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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 skeuomorph (also spelled skiamorph, / ˈ s k juː ə ˌ m ɔːr f, ˈ s k juː oʊ-/) [1] [2] is a derivative object that retains ornamental design cues (attributes) from structures that were necessary in the original. [3] Skeuomorphs are typically used to make something new feel familiar in an effort to speed understanding and acclimation.
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.
This is known as the SAS similarity criterion. [7] The "SAS" is a mnemonic: each one of the two S's refers to a "side"; the A refers to an "angle" between the two sides. Symbolically, we write the similarity and dissimilarity of two triangles ABC and A'B'C' as follows: [8]
For integer number of dimensions , when doubling a side of an object, copies of it are created, i.e. 2 copies for 1-dimensional object, 4 copies for 2-dimensional object and 8 copies for 3-dimensional object. For the Sierpiński triangle, doubling its side creates 3 copies of itself.