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Symmetry in Platonic Solids

Symmetry in Platonic Solids. Polyhedra. A polyhedron is a solid figure bounded by flat faces and straight edges, i.e., by polygons. Face  A face of a polyhedron is any of the plane surfaces forming a polyhedron. The faces of a polyhedron are polygons. Edge

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Symmetry in Platonic Solids

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  1. Symmetry in PlatonicSolids

  2. Polyhedra Regular Polyhedra and Symmetry A polyhedron is a solid figure bounded by flat faces and straight edges, i.e., by polygons. Face A face of a polyhedron is any of the plane surfaces forming a polyhedron. The faces of a polyhedron are polygons. Edge An edge of a polyhedron is any of the line segments making up the framework of a polyhedron. The edges are where the faces intersect each other. Vertex (pl. vertices) A vertex of a polyhedron is any of the corner points of a polyhedron, i.e., a point where three or more polygonal faces meet. Dihedral angle A dihedral angle is the space between two intersecting planes. Every polyhedron has a dihedral angle at every edge. Polyhedral angle A polyhedral angle is the space enclosed by three or more planes that intersect in a vertex. Every polyhedron has a polyhedral angle at every vertex.

  3. Activity 2 • In pairs, Ss match words to pictures: • T hands out the pictures, Ss write words next to appropriate pictures. • T asks “what is number 1” to individual Ss and check the answer, and so on. • Reading of the definitions previously defined: • UK Ssread the wordsexplainingwhere the stress goes. 3 Regular Polyhedra and Symmetry

  4. Regular Polyhedra Regular Polyhedra and Symmetry A regularpolyhedronis a polyhedronwhosefaces are congruent regular polygonswhich are assembled in the same way aroundeachvertex. Thismeansthat a regular polyhedronhasallcongruentdihedralanglesand allcongruentpolyhedralangles. The regular polyhedrathat can bebuiltare onlyfive. They are alsoknownasPlatonicSolids.

  5. Regular Polyhedra Are Only 5 Regular Polyhedra and Symmetry Let’s consider the followingconditions: At leastthreefacesof a regular polyhedronmustmeet at anygivenvertex. Thosefacescannotlie on the sameplane, therefore the sum of the anglesconverging on the samevertexmustbelowerthan 360°. The amplitudeof the anglesof a regular polygonincreaseswith the numberofitssides; weknowthat the amplitudeofeachinternal angle of a regular polygonwithnsidesis

  6. Activity 3 Trying constructions: • T divides Ss into 4 groups (8-9 Ss each). • T hands out paperboard polygons, each team receives a single shape (5 triangles, 4 squares, 4 pentagons, 3 hexagons). • Ss try to assemble polygons around a vertex to form a polyhedral angle. Describing polyhedra: • A volunteer for each team describes the polyhedral angles they could build, showing practical results. • Another volunteer explains how they could assemble them or the reason they couldn’t. 6 Regular Polyhedra and Symmetry

  7. Let’s analise all the possibile cases starting from the equilateral triangle • In anequilateraltriangleeach angle spans 60° • Ifthreetrianglesmeet at anyvertex, the sum of the converginganglesis 3·60° = 180° < 360° • Wehave a TETRAHEDRON, 4 faces, 4 vertices, 6 edges 7 Regular Polyhedra and Symmetry

  8. Once more we use equilateral triangles • If 4 trianglesmeet at anyvertex, wehave • 4·60° = 240° < 360° • OCTAHEDRON 8 faces, 6 vertices, 12 edges 8 Regular Polyhedra and Symmetry

  9. If 5 trianglesmeet at eachvertex, wehave: • 5·60° = 300° < 360° • ICOSAHEDRON, 20 faces, 12 vertices, 30 edges • Itcannotexistany more polyhedracomposedoftriangles, as 6·60° = 360° 9 Regular Polyhedra and Symmetry

