the illustrated figure), each bottom pentagon would be connected to a pentagonal face above it and to the right. The two chiral forms of {{math|J{{sub|48}}}} are not considered different Johnson solids.