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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.
Four numbering schemes for the uniform polyhedra are in common use, distinguished by letters: [C] Coxeter et al., 1954, showed the convex forms as figures 15 through 32; three prismatic forms, figures 33–35; and the nonconvex forms, figures 36–92.
Colored regions are cross-sections of the solid cone. Their boundaries (in black) are the named plane sections. A cross section of a polyhedron is a polygon. The conic sections – circles, ellipses, parabolas, and hyperbolas – are plane sections of a cone with the cutting planes at various different angles, as seen in the diagram at left.
V(3.4. 3 / 2 .4) π − π / 2 90° Hexahemioctacron (Dual of cubohemioctahedron) — V(4.6. 4 / 3 .6) π − π / 3 120° Octahemioctacron (Dual of octahemioctahedron) — V(3.6. 3 / 2 .6) π − π / 3 120° Small dodecahemidodecacron (Dual of small dodecahemidodecacron) — V(5.10. 5 / 4 ...
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 ...
This fact can be used to calculate the dihedral angles themselves for a regular or edge-symmetric ideal polyhedron (in which all these angles are equal), by counting how many edges meet at each vertex: an ideal regular tetrahedron, cube or dodecahedron, with three edges per vertex, has dihedral angles = / = (), an ideal regular octahedron or ...
An antiparallelogram is a special case of a crossed quadrilateral, with two pairs of equal-length edges. [3] In general, crossed quadrilaterals can have unequal edges. [ 3 ] Special forms of the antiparallelogram are the crossed rectangles and crossed squares, obtained by replacing two opposite sides of a rectangle or square by the two diagonals.
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.