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The Platonic solids, seen here in an illustration from Johannes Kepler's Mysterium Cosmographicum (1596), are an early example of exceptional objects. The symmetries of three-dimensional space can be classified into two infinite families—the cyclic and dihedral symmetries of n-sided polygons—and five exceptional types of symmetry, namely the symmetry groups of the Platonic solids.
The template to the right includes links to alphabetical lists of all mathematical articles. This article brings together the same content organized in a manner better suited for browsing. Lists cover aspects of basic and advanced mathematics, methodology, mathematical statements, integrals, general concepts, mathematical objects, and reference ...
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
The tribar/triangle appears to be a solid object, made of three straight beams of square cross-section which meet pairwise at right angles at the vertices of the triangle they form. The beams may be broken, forming cubes or cuboids. This combination of properties cannot be realized by any three-dimensional object in ordinary Euclidean space.
An object's degree of rotational symmetry is the number of distinct orientations in which it looks exactly the same for each rotation. Certain geometric objects are partially symmetrical when rotated at certain angles such as squares rotated 90°, however the only geometric objects that are fully rotationally symmetric at any angle are spheres ...
The template method is not always successful because members of a group may significantly differ visually from each other, and may look much different if viewed from different angles. To counter the problem of viewpoint, the visual system detects familiar components of an object in 3-dimensional space.
Breakdown of objects into geons. The recognition-by-components theory, or RBC theory, [1] is a process proposed by Irving Biederman in 1987 to explain object recognition. According to RBC theory, we are able to recognize objects by separating them into geons (the object's main component parts). Biederman suggested that geons are based on basic ...
An object behaves pathologically (or, somewhat more broadly used, in a degenerated way) if it either fails to conform to the generic behavior of such objects, fails to satisfy certain context-dependent regularity properties, or simply disobeys mathematical intuition. In many occasions, these can be and often are contradictory requirements ...