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Each polyhedron lies in Euclidean 4-dimensional space as a parallel cross section through the 600-cell (a hyperplane). In the curved 3-dimensional space of the 600-cell's boundary surface envelope, the polyhedron surrounds the vertex V the way it surrounds its own center. But its own center is in the interior of the 600-cell, not on its surface.
In geometry, a 4-polytope (sometimes also called a polychoron, [1] polycell, or polyhedroid) is a four-dimensional polytope. [2] [3] It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (), and cells ().
The tesseract is one of 6 convex regular 4-polytopes. In mathematics, a regular 4-polytope or regular polychoron is a regular four-dimensional polytope.They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.
A polyhedron that has a midsphere is said to be midscribed about this sphere. [1] When a polyhedron has a midsphere, one can form two perpendicular circle packings on the midsphere, one corresponding to the adjacencies between vertices of the polyhedron, and the other corresponding in the same way to its polar polyhedron, which has the same ...
In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry.In particular, all its elements or j-faces (for all 0 ≤ j ≤ n, where n is the dimension of the polytope) — cells, faces and so on — are also transitive on the symmetries of the polytope, and are themselves regular polytopes of dimension j≤ n.
If a plane intersects a solid (a 3-dimensional object), then the region common to the plane and the solid is called a cross-section of the solid. [1] A plane containing a cross-section of the solid may be referred to as a cutting plane. The shape of the cross-section of a solid may depend upon the orientation of the cutting plane to the solid.
It is one of only four 4-connected simplicial well-covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the regular octahedron, the pentagonal bipyramid, and an irregular polyhedron with 12 vertices and 20 triangular faces. [10]
The 16-cell is the second in the sequence of 6 convex regular 4-polytopes (in order of size and complexity). [a]Each of its 4 successor convex regular 4-polytopes can be constructed as the convex hull of a polytope compound of multiple 16-cells: the 16-vertex tesseract as a compound of two 16-cells, the 24-vertex 24-cell as a compound of three 16-cells, the 120-vertex 600-cell as a compound of ...