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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.
A central cross section of a regular tetrahedron is a square. The two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these planes intersects the tetrahedron the resulting cross section is a rectangle. [11] When the intersecting plane is near one of the edges the rectangle is long and skinny.
The rhombic dodecahedron forms the maximal cross-section of a 24-cell, and also forms the hull of its vertex-first parallel projection into three dimensions. The rhombic dodecahedron can be decomposed into six congruent (but non-regular) square dipyramids meeting at a single vertex in the center; these form the images of six pairs of the 24 ...
A section, or cross-section, is a view of a 3-dimensional object from the position of a plane through the object. A section is a common method of depicting the internal arrangement of a 3-dimensional object in two dimensions. It is often used in technical drawing and is traditionally crosshatched. The style of crosshatching often indicates the ...
It is common in mathematics publications that define the Borromean rings to do so as a link diagram, a drawing of curves in the plane with crossings marked to indicate which curve or part of a curve passes above or below at each crossing. Such a drawing can be transformed into a system of curves in three-dimensional space by embedding the plane ...
The diagram is named for Victor Schlegel, who in 1886 introduced this tool for studying combinatorial and topological properties of polytopes. In dimension 3, a Schlegel diagram is a projection of a polyhedron into a plane figure; in dimension 4, it is a projection of a 4-polytope to 3-space.
(The icosidodecahedron is the equatorial cross-section of the 600-cell, and the decagon is the equatorial cross-section of the icosidodecahedron.) These radially golden polytopes can be constructed, with their radii, from golden triangles which meet at the center, each contributing two radii and an edge.
As a simple, visible surface polyhedron, the outward form of the final stellation is composed of 180 triangular faces, which are the outermost triangular regions in the stellation diagram. These join along 270 edges, which in turn meet at 92 vertices, with an Euler characteristic of 2.