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The net has to be such that the straight line is fully within it, and one may have to consider several nets to see which gives the shortest path. For example, in the case of a cube , if the points are on adjacent faces one candidate for the shortest path is the path crossing the common edge; the shortest path of this kind is found using a net ...
The cone over a closed interval I of the real line is a filled-in triangle (with one of the edges being I), otherwise known as a 2-simplex (see the final example). The cone over a polygon P is a pyramid with base P. The cone over a disk is the solid cone of classical geometry (hence the concept's name). The cone over a circle given by
Blooming a regular dodecahedron. In the geometry of convex polyhedra, blooming or continuous blooming is a continuous three-dimensional motion of the surface of the polyhedron, cut to form a polyhedral net, from the polyhedron into a flat and non-self-overlapping placement of the net in a plane.
In the case of lines, the cone extends infinitely far in both directions from the apex, in which case it is sometimes called a double cone. Either half of a double cone on one side of the apex is called a nappe. The axis of a cone is the straight line passing through the apex about which the base (and the whole cone) has a circular symmetry.
For rigid origami (a type of folding that keeps the surface flat except at its folds, suitable for hinged panels of rigid material rather than flexible paper), the condition of Kawasaki's theorem turns out to be sufficient for a single-vertex crease pattern to move from an unfolded state to a flat-folded state. [22]
The polyhedron can be thought of as being folded from a sheet of paper (a net for the polyhedron) and it inherits the same geometry as the paper: for every point p within a face of the polyhedron, a sufficiently small open neighborhood of p will have the same distances as a subset of the Euclidean plane. The same thing is true even for points ...