Regular & semi-regular solids -...
Transcript of Regular & semi-regular solids -...
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Regular & semi-regular solids
WEEK 7
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THIS WEEK’S FOCUS
• Vertices, faces andedges
• The duals of thePlatonic solids
• Archimedean solids• Regular prisms and
anti-prisms
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DUAL OF A POLYHEDRON
• By the duality principle, for everypolyhedron, there exists anotherpolyhedron in which faces and polyhedronvertices occupy complementary locations.
• This polyhedron is known as the dual, orreciprocal. The process of taking the dualis also called reciprocation
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THE DUAL OF A CUBE
• The dual of a Platonicsolid can beconstructed by firstidentifying the centerof every face of theplatonic solid.
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• If two faces of theplatonic solid, P,share an edge,connect the centerpoints of those twofaces; these are theedges of its dual, P*.
THE DUAL OF A CUBE
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• The dual of a Platonicsolid is then formed.
• The number of facesand vertices isinterchanged, whilethe number of edgesstays the same.
DUAL OF CUBE: OCTAHEDRON
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DUAL OF THE CUBE
• The cube and theoctahedron form adual pair.
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DUAL OF OCTAHEDRON: CUBE
• The eight 3-sidedfaces of theoctahedron becomethe eight corners ofthe cube with 3 facesmeeting at each.
• Also observe that thetotal number of edgesremains unchanged. Exaggerated Illustration
The vertices of the cube should lie onthe plane of the triangular face!!
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DUAL OF THE DODECAHEDRON
THE ICOSAHEDRON
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THE TETRAHEDRON: SELF-DUAL
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THE TETRAHEDRON: SELF-DUAL
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ARCHIMEDEAN SOLIDS
• A convex semi-regular polyhedron
• Has regular polygonalsides of two or moretypes that meet in auniform patternaround each corner.
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ARCHIMEDEAN vs PLATONIC
• A convex semi-regular polyhedron
• Has regular polygonalsides of two or moretypes that meet in auniform patternaround each corner.
• A Platonic solid is aregular convexpolyhedron
• Has only one type ofpolygonal side
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THE CUBOCTAHEDRON
• The vertexconfiguration– (3.4.3.4)
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THE ICOSIDODECAHEDRON
• The vertexconfiguration– (3.5.3.5)
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THE RHOMBICOSIDODECAHEDRON
• The vertexconfiguration– (3.4.5.4)
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THE RHOMBICUBOCTAHEDRON
• The vertexconfiguration– (3.4.4.4)
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THE TRUNCATED CUBOCTAHEDRON
• The vertexconfiguration– (4.6.8)
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THE TRUNCATED ICOSIDODECAHEDRON
• The vertexconfiguration– (4.6.10)
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THE SNUB CUBE
• The vertexconfiguration– (3.3.3.3.4)
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THE SNUB DODECAHEDRON
• The vertexconfiguration– (3.3.3.3.5)
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THE TRUNCATED CUBE
• The vertexconfiguration– (3.8.8)
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THE TRUNCATED DODECAHEDRON
• The vertexconfiguration– (3.10.10)
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THE TRUNCATED ICOSAHEDRON
• The vertexconfiguration– (5.6.6)
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THE TRUNCATED OCTAHEDRON
• The vertexconfiguration– (4.6.6)
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THE TRUNCATED TETRAHEDRON
• The vertexconfiguration– (3.6.6)
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PRISMS
• In geometry, an n-sided prism is apolyhedron made ofan n-sided polygonalbase, a translatedcopy, and n facesjoining correspondingsides.
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PRISMS
• The joining facesare parallelograms.
• All cross-sectionsparallel to the basefaces are the same.
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ANTIPRISMS
• A semi-regular polyhedron constructed fromtwo n-sided polygons and 2n triangles.
• An antiprism is like a prism in that it contains twocopies of any chosen regular polygon
• It is unlike a prism in that one of the copies isgiven a slight twist relative to the other.
• The polygons are connected by a band oftriangles pointing alternately up and down.
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ANTIPRISMS
• At each vertex, three triangles and one ofthe chosen polygons meet.
• By spacing the two polygons at the properdistance, all the triangles becomeequilateral.
• Antiprisms are named square antiprisms,pentagonal antiprisms, and so on.
• The simplest, the triangular antiprism, isbetter known as the octahedron
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ANTIPRISMS
• In geometry, there is an infinite setof antiprisms formed by an even-numbered sequence of triangle sidesclosed by two polygon caps.
• If faces are all regular, it is a semiregularpolyhedron.
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ANTIPRISMS
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ANTIPRISMS