Polyhedra hatchet is a rather painful matter, come expert and novicealike. There is a details logic to specific aspects that the long conventionalnames, but there is likewise much i m sorry is impractical, ungeneralizable, andonly survives due to the fact that it is entrenched. Perhaps these names are being askedto perform too much: to succinctly explain the natural properties the a polyhedronand also specific of its relationship to other polyhedra.
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Manynames are built from Greek prefixesfor the number of sides and the source -hedron definition faces (literallymeaning "seat"). Because that example, dodeca-, definition 2+10, is used indescribing any 12-sided solid. The term constant indicates that thefaces and also vertex numbers 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 ns turns into an a in thisword only; elsewhere it remains i.) following this pattern, part 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 shape of the faces, come disambiguate betweentwo polyhedra through the same number of faces. For example, a rhombicdodecahedron has 12 rhombus-shaped faces. A pentagonalicositetrahedron has 24 (i.e., 20+4) five-sided faces. The term trapezoidalis standardly offered to refer to "kite-shaped" quadrilaterals, i beg your pardon havetwo bag of adjacent sides that equal size (and so are not trapezoidsby the contemporary American definition which needs that two opposite sidesbe parallel). Thus a trapezoidalicositetrahedron has 24 together faces. (This intake is no as odd as itmay an initial seem; a British meaning of trapezoid is "a quadrilateralfigure no two of who sides room parallel" --- Oxford English Dictionary.)The term -kis- refers to a the process of adding a new vertexat the facility of every face and using that to division each n-sided faceinto triangles. A prefix equivalent to n standardly preceedsthe kis. For example, the tetrakiscube is obtained from the cube by dividingeach square into 4 isosceles triangles. A pentakisdodecahedron is based upon the dodecahedron,but each pentagon is changed with 5 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 form of separation, personal, instance faces, but additionally to a basic polygon fromwhich specific infinite collection of unique polyhedra have the right to be constructed.For instance the pentagonal prismand hexagonal prism room two membersof an limitless series. Connected infinite collection are the antiprisms,and the dipyramids and trapezohedra.Many that the common polyhedron names originate in Kepler"sterminology and its translations native his Latin. The ax truncatedrefers to the process of cutting turn off corners. Compare for example the cubeand the truncated cube. Truncationadds a new face for each previously 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 provides all the faces continuous polygons, thatis commonly intended, yet this is only feasible in an easy symmetric cases.The hatchet snub have the right to refer to a chiral process of replacing eachedge v a pair the triangles, e.g., together a method of deriving what is usuallycalled the snub cube native thecube.The 6 square faces of the cube continue to be squares (but rotated slightly), the12 edges end up being 24 triangles, and the 8 vertices become an additional 8triangles. However, the same procedure applied to an octahedrongives the the same result: The 8 triangular faces of the octahedron remaintriangles (but rotated slightly), the 12 edges become 24 triangles, andthe 6 vertices become 6 squares. This is due to the fact that the cube and also octahedronare twin to each other. To emphasize this equivalence,it is much more logical to contact the an outcome a snubcuboctahedron however 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 called the snub dodecahedron, however bettercalled the snub icosidodecahedron.There are 4 Archimedean solidswhich each have two common names:The rhombi prefix suggests that some of the deals with (12 squares inthe an initial two cases, 30 squares in the critical two) are in the planes of therhombicdodecahedron (in the first two cases) and also the rhombictriacontahedron (in the last two cases). The usage of truncatedrather than good rhombi in two cases emphasizes a different relationship.However, it need to be observed that after truncating the vertices of a cuboctahedronor icosidodecahedron, some lengthadjustments have to be made prior to obtaining the objects called as theirtrunctations, because the truncation outcomes in rectangles, not squares.In the various other Archimedean solids v truncated in their names, noadjustment is necessary, therefore one can argue the the little and greatnames room preferable in the respect. On the other hand, the truncationdoes create their topological structure, and the state greatrhombicosidodecahedron and also greatrhombicuboctahedron are likewise used for other polyhedra.The ax stellated practically always describes a process of extendingthe challenge planes the a polyhedron right into a "star polyhedron." there areoften countless ways to perform this, resulting in various polyhedra i m sorry arenot constantly well identified with this nomenclature. Because that examples, seethe 59 stellations that the icosahedron.But be conscious that some authors have incorrectly offered the hatchet stellateto median "erect pyramids on all the encounters of a provided polyhedron," and also afew mathematicians have argued a stricter meaning of stellatebased on expanding a offered polyhedron"s edges rather than faces.The term compound refers toan interpenetrating collection of the same or connected polyhedra i ordered it in amanner which has some in its entirety polyhedral symmetry.The hatchet pseudoshows up in 2 "isomers" which are rearrangements the the piece of a morestandard polyhedron.Names for numerous of the nonconvex uniformpolyhedra and their duals have been in flux. The two booksby Wenninger which highlight these polyhedra list names greatly dueto Norman Johnson. The names evolved slightly between the two publications <1971,1983> and since. I have actually incorporated his most recent naming suggestionsat the time of this writing.Crystallographers use a slightly different collection of names because that certaincrystal forms.For a systematic method of naming a great many exciting symmetricpolyhedra, I choose 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. Right here is one such hecatohedron;it is a dipyramid. Think of the othertwo ways to assemble those exact same 100 triangles into a convex polyhedron.Answer: This and this.(Joe Malkevitch verified me the infinite family members that these members of.)Virtual Polyhedra, (c) 1996,GeorgeW. Hart