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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.
A Goldberg polyhedron is a dual polyhedron of a geodesic polyhedron. A consequence of Euler's polyhedron formula is that a Goldberg polyhedron always has exactly 12 pentagonal faces. Icosahedral symmetry ensures that the pentagons are always regular and that there are always 12 of them.
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 ...
Class II (b=c): {3,q+} b,b are easier to see from the dual polyhedron {q,3} with q-gonal faces first divided into triangles with a central point, and then all edges are divided into b sub-edges. Class III : {3, q +} b , c have nonzero unequal values for b , c , and exist in chiral pairs.
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.
Because all the faces of the cC have an even number of sides and are centrally symmetric, it is a zonohedron: Chamfered cube (3 zones are shown by 3 colors for their hexagons — each square is in 2 zones —.) The chamfered cube is also the Goldberg polyhedron GP IV (2,0) or {4+,3} 2,0, containing square and hexagonal faces.
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. For instance, while all the cross-sections of a ball are disks, [2] the cross-sections of a cube depend on how the cutting plane is related to the ...