Polyhedra terminology is a rather painful matter, come expert and also novicealike. Over there is a details logic to particular aspects that the long conventionalnames, but there is likewise much i m sorry is impractical, ungeneralizable, andonly survives because it is entrenched. Probably these names space being askedto carry out too much: come succinctly describe the innate properties the a polyhedronand also certain of its relationship to other polyhedra.

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Manynames are created from Greek prefixesfor the variety of sides and also the root -hedron definition faces (literallymeaning "seat"). For example, dodeca-, definition 2+10, is provided indescribing any 12-sided solid. The term constant indicates that thefaces and vertex figures are regularpolygons, e.g., to differentiate the regulardodecahedron (which is a Platonic solid)from the numerous dodecahedra. Similarly, icosi-,meaning 20, is offered in the 20-sided icosahedron,illustrated in ~ right. (Note: the i turns right into an a in thisword only; somewhere else it remains i.) adhering to this pattern, part authorscall the cube the hexahedron. The term-conta-refers to a group of ten, therefore a hexecontahedronhas 60 sides.Modifiers may describe the shape of the faces, to disambiguate betweentwo polyhedra v the same variety of faces. For example, a rhombicdodecahedron has actually 12 rhombus-shaped faces. A pentagonalicositetrahedron has 24 (i.e., 20+4) five-sided faces. The term trapezoidalis standardly provided to refer to "kite-shaped" quadrilaterals, i m sorry havetwo bag of surrounding sides that equal size (and so space not trapezoidsby the modern American definition which requires that two opposite sidesbe parallel). For this reason a trapezoidalicositetrahedron has 24 such faces. (This usage is not as odd as itmay an initial seem; a British an interpretation of trapezoid is "a quadrilateralfigure no 2 of who sides are parallel" --- Oxford English Dictionary.)The ax -kis- describes a the procedure of adding a new vertexat the center of every face and also using the to division each n-sided faceinto triangles. A prefix equivalent to n standardly preceedsthe kis. For example, the tetrakiscube is derived from the cube by dividingeach square into 4 isosceles triangles. A pentakisdodecahedron is based upon the dodecahedron,but each pentagon is replaced with five isosceles triangles. (Inthese cases, the tetra- and also penta- space redundant and in mostcases kis- alone would suffice.)Numerical modifiers choose pentagonal or hexagonal can refernot just to the shape of separation, personal, instance faces, but additionally to a basic polygon fromwhich particular infinite collection of distinct polyhedra deserve to be constructed.For example the pentagonal prismand hexagonal prism are two membersof an infinite series. Connected infinite collection are the antiprisms,and the dipyramids and also trapezohedra.Many that the usual polyhedron surname originate in Kepler"sterminology and its translations native his Latin. The hatchet truncatedrefers come the procedure of cutting off corners. Compare for instance the cubeand the truncated cube. Truncationadds a new face because that each formerly existing vertex, and also replaces n-gonswith 2n-gons, e.g., octagons instead of squares. If one can reduced offthe corners come a depth that makes all the faces continuous polygons, thatis generally intended, but this is only possible in an easy symmetric cases.The hatchet snub deserve to refer come a chiral process of replacing eachedge v a pair of triangles, e.g., together a method of deriving what is usuallycalled the snub cube from thecube.The 6 square encounters of the cube stay squares (but rotated slightly), the12 edges end up being 24 triangles, and also the 8 vertices become an additional 8triangles. However, the same procedure applied to an octahedrongives the the same result: The 8 triangular deals with of the octahedron remaintriangles (but rotated slightly), the 12 edges become 24 triangles, andthe 6 vertices come to be 6 squares. This is due to the fact that the cube and also octahedronare double to each other. To emphasize this equivalence,it is an ext logical to contact the an outcome a snubcuboctahedron but it may take a while for this name to be commonly adapted.Applying the analogous procedure to either the dodecahedron or the icosahedrongives the polyhedron usually dubbed the snub dodecahedron, however bettercalled the snub icosidodecahedron.There are 4 Archimedean solidswhich each have two usual names:The rhombi prefix indicates that several of the faces (12 squares inthe an initial two cases, 30 squares in the last two) room in the plane of therhombicdodecahedron (in the an initial two cases) and the rhombictriacontahedron (in the last two cases). The use of truncatedrather than an excellent rhombi in two cases emphasizes a different relationship.However, it need to be observed that after truncating the vertices the a cuboctahedronor icosidodecahedron, part lengthadjustments have to be made before obtaining the objects named as theirtrunctations, because the truncation outcomes in rectangles, not squares.In the various other Archimedean solids with truncated in their names, noadjustment is necessary, so one can argue that the little and greatnames space preferable in that respect. ~ above the other hand, the truncationdoes produce their topological structure, and also the state greatrhombicosidodecahedron and greatrhombicuboctahedron are additionally used for other polyhedra.The ax stellated almost always describes a process of extendingthe challenge planes the a polyhedron into a "star polyhedron." over there areoften many ways to do this, resulting in different polyhedra i beg your pardon arenot constantly well distinguished with this nomenclature. For examples, seethe 59 stellations of the icosahedron.But be aware that some authors have incorrectly used the term stellateto median "erect pyramids on all the deals with of a provided polyhedron," and also afew mathematicians have said a stricter meaning of stellatebased on prolonging a given polyhedron"s edges quite than faces.The term link refers toan interpenetrating set of similar or associated polyhedra i ordered it in amanner which has some all at once polyhedral symmetry.The term pseudoshows up in 2 "isomers" which are rearrangements of the pieces of a morestandard polyhedron.Names for plenty of of the nonconvex uniformpolyhedra and their duals have remained in flux. The two booksby Wenninger which illustrate these polyhedra perform names largely dueto Norman Johnson. The names progressed slightly between the two books <1971,1983> and since. I have incorporated his most recent specify name suggestionsat the time of this writing.Crystallographers use a contempt different set of names for certaincrystal forms.For a systematic technique of naming a an excellent many amazing symmetricpolyhedra, I prefer 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 created of 100 isoscelestriangles in (at least) three different ways. Here is one together hecatohedron;it is a dipyramid. Think that the othertwo ways to rally those very same 100 triangles into a convex polyhedron.Answer: This and this.(Joe Malkevitch proved me the infinite families that this members of.)Virtual Polyhedra, (c) 1996,GeorgeW. Hart