Polyhedra ax is a somewhat painful matter, come expert and novicealike. There is a certain logic to certain aspects that the long conventionalnames, but there is likewise much i beg your pardon is impractical, ungeneralizable, andonly survives due to the fact that it is entrenched. Maybe these names are being askedto execute too much: to succinctly define the inherent properties the a polyhedronand also details of its relationship to various other polyhedra.
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Manynames are constructed from Greek prefixesfor the variety of sides and also the root -hedron meaning faces (literallymeaning "seat"). Because that example, dodeca-, meaning 2+10, is used indescribing any type of 12-sided solid. The term regular indicates the thefaces and vertex figures are regularpolygons, e.g., to distinguish the regulardodecahedron (which is a Platonic solid)from the plenty of dodecahedra. Similarly, icosi-,meaning 20, is used in the 20-sided icosahedron,illustrated at right. (Note: the i turns right into an a in thisword only; somewhere else it remains i.) complying with this pattern, some authorscall the cube the hexahedron. The term-conta-refers come a team of ten, for this reason a hexecontahedronhas 60 sides.Modifiers may describe the form of the faces, to disambiguate betweentwo polyhedra v the same number of faces. For example, a rhombicdodecahedron has 12 rhombus-shaped faces. A pentagonalicositetrahedron has actually 24 (i.e., 20+4) five-sided faces. The term trapezoidalis standardly used to describe "kite-shaped" quadrilaterals, i m sorry havetwo pairs of surrounding sides that equal length (and so are not trapezoidsby the modern-day American meaning which requires that 2 opposite sidesbe parallel). Hence a trapezoidalicositetrahedron has actually 24 such faces. (This intake is not as odd as itmay first seem; a British an interpretation of trapezoid is "a quadrilateralfigure no two of who sides are parallel" --- Oxford English Dictionary.)The ax -kis- describes a the process of including a brand-new vertexat the facility of every face and also using that to division each n-sided faceinto triangles. A prefix corresponding to n standardly preceedsthe kis. Because that example, the tetrakiscube is acquired from the cube by dividingeach square into four isosceles triangles. A pentakisdodecahedron is based upon the dodecahedron,but every pentagon is replaced with five isosceles triangles. (Inthese cases, the tetra- and penta- room redundant and in mostcases kis- alone would suffice.)Numerical modifiers favor pentagonal or hexagonal can refernot just to the form of individual faces, but additionally to a basic polygon fromwhich details infinite collection of one-of-a-kind polyhedra deserve to be constructed.For instance the pentagonal prismand hexagonal prism are two membersof an boundless series. Associated infinite series are the antiprisms,and the dipyramids and trapezohedra.Many that the typical polyhedron names originate in Kepler"sterminology and its translations from his Latin. The term truncatedrefers come the process of cutting turn off corners. To compare for instance the cubeand the truncated cube. Truncationadds a brand-new face because that each previously existing vertex, and replaces n-gonswith 2n-gons, e.g., octagons instead of squares. If one can reduced offthe corners come a depth that renders all the faces consistent polygons, thatis typically intended, however this is only feasible in simple symmetric cases.The ax snub can refer come a chiral process of replacing eachedge v a pair of triangles, e.g., as a means of deriving what is usuallycalled the snub cube from thecube.The 6 square deals with of the cube stay squares (but rotated slightly), the12 edges end up being 24 triangles, and also the 8 vertices become secondary 8triangles. However, the same procedure applied come an octahedrongives the similar result: The 8 triangular encounters of the octahedron remaintriangles (but rotated slightly), the 12 edges come to be 24 triangles, andthe 6 vertices come to be 6 squares. This is due to the fact that the cube and also octahedronare dual to each other. Come emphasize this equivalence,it is more logical to call the an outcome a snubcuboctahedron however it might take a while because that this surname to be commonly adapted.Applying the analogous process to either the dodecahedron or the icosahedrongives the polyhedron usually called the snub dodecahedron, yet bettercalled the snub icosidodecahedron.There are 4 Archimedean solidswhich each have actually two usual names:The rhombi prefix suggests that some of the deals with (12 squares inthe first two cases, 30 squares in the last two) room in the plane of therhombicdodecahedron (in the very first two cases) and also the rhombictriacontahedron (in the last two cases). The usage of truncatedrather than great rhombi in two situations emphasizes a different relationship.However, it should be observed that after truncating the vertices that a cuboctahedronor icosidodecahedron, some lengthadjustments need to be made prior to obtaining the objects named as theirtrunctations, since the truncation outcomes in rectangles, no squares.In the other Archimedean solids through truncated in their names, noadjustment is necessary, therefore one might argue the the little and greatnames space preferable in the respect. ~ above the various other hand, the truncationdoes produce their topological structure, and also the state greatrhombicosidodecahedron and greatrhombicuboctahedron are likewise used for other polyhedra.The term stellated practically always refers to a procedure of extendingthe confront planes that a polyhedron into a "star polyhedron." over there areoften countless ways to execute this, resulting in various polyhedra which arenot always well differentiated with this nomenclature. For examples, seethe 59 stellations that the icosahedron.But be mindful that some authors have actually incorrectly supplied the ax stellateto median "erect pyramids on every the faces of a given polyhedron," and afew mathematicians have argued a stricter meaning of stellatebased on extending a offered polyhedron"s edges fairly than faces.The term compound refers toan interpenetrating set of similar or associated polyhedra i ordered it in amanner which has some as whole polyhedral symmetry.The hatchet pseudoshows up in 2 "isomers" which are rearrangements that the pieces of a morestandard polyhedron.Names for numerous of the nonconvex uniformpolyhedra and their duals have remained in flux. The 2 booksby Wenninger which illustrate these polyhedra perform names greatly dueto Norman Johnson. The names evolved slightly in between the two books <1971,1983> and since. I have incorporated his many recent naming suggestionsat the time of this writing.Crystallographers use a contempt different set of names for certaincrystal forms.For a systematic technique of naming a great many exciting symmetricpolyhedra, I favor John Conway"s notation.Exercise: Name this,this,this, and also this.Exercise: Hecatomeans 100.
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A convex hecatohedron deserve to be built of 100 isoscelestriangles in (at least) three various ways. Below is one together hecatohedron;it is a dipyramid. Think that the othertwo methods to assemble those very same 100 triangles right into a convex polyhedron.Answer: This and also this.(Joe Malkevitch proved me the infinite households that these members of.)Virtual Polyhedra, (c) 1996,GeorgeW. Hart