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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.
A more unusual measurement for firewood is the "rick" or face cord. It is stacked 16 inches (40.6 cm) deep with the other measurements kept the same as a cord, making it 1 ⁄ 3 of a cord; however, regional variations mean that its precise definition is non-standardized. [44]
Every kite is an orthodiagonal quadrilateral, meaning that its two diagonals are at right angles to each other. Moreover, one of the two diagonals (the symmetry axis) is the perpendicular bisector of the other, and is also the angle bisector of the two angles it meets. [1] Because of its symmetry, the other two angles of the kite must be equal.
The Euler angles are three angles introduced by Leonhard Euler to describe the orientation of a rigid body with respect to a fixed coordinate system. [ 1 ] They can also represent the orientation of a mobile frame of reference in physics or the orientation of a general basis in three dimensional linear algebra .
The subject codes so listed are used by the two major reviewing databases, Mathematical Reviews and Zentralblatt MATH. This list has some items that would not fit in such a classification, such as list of exponential topics and list of factorial and binomial topics , which may surprise the reader with the diversity of their coverage.
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 ...
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 ...