Regular & semi-regular solidsARCHIMEDEAN vs PLATONIC • A convex semi-regular polyhedron •...

Regular & semi-regular solids

WEEK 7

THIS WEEK’S FOCUS

• Vertices, faces andedges

• The duals of thePlatonic solids

• Archimedean solids• Regular prisms and

anti-prisms

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

THE DUAL OF A CUBE

• The dual of a Platonicsolid can beconstructed by firstidentifying the centerof every face of theplatonic solid.

• 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

• The dual of a Platonicsolid is then formed.

• The number of facesand vertices isinterchanged, whilethe number of edgesstays the same.

DUAL OF CUBE: OCTAHEDRON

DUAL OF THE CUBE

• The cube and theoctahedron form adual pair.

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!!

DUAL OF THE DODECAHEDRON

THE ICOSAHEDRON

THE TETRAHEDRON: SELF-DUAL

THE TETRAHEDRON: SELF-DUAL

ARCHIMEDEAN SOLIDS

• A convex semi-regular polyhedron

• Has regular polygonalsides of two or moretypes that meet in auniform patternaround each corner.

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

THE CUBOCTAHEDRON

• The vertexconfiguration– (3.4.3.4)

THE ICOSIDODECAHEDRON

• The vertexconfiguration– (3.5.3.5)

THE RHOMBICOSIDODECAHEDRON

• The vertexconfiguration– (3.4.5.4)

THE RHOMBICUBOCTAHEDRON

• The vertexconfiguration– (3.4.4.4)

THE TRUNCATED CUBOCTAHEDRON

• The vertexconfiguration– (4.6.8)

THE TRUNCATED ICOSIDODECAHEDRON

• The vertexconfiguration– (4.6.10)

THE SNUB CUBE

• The vertexconfiguration– (3.3.3.3.4)

THE SNUB DODECAHEDRON

• The vertexconfiguration– (3.3.3.3.5)

THE TRUNCATED CUBE

• The vertexconfiguration– (3.8.8)

THE TRUNCATED DODECAHEDRON

• The vertexconfiguration– (3.10.10)

THE TRUNCATED ICOSAHEDRON

• The vertexconfiguration– (5.6.6)

THE TRUNCATED OCTAHEDRON

• The vertexconfiguration– (4.6.6)

THE TRUNCATED TETRAHEDRON

• The vertexconfiguration– (3.6.6)

PRISMS

• In geometry, an n-sided prism is apolyhedron made ofan n-sided polygonalbase, a translatedcopy, and n facesjoining correspondingsides.

PRISMS

• The joining facesare parallelograms.

• All cross-sectionsparallel to the basefaces are the same.

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.

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

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.

ANTIPRISMS

ANTIPRISMS