  10. Now we use squares; each angle spans 90° • Ifthreesquaresmeet at anyvertex, the convergingangles sum to 3·90° = 270° < 360° • The CUBE , or hexahedron, has 6 faces, 8 vertices, 12 edges • Itcannotexistany more polyhedramade up ofsquares, as4·90° = 360° 10 Regular Polyhedra and Symmetry

  11. Let’s use pentagons; each angle spans 108° • Ifthreepentagons converge on eachvertex, the sum of the anglesis 3·108° = 324° < 360° • the DODECAHEDRONhas 12 faces, 20 vertices, 30 edges • No more polyhedramade up bypentagons can exist, as4·108° = 432° > 360° 11 Regular Polyhedra and Symmetry

  12. We can tryto pass toexaminehexagons, but… • Itcannotexistany more polyhedra, as the anglesof a hexagon are 120°, and 3·120° = 360° Dodechaedronvacuum (Leonardo da Vinci – De divina proportione) 12 Regular Polyhedra and Symmetry

  13. D&D The shapesof the dicesof some popularrole play follow the regular polyhedra Dungeons & Dragonsdices 13 Regular Polyhedra and Symmetry

  14. Activity 4 • Other name of cube • The regular polyhedron having four face • A polyhedron that has a polygonal base, whose vertices are joined to a single vertex • The regular polyhedron bounded by 20 triangles • A point where edges meet • The faces of a dodecahedron • Each face of a octahedron • The segment where two faces meet • The part of space between two half-planes • The regular polygon whose angles span 120° • The segments bounding a polygon 14 Regular Polyhedra and Symmetry

  15. Euler’s Formula Regular Polyhedra and Symmetry Euler’s formula bounds the numberoffaces (f), vertices (v) and edges (e) of a regular polyhedron. Foranypolyhedronwehave f + v = e + 2

  16. Demonstration of Euler’s Formula • Let’s consider a regular solidwhosefaces are polygonswithnsides and nvertices • Ifwe start with a single face, itisobviously f = 1v = ne = n • We can immediatelyverifythat f + v – e = 1 16 Regular Polyhedra and Symmetry

  17. We add now a second facef’ = f + 1 • The numberofedgesshouldincreasebyn, butoneedgeissharedby the twofigures: e’ = e + n - 1 • the numberofverticesshouldincreasebyn, buttwovertices are in common v’ = v + n - 2 • f’+v’-e’ = (f+1)+(v+n-2)-(e+n-1) f’ + v’ - e’ = 1 17 Regular Polyhedra and Symmetry

  18. Let’s add a third face f” = f’ + 1 • The number of edges should increase by n, but two edges are in common e” = e’ + n - 2 • the number of vertices should increase by n, but three vertices are shared v” = v’ + n - 3 • f”+v”-e” = (f’+1)+(v’+n-3)-(e’+n-2) f”+ v”– e” = 1 18 Regular Polyhedra and Symmetry

  19. Regular Polyhedra and Symmetry As we have seen, if we add other faces, it always remains f + v – e = 1 But pay attention! The polyhedron is not yet closed!

  20. Finally, we close the polyhedron with the last face • The numberoffacesincreasesagainby 1. • But the numbersofedges and vertices don’t increase, aswe can use the alreadyexistingones. • Therefore f + v – e = 2 20 Regular Polyhedra and Symmetry

  21. Determination of the number offaces, edges and vertices Regular Polyhedra and Symmetry Let’s calln the numberofsides or verticesofeachpolygonal face, and m the numberoffacesthatmeet at the samevertexof the polyhedron. The total numbervofverticesshouldbef·n, butwemustconsiderthatmpolygons converge on the samevertex, so v = f·n/ m

  22. Regular Polyhedra and Symmetry The total numbereofedgesshouldbef·n, butwemustconsiderthat2polygonsmeet at the sameedge, so e =f·n/ 2 ByEuler’s formula, f + f·n/m - f·n/2 = 2 Solvingbyf, wehave

