The term "4-polytope stubs" does not appear to be a standard term in mathematics or geometry as of my last knowledge update. However, it seems to suggest a focus on properties or structures related to 4-dimensional polytopes (also known as 4-polytopes). A **4-polytope** is a four-dimensional generalization of a polytope, which can be thought of as a shape in four-dimensional space.
The term "57-cell" can refer to a specific type of mathematical object in the field of geometry, particularly in the study of higher-dimensional polytopes. In this context, a "cell" refers to a higher-dimensional analogue of a polygon or polyhedron.
A cubic cupola is a type of geometric structure that can be described as a polyhedron. In the context of architecture and geometry, a cupola generally refers to a small dome that is often placed on top of a building. However, a "cubic cupola" specifically refers to a version that takes the form of a cubic shape.
A cubic pyramid, also known as a square pyramid, is a three-dimensional geometric shape that consists of a square base and four triangular faces that converge at a single point called the apex. Here are some key characteristics of a cubic pyramid: 1. **Base**: The base of the pyramid is a square, which means that all four sides are equal in length and all angles are right angles (90 degrees).
A cubical bipyramid is a polyhedron that is constructed by connecting the apexes of two square pyramids at their bases, where the base of each pyramid is a square. This structure contains two square faces at the ends, and four triangular faces that connect the corners of the square base to the apexes. The cubical bipyramid has the following characteristics: - It has 8 faces (2 square faces and 6 triangular faces). - It has 12 edges.
A cuboctahedral prism is a type of polyhedron that can be described as a prism whose bases are cuboctahedra. The cuboctahedron is a three-dimensional shape that has 8 triangular faces and 6 square faces, with a total of 12 edges and 12 vertices.
A cuboctahedral pyramid is a geometric structure that can be visualized as a pyramid whose base is a cuboctahedron. To break this down further: 1. **Cuboctahedron**: This is a convex polyhedron with 8 triangular faces and 6 square faces, and it has 12 edges and 12 vertices. It can be thought of as the intersection of a cube and an octahedron.
A dodecahedral bipyramid is a polyhedron formed by connecting two regular dodecahedra (which are 12-faced polyhedra with regular pentagonal faces) at their bases. It can also be viewed as a bipyramid with a dodecahedron as its base, which consists of 12 pentagonal faces.
A dodecahedral cupola is a type of geometric solid that is formed by combining two elements: a dodecahedron and a cupola. The dodecahedron is a polyhedron with 12 pentagonal faces, while a cupola is a type of dome shape that typically consists of a polygonal base and a set of triangular faces that converge at a point above the base.
A dodecahedral pyramid is a three-dimensional geometric figure that consists of a regular dodecahedron (a polyhedron with twelve flat faces that are regular pentagons) as its base, with triangular faces rising to a single apex point above the base. To understand the structure of a dodecahedral pyramid: 1. **Base**: The base is a regular dodecahedron, which has 12 pentagonal faces, 20 vertices, and 30 edges.
The Grand 120-cell is a four-dimensional convex polytope, which is one of the higher-dimensional analogs of three-dimensional shapes. It is part of a class of polytopes known as "regular polytopes" in four dimensions, specifically a type of "uniform 4-polytope". The Grand 120-cell is an extension of the 120-cell, one of the six regular convex 4-polytopes.
The Grand 600-cell, also known as the Grand 600-cell honeycomb, is a type of polytopal structure in higher-dimensional geometry. The term generally refers to a specific configuration related to the 600-cell, which is a convex four-dimensional polytope, also known as a 4-dimensional regular simplex or a 600-cell polytope. The 600-cell itself has 600 tetrahedral cells, and it is one of the six regular convex 4-polytopes.
The Grand Stellated 120-Cell is a complex mathematical structure in the realm of higher-dimensional geometry, specifically in four-dimensional space. It is categorized as a type of polytopes, which are the higher-dimensional analogues of polygons (2D) and polyhedra (3D).
The Great 120-cell, also known as the grand 120-cell or the great 120-cell, is a four-dimensional polytope that is part of the category of regular polytopes. Specifically, it is one of the six convex regular 4-polytopes and is classified as a honeycomb of an icosahedral structure. Here are some key characteristics of the Great 120-cell: 1. **Dimensions**: It exists in four-dimensional space (4D).
