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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.
As well as being interesting mathematical objects, they also occur in nature. Above is a photograph of a Romanesco broccoli set against a black background. Self-similarity can be observed in the inflorescence (the bud), which is approximately a scaled version of the broccoli as a whole, each of which are made up of yet more scaled versions of ...
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 ...
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.
Also called infinitesimal calculus A foundation of calculus, first developed in the 17th century, that makes use of infinitesimal numbers. Calculus of moving surfaces an extension of the theory of tensor calculus to include deforming manifolds. Calculus of variations the field dedicated to maximizing or minimizing functionals. It used to be called functional calculus. Catastrophe theory a ...
Aspects include "counting" the objects satisfying certain criteria (enumerative combinatorics), deciding when the criteria can be met, and constructing and analyzing objects meeting the criteria (as in combinatorial designs and matroid theory), finding "largest", "smallest", or "optimal" objects (extremal combinatorics and combinatorial ...
Complex numbers are all fun and games until someone loses an i. That's when things get real. Minkowski's question-mark function: A function with an unusual notation and possessing unusual fractal properties. Monty Hall problem: The counter-intuitive way to prevail when playing Let's Make a Deal. Moving sofa problem
Any two pairs of angles are congruent, [4] which in Euclidean geometry implies that all three angles are congruent: [a] If ∠BAC is equal in measure to ∠B'A'C', and ∠ABC is equal in measure to ∠A'B'C', then this implies that ∠ACB is equal in measure to ∠A'C'B' and the triangles are similar. All the corresponding sides are ...