12. Polyhedra 2005. 6

Polyhedra All possible polyhedra are defined by only the three simplest of geometric elements (points, lines, and planes) Contents Regular polyhedra Semi-Regular polyhedra Dual polyhedra Star Polyhedra Nets The convex Hull of a polyhedron Euler’s Formula The Connectivity Matrix

Definition Polyhedron A multifaceted 3D solid bounded by a finite connected set of plane polygons Every edge of each polygon belongs to one other polygon The polygon faces form a closed surface, dividing space into two regions The interior of the polyhedron and the exterior All face of a polyhedron are plane polygons All its edges are straight line segments Each polyhedral edge is shared by exactly two polygonal faces Simplest possible polyhedron (fig. 12.1) tetrahedron(사면체) with 4 faces

Definition 3 geometric elements define all polyhedra in space Vertices(V), edges(E), and faces(F) Each vertex is surrounded by an equal number of edges and faces Each edge is bounded by two vertices and two faces Each face is bounded by a closed loop of coplanar edges that form a polygon Half-planes: any straight line in the plane divides the plane into two half planes Dihedral angle: Angle b/w faces that intersect at a common edge Two half planes extending from a common line form a dihedral angle

Definition Polyhedral angle Three or more planes intersecting at a common point form a polyhedral angle The common point is the vertex of this angle The intersection of the planes are the edges of the angle The parts of the planes lying b/w the edge are the faces of the angle face angle of the polyhedral angle The angle formed by adjacent edges For any polyhedral angle There is an same number of edges, faces, face angles, and dihedral angles Ex) Cube: trihedral angle  a polyhedral angle with 3 faces

Definition Comparison An angle of 360o surrounds a point in the plane The sum of the face angles around a vertex of a polyhedron Angular deficit (결손): defined as difference b/w the sum of the face angles surrounding the vertex and 360o Total angular deficit : the sum of the angular deficits over all the vertices of a polyhedron The smaller the angular deficit, the more sphere-like the polyhedron Regular polyhedron (=simple polyhedron) is homeomorphic to a sphere Homeomorphics If their bounding surfaces can be deformed into one another without cutting or gluing (= They are topologically equivalent)

The regular Polyhedra The regular Polyhedra A convex polyhedron is a regular polyhedron if the following condition are true All face polygons are regular Equal edge and interior angles All face polygons are congruent(=identical) All vertices are identical All dihedral angles are equal Ex) Cube: All its face are identical All its edge are of equal length In 3D space we can construct only 5 regular polyhedra Tetrahedron(4면체), hexahedron(=cube) (6면체), octahedron(8면체), dodecahedron,(12면체) icosahedron(20면체)

The regular Polyhedra The sum of all face angles The sum of all face angles at a vertex of a convex polygon is always less than 2PI Otherwise If the sum of the angles = 2PI, then the edges meeting at the vertex are coplanar If the sum of the angles > 2PI, then some of the edges at vertex are reentrant(오목한) and the polyhedron is concave Characteristic properties of the 5 regular polyhedra (Table 12.1-2) e: the length of the edge, RI: the radius of the inscribed sphere RC: the radius of the circumscribed sphere Theta: dihedral angle Vertex coordinate for each of the regular polyhedra (Table 12.3-7)

Semiregular Polyhedra If we relax condition 2 and 4 All face polygons are congruent(=identical) All dihedral angles are equal Infinite number of polyhedra is possible Archimedean polyhedron (13개) Faces are regular polygons and equilateral angle If we relax condition 1 and 3 All face polygons are regular All vertices are identical Another Infinite set of polyhedra is possible If we appropriately truncate the five regular polyhedra Generate all the semiregular polyhedra exept 2 snub form (Figure 12.4-5)

Semiregular Polyhedra (Examples) Archimedean semiregular polyhedra

Dual Polyhedra Dual Polyhedra Two polyhedra are dual If the vertices of one can be put into a 1-to-1 correspondence with the center of the faces of the other If we connect the centers of the faces of one of them with line segments, we obtain the edges of the other The number of faces of one becomes the number of vertives of the other Total number of edges does not change Ex) The octahedron and cube are dual. (Table 12.9) icoshedron and dodecahedron, tetrahedron is self dual

Regular polygon and star polygon

Regular polygon and star polygon

Star Polyhedra Star Polyhedra If we extend the edges of a regular polygon with five or more edge it will enclose additional region of the plane and form a star or stellar polygon This does not work for cubes Their faces interpenetrate They are not topologically simple Euler’s Formula dose not apply

Nets Nets (fig. 12.6-7) By careful cutting and unfolding, we can open up and flatten out a polyhedron  a net of the polyhedron It lies in a plane No single, unique net for a particular polyhedron

The convex Hull of a Polyhedron A polyhedral convex hull is a 3D analog of the convex hull for a polygon The convex hull of a convex polyhedron Identical to the polyhedron itself The convex hull of a concave polyhedron By wrapping it in a rubber sheet (Figure 12.8)

Euler’s Formula for Simple Polyhedra Euler’s Formula (fig. 12.9) V – E + F = 2 Vertices (V), Edges (E), Faces (F) Ex) a cube  8 – 12 + 6 =2 a octahedron  6 – 12 + 8 = 2 All faces must be bounded by a single ring of edges, with no holes in the faces The polyhedron must have no holes through it Each edge is shared by exactly two faces and is terminated by a vertex at each end At least three edges must meet at each vertex

Euler’s Formula for Simple Polyhedra Ludwig Schlafi’s formula Euler’s formula is only a special case of this formula 1. An edge, or one-dimensional polytope, has a vertex at each end: N0 = 2 2. A polygonal face, or two-dimensional polytope, has as many vertices as edges: N0 – N1 = 0 3. A polyhedron, or three-dimensional polytope, satisfies Euler formula: N0 – N1 + N2 = 2 4. Four-dimensional polytope satisfies N0 – N1 + N2 – N3 = 2 5. Any simply-connected n-dimensional polytope satisfies N0 – N1 + … + (-1)n-1Nn-1 = 1 – (-1)n polytope is the general term of the sequence-point, segment, polygon, polyhedron, and so on. Also he invented the symbol {p, q} for the regular polyhedron whose face are p-gons, q meeting at each vertex, or the polyhedron with face {p} and veretx figure {q}

The Connectivity Matrix A two-dimensional list or table that describes how vertices are connected by edges to form a polyhedron square matrix with as many rows and columns as vertices Symmetric matrix about its main diagonal, which is comprised of all zeros (twice as much information as necessary) If element aij = 1, then vertices I and j are connected by an edge If element aij = 0, then vertices I and j are not connected Ex) fig 12.13-14 What do we do about the faces?? Form a matrix with each row containing the vertex sequence bounding a face (counterclockwise order  outward from the interior of the polyhedron)

Halfspace Representation of Polyhedra Represent a convex polyhedron with n faces by a consistent system of n equations constructed as follows Aix+ Biy + Ciz + Di >0 Any point that satisfies all n inequalities lies inside the polyhedron Ex) a cube : x > 0 -x + 4 > 0 y > 0 -y + 4 > 0 z > 0 -z + 4 > 0

Halfspace Representation of Polyhedra Definition of the polyhedron P Ex) four-side polyhedron (figure 12.15)

Maps of Polyhedra Schlegel diagram or map A special two-dimensional image of a polyhedron Projecting its edges onto a plane from a point directly above the center of one of its face Tetrahedron Hexahedron (=cube) octahedron dodecahedron icosahedron