The great duoantiprism is a type of convex polyhedron that is part of the category of Archimedean solids. It is characterized by its unique structure, which consists of two layers of triangular faces. The solid can be viewed as a combination of a duoantiprism and an additional layer of triangular faces that create an intricate arrangement.
The Great Grand 120-cell is a four-dimensional convex polytopic figure, which is part of a family of polytopes in higher dimensions. To understand it, we first need to break down what a "120-cell" is and then explore the "Great Grand" aspect. ### 120-cell The 120-cell, or hexacosichoron, is one of the six regular convex 4-polytopes (also known as polychora) in four-dimensional space.
The great icosahedral 120-cell (also known as the great icosahedron or the 120-cell) is a four-dimensional polytope, belonging to the family of regular polytopes. It is one of the six convex regular 4-polytopes known as the "4D polytopes," and it is specifically classified as a regular 120-cell.
The Great Stellated 120-cell is one of the fascinating four-dimensional polytopes in the realm of higher-dimensional geometry. Specifically, it is one of the uniform 4-polytopes, and it belongs to the family of polytopes known as the stellar polytopes.
The icosahedral 120-cell, also known as the icosahedral honeycomb or 120-cell, is one of the six regular polytopes in four-dimensional space. It is a four-dimensional analog of the platonic solids and features a highly symmetric structure.
An icosahedral bipyramid is a polyhedral shape that can be visualized as two identical icosahedra joined at their bases. This shape consists of 12 vertices, 30 edges, and 20 triangular faces. The vertices of an icosahedral bipyramid can be grouped into two sets: six at the top and six at the bottom, with each set forming the vertices of an individual triangular face.
An icosahedral prism is a three-dimensional geometric shape that combines the properties of an icosahedron and a prism. An icosahedron is a polyhedron with 20 triangular faces, 12 vertices, and 30 edges. A prism, in general, is a solid shape with two parallel bases that are congruent polygons, and rectangular faces connecting the corresponding sides of the bases.
An icosahedral pyramid is a geometric structure that can be described as a pyramid whose base is an icosahedron—a polyhedron with 20 triangular faces. In this context, the term "pyramid" refers to a shape formed by connecting a point (the apex) to each vertex of the base, which in this case is the icosahedron.
An icosidodecahedral prism is a type of polyhedral solid that can be classified as a prism. More specifically, it is formed by taking two identical icosidodecahedron bases and connecting them with rectangular faces. The icosidodecahedron is a convex Archimedean solid made up of 20 equilateral triangular faces and 12 regular pentagonal faces, with 30 edges and 60 vertices.
An octahedral cupola is a type of geometric shape that can be classified as a part of the family of cupolae. It is formed by taking an octagonal base and placing it on top of a square prism (or frustum), which creates a structure resembling a dome on top of a flat base. In more detail, a cupola is a solid that consists of a polygonal base and triangular faces that connect the base to another polygonal face above.
An octahedral pyramid is a three-dimensional geometric figure formed by extending the apex (top point) of a pyramid to the center of an octahedron. An octahedron itself is a polyhedron composed of eight triangular faces.
A polytetrahedron generally refers to a geometric figure that is a higher-dimensional analogue of a tetrahedron. 1. **Tetrahedron in 3D**: A tetrahedron is a three-dimensional shape (a polyhedron) with four triangular faces, six edges, and four vertices. 2. **Generalization to Higher Dimensions**: In higher dimensions, a polytetrahedron can be thought of as the simplest form of a polytope in that dimension.
A rhombicosidodecahedral prism is a three-dimensional geometric solid formed by extending the two-dimensional shape of a rhombicosidodecahedron vertically along a third axis, creating a prism. To break this down a bit: 1. **Rhombicosidodecahedron**: This is one of the Archimedean solids and is characterized by its 62 faces: 20 regular triangles, 30 squares, and 12 regular pentagons.
The small stellated 120-cell is a four-dimensional convex uniform polytope, which is an example of a 120-cell, a higher-dimensional analogue of a polyhedron in three dimensions. Specifically, this polytope is a member of the family of polytopes known as the "120-cells" or "120-vertex polytopes.
