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Conversely, a given net may fold into more than one different convex polyhedron, depending on the angles at which its edges are folded and the choice of which edges to glue together. [6] If a net is given together with a pattern for gluing its edges together, such that each vertex of the resulting shape has positive angular defect and such that ...
A sphere of radius r has surface area 4πr 2.. The surface area (symbol A) of a solid object is a measure of the total area that the surface of the object occupies. [1] The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the definition of arc length of one-dimensional curves, or of the surface area for polyhedra (i.e., objects with ...
Graphs of surface area, A against volume, V of all 5 Platonic solids and a sphere by CMG Lee, showing that the surface area decreases for rounder shapes, and the surface-area-to-volume ratio decreases with increasing volume. The dashed lines show that when the volume increases 8 (2³) times, the surface area increases 4 (2²) times.
A net = is said to be frequently or cofinally in if for every there exists some such that and . [5] A point is said to be an accumulation point or cluster point of a net if for every neighborhood of , the net is frequently/cofinally in . [5] In fact, is a cluster point if and only if it has a subnet that converges to . [6] The set of all ...
group [6] Truncated tetrahedron: 3.6.6: 4 triangles 4 hexagons: 18 12 T d: Cuboctahedron: 3.4.3.4: 8 triangles 6 squares: 24 12 O h: Truncated cube: 3.8.8: 8 triangles 6 octagons: 36 24 O h: Truncated octahedron: 4.6.6: 6 squares 8 hexagons 36 24 O h: Rhombicuboctahedron: 3.4.4.4: 8 triangles 18 squares 48 24 O h: Truncated cuboctahedron: 4.6.8 ...
If a surface has constant Gaussian curvature, it is called a surface of constant curvature. [52] The unit sphere in E 3 has constant Gaussian curvature +1. The Euclidean plane and the cylinder both have constant Gaussian curvature 0. A unit pseudosphere has constant Gaussian curvature -1 (apart from its equator, that is singular).
The area density (also known as areal density, surface density, superficial density, areic density, mass thickness, column density, or density thickness) of a two-dimensional object is calculated as the mass per unit area. The SI derived unit is the "kilogram per square metre" (kg·m −2).
Schmidt net, used for making plots of the Lambert azimuthal projection. The Schmidt net is a manual drafting method for the Lambert azimuthal equal-area projection using graph paper. It results in one lateral hemisphere of the Earth with the grid of parallels and meridians. The method is common in geoscience.