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The old UBC (Uniform Building Code) and newer ICC (IBC and IRC) [4] codes state that a 4” sphere shall not pass through any portion of a barrier on a guardrail. In a horizontal or vertical cable rail, the cables, once tensioned must be rigid enough to prevent a 4-inch sphere passing through it.
Before Smale proved this theorem, mathematicians became stuck while trying to understand manifolds of dimension 3 or 4, and assumed that the higher-dimensional cases were even harder. The h -cobordism theorem showed that (simply connected) manifolds of dimension at least 5 are much easier than those of dimension 3 or 4.
Codes also generally require that there be a 1 + 1 ⁄ 2 inches (38 mm) clearance between the underside of the handrail and any obstruction—including the horizontal bracket arm. There is an allowance however for variations in the handrail size—for every 1 ⁄ 2 inch (13 mm) of additional perimeter dimension over 4 inches (102 mm), 1 ⁄ 8 ...
Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points u, v, and w on the sphere (shown at right). If the lengths of these three sides are a (from u to v ), b (from u to w ), and c (from v to w ), and the angle of the corner opposite c is C , then the (first) spherical ...
Building codes also require that no opening in a guard be of a size such that a 4-inch (10 cm) sphere may pass. There are three exceptions according to the 2003 International Building Code Section 1012.3 which allow openings to not exceed 8 or 21 inches (20 or 53 cm) depending on occupancy groups or special areas.
The most common residential deck railing design is built on-site using pressure treated lumber, with the vertical balusters regularly spaced to meet building code. [1] Wood railing could be in different styles such as Victorian, Chippendale railing and others. [2] A popular alternative to wood railing is composite lumber and PVC railing. [3] [4 ...
The Lebedev grid points are constructed so as to lie on the surface of the three-dimensional unit sphere and to be invariant under the octahedral rotation group with inversion. [4] For any point on the sphere, there are either five, seven, eleven, twenty-three, or forty-seven equivalent points with respect to the octahedral group, all of which ...
This is an example of a subdivision rule arising from a finite universe (i.e. a closed 3-manifold). In mathematics, a finite subdivision rule is a recursive way of dividing a polygon or other two-dimensional shape into smaller and smaller pieces. Subdivision rules in a sense are generalizations of regular geometric fractals.