A snub cubic prism is a type of polyhedral shape that can be classified among the Archimedean solids. It is formed by taking a cube and "snubbing" or truncating its edges by adding a triangular prism on each edge. This results in a hybrid shape that retains some characteristics of a cube while also incorporating elements of the triangular prism. More specifically, a snub cubic prism can be considered as consisting of: 1. **Vertices**: It has 12 vertices.
A snub dodecahedral prism is a type of three-dimensional geometric shape that can be classified as a prism. More specifically, it is constructed by taking a snub dodecahedron as its base and extending that shape vertically to form the prism. ### Characteristics of a Snub Dodecahedral Prism: 1. **Base Shape**: The snub dodecahedron is a convex polyhedron with 12 regular pentagons and 20 equilateral triangles.
The stellated rhombic dodecahedral honeycomb is a three-dimensional arrangement of space-filling cells that composed of stellated rhombic dodecahedra. A honeycomb, in geometrical terms, refers to a structure comprised of repeating units that completely fill space without any gaps. In the case of the stellated rhombic dodecahedral honeycomb, the basic unit cell is a stellated rhombic dodecahedron.
A tetrahedral bipyramid is a type of geometric shape that consists of two tetrahedra joined at their bases, resulting in a figure with six vertices, nine edges, and four triangular faces. It is classified as a polyhedron and can be visualized as forming a bipyramidal structure by connecting the apex (top vertex) of one tetrahedron to the apex of another.
A tetrahedral cupola is a type of geometric solid that features characteristics of both a tetrahedron and a cupola. It can be understood as a combination of two shapes: 1. **Tetrahedron**: A polyhedron with four triangular faces, six edges, and four vertices. 2. **Cupola**: A polyhedron formed by the combination of a polygonal base and two congruent polygonal faces on top, typically resulting in a shape that has an apex.
The Triakis truncated tetrahedral honeycomb is a type of honeycomb structure in three-dimensional space formed by a specific arrangement of truncated tetrahedra and triangular prisms. In more detail: - A **honeycomb** refers to a repetitive, tessellated arrangement in which space is filled with a defined geometric shape without any gaps.
The term "trigonal trapezohedral honeycomb" refers to a type of tessellation or honeycomb structure in three-dimensional space. This particular arrangement is part of the broader study of geometric and topological structures. Essentially, it relates to how certain shapes can fill space without gaps or overlaps.
A truncated cubic prism is a geometric shape that can be described as a prism with its top and bottom faces being truncated (cut off) in such a way that the shape retains a polygonal top and a polygonal bottom, typically with the base being a rectangle or square.
A truncated dodecahedral prism is a type of geometric solid that is a combination of two distinct shapes: a truncated dodecahedron and a prism. To break it down: 1. **Truncated Dodecahedron**: This is a convex polyhedron with 12 regular pentagonal faces, where each vertex of the original dodecahedron has been truncated (flattened) to create additional faces.
A truncated icosahedral prism is a three-dimensional geometric shape that extends a truncated icosahedron along a perpendicular axis, forming a prism. To understand this shape, we need to break it down into its components: 1. **Truncated Icosahedron**: This is a well-known Archimedean solid that consists of 12 regular pentagonal faces and 20 regular hexagonal faces.
A truncated icosidodecahedral prism is a three-dimensional geometric shape that is a type of prism based on a truncated icosidodecahedron. To understand this shape, let's break down the components: 1. **Truncated Icosidodecahedron**: This is a convex Archimedean solid that is formed by truncating (cutting off) the vertices of a regular icosidodecahedron.
A truncated octahedral prism refers to a geometric figure that combines elements of a truncated octahedron and a prism structure. 1. **Truncated Octahedron**: A truncated octahedron is a type of Archimedean solid that has 8 regular hexagonal faces and 6 square faces. It is created by truncating (or cutting off) the corners of a regular octahedron.
A truncated tetrahedral prism is a three-dimensional geometric shape that is formed by extending a truncated tetrahedron along a perpendicular axis to create a prism. To clarify each component: 1. **Truncated Tetrahedron**: This is a type of polyhedron that results from truncating (or cutting off) the corners (vertices) of a regular tetrahedron.
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