Higher Dimensional Polytope Nets
The geometric concept of a net can be extended to higher dimensions.
Tesseract |
Truncated tesseract |
24-cell |
For example, a net of a polychoron, or four-dimensional polytope, is composed of polyhedral cells that are connected by their faces and all occupy the same three-dimensional space, just as the polygon faces of a net of a polyhedron are connected by their edges and all occupy the same plane. The above net of the tesseract, the four-dimensional hypercube, is used prominently in a painting by Salvador Dalí, Crucifixion (Corpus Hypercubus) (1954).
Whether or not every four-dimensional polytope may be cut along the two-dimensional faces shared by its three-dimensional facets, and unfolded into 3D to a single nonoverlapping polyhedron (as in the above unfolding of the tesseract), remains unknown, as does the corresponding question in higher dimensions.
Read more about this topic: Net (polyhedron)
Famous quotes containing the words higher, dimensional and/or nets:
“To higher or lower ends, they [the majority of mankind] move too often with something of a sad countenance, with hurried and ignoble gait, becoming, unconsciously, something like thorns, in their anxiety to bear grapes; it being possible for people, in the pursuit of even great ends, to become themselves thin and impoverished in spirit and temper, thus diminishing the sum of perfection in the world, at its very sources.”
—Walter Pater (1839–1894)
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“Shakespearean fish swam the sea, far away from land;
Romantic fish swam in nets coming to the hand....”
—William Butler Yeats (1865–1939)