Polyhedra terminology is a somewhat painful matter, to professional and novicealike. There is a certain logic to specific elements of the long conventionalnames, but there is likewise much which is imvaluable, ungeneralizable, andonly survives because it is entrenched. Perhaps these names are being askedto perform also much: to succinctly describe the natural properties of a polyhedronand likewise certain of its relationships to various other polyhedra.
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Manynames are created from Greek prefixesfor the variety of sides and also the root -hedron interpretation faces (literallydefinition "seat"). For instance, dodeca-, meaning 2+10, is used indescribing any type of 12-sided solid. The term continual shows that thedeals with and vertex numbers are regularpolygons, e.g., to differentiate the regulardodecahedron (which is a Platonic solid)from the many dodecahedra. Similarly, icosi-,interpretation 20, is offered in the 20-sided icosahedron,illustrated at best. (Note: the i turns right into an a in thisword only; somewhere else it remains i.) Following this pattern, some authorscall the cube the hexahedron. The term-conta-describes a team of ten, so a hexecontahedronhas 60 sides.Modifiers may explain the form of the faces, to disambiguate between2 polyhedra through the very same number of encounters. For example, a rhombicdodecahedron has actually 12 rhombus-shaped encounters. A pentagonalicositetrahedron has 24 (i.e., 20+4) five-sided encounters. The term trapezoidalis standardly provided to refer to "kite-shaped" quadrilaterals, which havetwo pairs of surrounding sides of equal size (and also so are not trapezoidsby the modern American meaning which needs that 2 opposite sidesbe parallel). Thus a trapezoidalicositetrahedron has actually 24 such encounters. (This consumption is not as odd as itmay first seem; a British definition of trapezoid is "a quadrilateralfigure no 2 of whose sides are parallel" --- Oxford English Dictionary.)The term -kis- refers to a the procedure of including a new vertexat the facility of each face and also using it to divide each n-sided faceright into triangles. A preresolve corresponding to n standardly preceedsthe kis. For instance, the tetrakiscube is derived from the cube by dividingeach square right into 4 isosceles triangles. A pentakisdodecahedron is based upon the dodecahedron,but each pentagon is reput via five isosceles triangles. (Inthese situations, the tetra- and penta- are redundant and in mostcases kis- alone would certainly suffice.)Numerical modifiers prefer pentagonal or hexagonal can refernot just to the form of individual encounters, yet also to a base polygon fromwhich certain limitless series of one-of-a-kind polyhedra have the right to be created.For instance the pentagonal prismand also hexagonal prism are 2 membersof an limitless series. Related boundless series are the antiprisms,and also the dipyramids and also trapezohedra.Many of the prevalent polyhedron names originate in Kepler"sterminology and its translations from his Latin. The term truncatedrefers to the process of cutting off corners. Compare for instance the cubeand the truncated cube. Truncationadds a brand-new face for each formerly existing vertex, and reareas n-gonswith 2n-gons, e.g., octagons instead of squares. If one deserve to reduced offthe corners to a depth that provides all the deals with continual polygons, thatis usually intended, but this is just possible in easy symmetric situations.The term snub have the right to describe a chiral procedure of replacing eachedge through a pair of triangles, e.g., as a way of deriving what is usuallydubbed the snub cube from thecube.The 6 square deals with of the cube remain squares (but rotated slightly), the12 edges come to be 24 triangles, and also the 8 vertices come to be a secondary 8triangles. However, the very same process used to an octahedronoffers the identical result: The 8 triangular deals with of the octahedron remaintriangles (but rotated slightly), the 12 edges come to be 24 triangles, andthe 6 vertices end up being 6 squares. This is bereason the cube and octahedronare dual to each other. To emphadimension this equivalence,it is more logical to speak to the outcome a snubcuboctahedron yet it might take a while for this name to be widely adapted.Applying the analogous process to either the dodecahedron or the icosahedronoffers the polyhedron usually referred to as the snub dodecahedron, but betterreferred to as the snub icosidodecahedron.There are four Archimedean solidswhich each have actually 2 common names:The rhombi presettle shows that some of the faces (12 squares inthe first two instances, 30 squares in the last two) are in the planes of therhombicdodecahedron (in the first 2 cases) and also the rhombictriacontahedron (in the last two cases). The use of truncatedrather than great rhombi in two situations emphasizes a various relationship.However before, it should be oboffered that after truncating the vertices of a cuboctahedronor icosidodecahedron, some lengthadjustments need to be made prior to obtaining the objects called as theirtrunctations, because the truncation results in rectangles, not squares.In the other Archimedean solids through truncated in their names, noadjustment is vital, so one can argue that the small and greatnames are preferable in that respect. On the other hand also, the truncationdoes produce their topological framework, and also the terms greatrhombicosidodecahedron and also greatrhombicuboctahedron are additionally used for various other polyhedra.The term stellated almost always refers to a process of extendingthe confront planes of a polyhedron into a "star polyhedron." Tbelow areregularly many means to carry out this, causing different polyhedra which arenot constantly well distinguimelted with this nomenclature. For examples, seethe 59 stellations of the icosahedron.But be conscious that some authors have actually incorrectly used the term stellateto suppose "erect pyramids on all the faces of a offered polyhedron," and afew mathematicians have actually said a stricter definition of stellatebased on extending a offered polyhedron"s edges fairly than encounters.The term compound refers toan interpenetrating set of similar or connected polyhedra arranged in amanner which has some as a whole polyhedral symmetry.The term pseudoreflects up in 2 "isomers" which are rearrangements of the pieces of a moreconventional polyhedron.Names for many of the nonconvex uniformpolyhedra and also their duals have actually been in flux. The 2 booksby Wenninger which show these polyhedra list names greatly dueto Normale Johnson. The names developed slightly in between the 2 books <1971,1983> and also considering that. I have actually incorporated his many recent naming suggestionsat the moment of this creating.Crystallographers use a slightly different set of names for certaincrystal develops.For a organized approach of naming an excellent many interesting symmetricpolyhedra, I choose John Conway"s notation.Exercise: Name this,this,this, and also this.Exercise: Hecatoindicates 100.
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A convex hecatohedron have the right to be created of 100 isoscelestriangles in (at least) three various ways. Here is one such hecatohedron;it is a dipyramid. Think of the othertwo means to assemble those exact same 100 triangles right into a convex polyhedron.Answer: This and this.(Joe Malkevitch showed me the unlimited family members that these members of.)Virtual Polyhedra, (c) 1996,GeorgeW. Hart