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Tilings and Polyhedra

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Tilings and Polyhedra REGULAR TILINGS Description A regular tiling is a tiling of the plane consisting of multiple copies of a single regular p olygon meeting edge to edge Howmany can you construct Comments While these tilings can b e easily found by trial and error and all are quite familiar they do give some insightinto the geometry of regular p olygons Hints There are only three regular tilings References Kappra Connections The Geometric Bridge Between Art and Science McGrawHill Chapter REGULAR TILINGS Illustration Kappra p SEMIREGULAR TILINGS Description A semiregular tiling is a tiling of the plane consisting of at least two dierent typ es of regular p olygons In addition the pattern of p olygons meeting at a common juncture p oint or vertex must b e the same for each p oint How many can you construct Comments Some of these tilings are not immediately obvious You can calculate or eliminate some p ossibilities by making a table of the measures of the interior angles of regular p olygons since the sum of these angles for the p olygons meeting at a common vertex must b e degrees Hints There are only eight semiregular tilings References Kappra Connections The Geometric Bridge Between Art and Science McGrawHill Chapter SEMIREGULAR TILINGS Illustration Kappra p PLATONIC SOLIDS Description A Platonic solid has the prop erty that eachfaceisanidentical regular p oly gon and that the same numb er of p olygons meet at each corner Can you construct them all Comments These are ancient ob jects of study Plato naturally is reputed to have investigated them and they are discussed in Euclids Elements but they may b e quite older Hints There are only ve Platonic solids Tetrahedron Four equilateral triangles three meeting at each corner Cub e Six squares three meeting at each corner Octahedron Eight equilateral triangles four meeting at each corner Do decahedron Twelve regular p entagons three meeting at each corner Icosahedron Twenty equilateral triangles ve meeting at each corner References Holden Shap es Space and SymmetryDover Wells The Penguin Dictionary of Curious and Interesting GeometryPenguin pp PLATONIC SOLIDS Illustration Wells p SEMIREGULAR SOLIDS Description A semiregular solid has the prop ertythateach face is a regular p olygon but that there are at least two dierent kinds of faces In addition the same pattern of faces must meet at each corner How manycanyou construct Comments According to Heron and Pappus Archimedes wrote ab out these solids but unfortunately the work has b een lost These are b eautiful ob jects with very app ealing symmetry Hints Taketwo copies of any regular p olygon and use them as the top and b ottom of a drum ringed with squares This is a prism Now remove the squares and use a strip of equilateral triangles to form the sides of the drum This is an antiprism There is an innite numb er of prisms and antiprismsone for eachchoice of base p olygon Apart from these there are exactly thirteen semiregular solids also called Archimedean solids One of them is made with hexagons and p entagons and is the template for a so ccer ball as well as a representation of a recently discovered form of carb on C known as Buckminsterfullerene References Holden Shap es Space and SymmetryDover Pearce and Pearce Polyhedra Primer Dale Seymour pp Wells The Penguin Dictionary of Curious and Interesting GeometryPenguin pp SEMIREGULAR SOLIDS Illustration Wells pp PearcePearce p THE DELTAHEDRA Description A deltahedron is a p olyhedron made solely from equilateral triangles There is no requirement that the same numb er of triangles meet at every corner but the ob ject must b e convex ie it should have no indentations and two triangles that meet along an edge must not lie in the same plane Howmany can you construct Comments It is conjectured that the name comes from the fact that the Greek letter lo oks like a triangle Hints Try to rst understand whyevery deltahedron has an even numb er of faces Exp er iment to see if you can determine the minimum and maximum numb er of faces Curiously one p otential deltahedron is missing References Holden Shap es Space and SymmetryDover p Pearce and Pearce Polyhedra Primer Dale Seymour p THE DELTAHEDRA Illustration PearcePearce p .
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