  23. Faces, edges and vertices 4m 2m-2n-nm 2mn 2m-2n-nm 4n 2m-2n-nm Regular Polyhedra and Symmetry So wehave f = e = v =

  24. Activity 5 • Ss move to match cards • T hands out the cards with pictures and words to Ss; • Ss move to match the cards, forming group of 6; • each group read the characteristics of their polyhedron; • T asks Ss to feed back their answers; • UK Ss correct the answers. 24 Regular Polyhedra and Symmetry

  25. Activity 6 • Groups move to match polyhedra: • Previous groups must move to match polyhedra; • Number of faces of one group must match the number of vertex of another; • UK Ss describe the groups; • What are dual polyhedra? 25 Regular Polyhedra and Symmetry

  26. DualPolyhedra • Cube and octahedron are each other dualpolyhedra. • The centre of the faces of a cube are the vertices of an octahedron and viceversa. • The tetrahedron is dual to itself. 26 Regular Polyhedra and Symmetry

  27. Also the dodecahedron and the icosahedron are dual solids. • In this image, the edges of the dodecahedron are perpendicular to the faces of the icosahedron in their middle point, and viceversa. • We can also build the two polyhedra in such a way that the vertices of the dodecahedron coincide with the centres of the faces of the icosahedron or viceversa. 27 Regular Polyhedra and Symmetry

  28. The PlatonicSolids Procluswas a historianofmathematicsof the 5thcenturyB.C. and a discipleof the neo-Platonicphilosophy. HecreditedPythagoraswith the discoveryof the five regular polyhedra. The lackofsourcesthat can beattributedtoPythagorasmakesdifficulttodemonstratethisthesis, nevertheless in Plato’s Timeo , only a bit laterthanPythagoras, we can find a precise descriptionof the five regular bodies, i.e., the onlypossiblepolyhedracomposedofregular polygonshavingequalfaces, edges and solidangles. Platowillusethisextraordinarydiscoveryas a symbologyof the universe and ofitsbasicelements: fire(tetrahedron), earth(cube), air (octahedron) and water (icosahedron). The fifth regular polyhedron, the dodecahedron, wasassociatedbyAristotleto the aithêr(ether) or quintessencethatpermeates the wholeuniverse. Thismetaphorehas some mathematicalsense, giventhat the only regular polyhedronwhereitispossibletoinscribe the remainingfourisindeed the dodecahedron. 28 Regular Polyhedra and Symmetry

  29. Melancolia I • A wellknownexampleof interest for the solids can befound in the work of the German Renaissance master AlbrechtDürer. • Among the elementsappearing in his 1514 engravingMelancolia I , thereis a solidthatinterested, and stillinterests, artists and scientists, evenifitisnot easy tounderstand the prospecttve the solidhasbeenrepresentedwith. • The geometry and architectureinstrumentslie on the groundsurrounding the musive figure of the Melanconia, unused. The scale pans are empty, none climbs the ladder, the sleeping greyhoundisstarving, the wingedcherubiswaitingtobedictated, while the passingtimeisshownby the hourglass. • The sphere and the truncatedtetrahedronsuggest the mathematicalbasisofarchitecture. The scene appearstobesoffusedwithmoon light. The moonrainbow, perhapsincluding a comet, can symbolise the hope in a better mood. 29 Regular Polyhedra and Symmetry

  30. Magic Squares On the right upper part ofDürer’s engravingone can see a magicsquare, whichisconsideredoneof the first examplesofsuchanobject in the Western world. Activity 6a Find out allsymmetries in Dürer’s magicsquare! 30 Regular Polyhedra and Symmetry

  31. Symmetries in Dürer’s MagicSquare There are alsomanygroupsoffourcellswhichalways sum to34 The squareissaidtobesymmetricaseachcell and itssymmetriconewithrespectto the centrealways sum to 17. The columns The rows The principaldiagonals The four corner cells The 4 centralcells The 2x2 squares in the corners. Finally, Dürermanaged the square in such a way that the twocentralcellsof the bottomrowshow the yearof the engraving (1514). 31 Regular Polyhedra and Symmetry

