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A cuboid is a convex polyhedron whose polyhedral graph is the same as that of a cube. [1] [2] General cuboids have many different types. When all of the rectangular cuboid's edges are equal in length, it results in a cube, with six square faces and adjacent faces meeting at right angles.
Common net of a 1x1x5 and 1x2x3 cuboid. Common nets of cuboids have been deeply researched, mainly by Uehara and coworkers. To the moment, common nets of up to three cuboids have been found, It has, however, been proven that there exist infinitely many examples of nets that can be folded into more than one polyhedra. [10]
A rectangular cuboid with integer edges, as well as integer face diagonals, is called an Euler brick; for example with sides 44, 117, and 240. A perfect cuboid is an Euler brick whose space diagonal is also an integer. It is currently unknown whether a perfect cuboid actually exists. [6] The number of different nets for a simple cube is 11 ...
Nets of a cube. An elementary way to construct a cube is using its net, an arrangement of edge-joining polygons constructing a polyhedron by connecting along the edges of those polygons. Eleven nets for the cube are shown here. [24] In analytic geometry, a cube may be constructed using the Cartesian coordinate systems.
Many different nets can exist for a given polyhedron, depending on the choices of which edges are joined and which are separated. The edges that are cut from a convex polyhedron to form a net must form a spanning tree of the polyhedron, but cutting some spanning trees may cause the polyhedron to self-overlap when unfolded, rather than forming a ...
Alternatively, it can be described by unfolding the cuboid into a net and finding a shortest path (a line segment) on the resulting unfolded system of six rectangles in the plane. Different nets produce different segments with different lengths, and the question becomes one of finding a net whose segment length is minimum. [2]
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All 11 unfoldings of the cube. A polyhedral net for the cube is necessarily a hexomino, with 11 hexominoes (shown at right) actually being nets. They appear on the right, again coloured according to their symmetry groups. A polyhedral net for the cube cannot contain the O-tetromino, nor the I-pentomino, the U-pentomino, or the V-pentomino.