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The term "paragyrate diminished rhombicosidodecahedron" refers to a specific type of geometric polyhedron that is derived from the rhombicosidodecahedron, one of the Archimedean solids. 1. **Rhombicosidodecahedron**: This is a convex polyhedron with 62 faces (20 regular triangles, 30 squares, and 12 regular pentagons), 120 edges, and 60 vertices.

A pentadecahedron is a 3-dimensional geometric shape that has 15 faces. In geometry, polyhedra are categorized by the number of faces, and a pentadecahedron specifically consists of 15 polygonal faces. The exact configuration of these faces can vary, as there are different types of pentadecahedra, depending on the arrangement and shape of the polygons used (triangles, quadrilaterals, etc.).

A pentagonal antiprism is a type of geometric solid that belongs to a family known as antiprisms. It is constructed by taking two pentagonal bases that are parallel to each other and connected by a series of triangular faces. The triangular faces are arranged around the sides of the bases and are oriented such that they provide a twist between the two bases.

The pentagonal gyrobicupola is a type of polyhedron that belongs to the category of Archimedean solids. It is formed by arranging two pentagonal cupolas back-to-back, with a rotational symmetry about the vertical axis. Key characteristics of the pentagonal gyrobicupola include: 1. **Faces**: It consists of 10 triangular faces and 2 non-pentagonal cupolas, which contribute to a total of 12 faces.

The term "pentagonal gyrocupolarotunda" refers to a specific type of convex uniform polyhedron in the category of Archimedean solids. It is one of the many complex shapes that can be constructed using a combination of polygons and curved surfaces. The pentagonal gyrocupolarotunda features pentagonal faces and has some unique characteristics, such as its rotational symmetry.

The pentagonal hexecontahedron is a type of convex polyhedron, specifically a member of the category of Archimedean solids. It is defined by its 60 faces, which are all regular pentagons. The name "hexecontahedron" derives from the Greek prefix "hex-" meaning sixty, and "-hedron" meaning face. The pentagonal hexecontahedron features a high level of symmetry and is characterized by its vertices and edges.

The pentagonal orthobicupola is a type of convex polyhedron that is categorized among the Archimedean solids. It can be defined by its specific geometric properties as follows: 1. **Faces**: The pentagonal orthobicupola consists of 20 triangular faces and 12 regular pentagonal faces. 2. **Vertices**: It has a total of 60 vertices. 3. **Edges**: There are 90 edges in total.

The pentagonal orthobirotunda is a type of convex polyhedron in geometry. Specifically, it is one of the Archimedean solids, characterized by its vertex configuration and symmetry. Here are some key features of the pentagonal orthobirotunda: 1. **Faces**: It has 20 faces comprised of 10 triangles and 10 pentagons. 2. **Vertices**: The orthobirotunda has 30 vertices.

The pentagonal orthocupolarotunda is a type of convex polyhedron that belongs to the family of Archimedean solids. It can be described as a member of the broader category of polyhedra that exhibit a combination of regular polygons for their faces. Specifically, the pentagonal orthocupolarotunda features: - **Vertices**: It has 60 vertices. - **Edges**: It consists of 100 edges.

A pentagonal prism is a three-dimensional geometric shape that consists of two parallel pentagonal bases connected by five rectangular lateral faces. It is a type of prism, which means that its cross-section (the shape of the base) is constant along its height. Here are some key characteristics of a pentagonal prism: 1. **Bases**: There are two pentagonal bases situated parallel to each other.

A pentagrammic antiprism is a type of geometric solid that belongs to the family of antiprisms. Specifically, it is a variation in which the two polygonal bases are pentagrams (star polygons with five points) instead of the regular polygons found in standard antiprisms. ### Properties of a Pentagrammic Antiprism: 1. **Faces**: It has 10 triangular lateral faces that connect the vertices of the two pentagram bases.

The "Pentagrammic crossed-antiprism" is a type of polyhedron that belongs to the family of antiprisms. Specifically, it is a variation of the antiprism that involves a pentagram (a five-pointed star) instead of a regular polygon as its base faces. In geometrical terms, a crossed-antiprism consists of two parallel, congruent bases that are polygonal faces, connected by a set of triangular faces.

A pentagrammic prism is a type of three-dimensional geometric figure (a polyhedron) that consists of two parallel pentagrammic bases connected by rectangular sides. Here’s a breakdown of the components: 1. **Pentagram**: A pentagram is a five-pointed star formed by extending the sides of a regular pentagon. It has five vertices and five edges, and it can be drawn continuously without lifting the pen.

A pentahedron is a type of polyhedron that has five faces. The term is derived from the Greek prefix "penta-", meaning five, and "hedron," which refers to a face or surface. In three-dimensional geometry, the most common type of pentahedron is the triangular prism, which has two triangular faces and three rectangular faces. Other forms of pentahedra can include various combinations of face shapes as long as the total number of faces equals five.

The Pentakis snub dodecahedron is a type of convex polyhedron and a member of the Archimedean solids. It can be described in a few ways: 1. **Description**: The Pentakis snub dodecahedron is derived from the regular dodecahedron by adding a pyramidal "cap" on each of its pentagonal faces.

The term "prismatic compound of antiprisms" refers to a specific geometric arrangement involving multiple antiprismatic shapes combined in a structured way. **Antiprisms** are polyhedra characterized by two parallel, congruent bases (usually polygons) connected by an alternating band of triangular faces. They can be visualized as a type of prism with a twist, where the top and bottom faces are rotated relative to each other.

The term "prismatic compound of antiprisms" typically refers to a configuration that combines features of antiprisms with some aspects of prismatic structures. Antiprisms are polyhedra consisting of two parallel polygonal faces (the "bases") connected by an alternating band of triangular faces.

A prismatic compound of prisms refers to a geometric arrangement or structure made up of multiple prisms that interact with light in interesting ways. In optics, a prism is a transparent optical element that refracts light. When multiple prisms are combined, they can create a prismatic compound that manipulates light in complex ways, potentially leading to various optical effects, such as dispersion (separating light into its constituent colors), total internal reflection, or altering the direction of light beams.

The term "prismatic compound of prisms with rotational freedom" refers to a type of geometric or mathematical structure wherein multiple prisms are combined in such a way that they can rotate relative to one another. Let's break down the components of the concept: 1. **Prism**: A prism is a solid shape that has two identical bases connected by rectangular sides. The most common prisms are triangular prisms, rectangular prisms, and pentagonal prisms.

A prismatoid is a specific type of polyhedron that can be considered as a generalized prism. In geometry, a prismatoid is defined as a three-dimensional solid that has two parallel faces (called bases) that can be any polygon and all other faces that are trapezoidal or triangular. Essentially, it has a structure where the top and bottom faces are connected in such a way that they aren't necessarily congruent or identical in shape.