  32. SymmetryGroups • Here by a symmetry of a figure we mean any rigid transformation which takes the figure into itself, i.e., an Euclidean isometry. • In mathematics, the concept of symmetry is studied with the notion of a mathematical group. Every polyhedron has an associated symmetry group, which is the set of all transformations which leave the polyhedron invariant. • Possible symmetries are rotations, reflections, translation, and their compositions. Translations are meaningless for regular polyhedra, as they have a centre (of the inscribed or circumscribed sphere) that must remain fixed. • The order of the symmetry group is the number of symmetries of the polyhedron. One often distinguishes between the full symmetry group, which includes rotations and reflections, and the proper (or chiral) symmetry group, which includes only rotations. 32 Regular Polyhedra and Symmetry

  33. Regular Polyhedra and Symmetry The symmetry groups of the Platonic solids are known as polyhedral groups. There are only three symmetry groups associated with the Platonic solids rather than five, since the symmetry group of any polyhedron coincides with that of its dual. The three polyhedral groups are: • the tetrahedral group T; • the octahedral group O(which is also the symmetry group of the cube); • the icosahedral group I (which is also the symmetry group of the dodecahedron). All Platonic solids except the tetrahedron are centrally symmetric, meaning they are preserved under reflection through the origin.

  34. Activity 7 Mutualdictation Symmetrygroups 34 Regular Polyhedra and Symmetry

  35. The Octahedral Group • The cube and the octahedronbelongtothisgroup. Let’s consider the rotationsof a cube. • Any rotation willoccuraroundanaxispassingthrough the centreof the solid; thislinewillintersect the cube in twopoints, A and B. • Toleave the cubeunchanged, the points A and B can onlylie on the followingpositions: 35 Regular Polyhedra and Symmetry

  36. Opposedvertices Centresoftwoopposedfaces Middle pointsoftwoopposededges 36 Regular Polyhedra and Symmetry

  37. Todetect the reflections, onemustfind the symmetryplanesof the cube Planes parallel totwoopposedfaces Planescontainingtwoopposededges 37 Regular Polyhedra and Symmetry

  38. Activity 8 Count the symmetries: - Ss form 5 groups; - Ss must find out the numbers of rotations and reflections of the octahedral group. - A volunteer for each group reads the types and number of symmetries; - T check out the answers. 38 Regular Polyhedra and Symmetry

  39. Opposedvertices • There are only 4 linesofsuch a kind. • Foreachlinethere are tworotations, otherthanidentity, thatleave the cubeinvariant: oneof 120° and oneof 240°. • In total, 8 rotations. Centresoftwoopposedfaces • Theselines are 3 • Foranyofthem, the cubeisleftunchangedrotatingof 90°, 180°, 270°. • In total, 9 rotations. • Middle pointsoftwoopposededges • There are 6 linesofthistype. • Rotatingaroundeachofthemof 180°, the cubemapsontoitself. • In total, 6 rotations. So wehave 24 rotations, includingidentity 39 Regular Polyhedra and Symmetry

  40. Todetect the reflections, onemustfind the symmetryplanesof the cube, that are nine in total: • Three planes parallel totwoopposedfaces • Sixplanescontainingtwoopposededges Wehave 9 reflections in total 40 Regular Polyhedra and Symmetry

  41. RotationalReflections Regular Polyhedra and Symmetry Wemustalso account foranypossiblecompositionof a reflectionwith a non-identical rotation thatleaves the cubeunchanged, excluding the equivalentcases. The first ofsuchsymmetriesiscalledantipodalapplication, and itis the symmetrywithrespectto the centre. Thissymmetrymapsanyvertexontoitsopposedone, and isobtainedbycomposing a reflectionofanysymmetryplanetrough the centrewith a rotation of 180°, whoseaxisisorthogonalto the selectedplane. Everyotherrotationalreflectionmustbelookedforstartingfrom the alreadyknowngrouprotations, butleaving out allthoseof 180°, astheircompositionwith a reflectionwillleadagainto the antipodalapplication.

  42. Accounting forantipodalapplication, wehave 15 rotationalreflections Regular Polyhedra and Symmetry Axisjoiningtwoopposedvertices (v, v’): • the cubecuts the reflectionplanewith a regular hexagonwhoseverticeslie on the middle pointsof the sixedgesnotadjacenttov, v‘; • the possiblerotations are of 120° and 240°; • 8 symmetriesforthis case. Axisjoining the middle points of two opposed edges: • The onlypossible rotation spans 180°, henceitisdiscarded. Axisjoining the centres of two opposed faces: • the reflectionplaneisparallelto the selectedfaces; • not accounting for the oneof 180°, wehave the rotationsof 90° and 270°; • 6 symmetriesforthis case.

  43. Orders of the Octahedral Group Regular Polyhedra and Symmetry Resuming: • 24 rotations • 9 reflections • 15 rotationalreflections Thatmakes 48 possiblesymmetries, thatis the orderof the full octahedralgroupOh. Not accounting forreflections, wehave the orderof the properoctahedralgroupO, thatis 24.

  44. The Icosahedral Group Let’s considerall the possiblerotations: Regular Polyhedra and Symmetry Axisjoiningtwoopposedvertices • Withrespecttotheselines, the rotationstaking the icosahedronintoitself are the multiplesof 360°/5, namely, 72°, 144°, 216° and 288°. • Theseaxis are six, hencewehave 24 rotations.

  45. Axisjoining the centresoftwoopposedfaces • Theselines are 10 • Foranyofthem, only the rotationsof 120° and 240° take the solidintoitself. • Thisleadsto 20 rotations. • Axisjoining the middle pointsoftwoopposededges • Foranyofthese 15 lines, thereisonlyone rotation otherthan the identitymapping the solidontoitself, thatis a rotation of 180°. • Thereforewehave 15 rotations. • So wehave 60 rotations, includingidentity 45 Regular Polyhedra and Symmetry

  46. Orders of the Icosahedral Group Regular Polyhedra and Symmetry One can demonstratethatreflections and rotationalreflectionsthatmap the icosahedronontoitself are 60. The orderof the full icosahedralgroupIh (aswellasdodecahedral) istherefore 120. Not accounting forreflections, wehavethat the orderof the propericosahedralgroupIis 60.

  47. The Tetrahedral Group • Now the situation isdifferent, asthere are neitherverticesnoropposedfaces; therefore, Tdoesnotcontain the antipodalapplication. • Let’s consider the rotations: • Axisjoining the middle points of two opposed edges: • The solidreturnstoitselffor a rotation of 180°. • There are 3 ofsuchlines, so 3 rotations. • Axisjoining the centre of a face to the opposed vertex: • The tetrahedronreturnstoitselfforrotationsof 120° and 240*. • There are 4 ofsuchlines, so 8 rotations. 47 Regular Polyhedra and Symmetry

  48. Reflections Rotationalreflections Any rotation axisjoins the middle pointsoftwooppositeedges. The reflectionplaneisorthogonaltothatlinethrough the centreof the tetrahedron. There are threeaxis and the rotation can beof 90° or 270°. Wehave 6 rotationalreflexions. • Anysymmetryplaneisdetectedbyanedge and the middle pointof the opposededge. • Theseplanes are six. 48 Regular Polyhedra and Symmetry

  49. Orders of the Tetrahedral Group Regular Polyhedra and Symmetry The orderof the full tetrahedralgroupTdistherefore 24. Not accounting forreflections, wehavethat the orderof the propertetrahedralgroupTis 12.

  50. Symmetry Properties of Platonic Solids Regular Polyhedra and Symmetry It is of interest to note that the orders of the proper are precisely twice the number of edges in the respective polyhedra (12, 24 and 60, respectively). The orders of the full symmetry groups are twice as much again (24, 48 and 120). The symmetry groups listed in the next table are the full groups with the rotational subgroups given in parenthesis (likewise for the number of symmetries). The Schläfli symbol shows the couple { p, q } where p is the number of edges for each face, and q the number of faces meeting at the same vertex.

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