1. 12. Polyhedra

2. 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

3. 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

4. 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

5. 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

6. 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)

7. 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)

8. 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 )
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 )

9. 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 )

10. Semiregular Polyhedra (Examples)
Archimedean semiregular polyhedra

11. 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

12. Regular polygon and star polygon

13. Regular polygon and star polygon

14. 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

15. Nets
Nets (fig )
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

16. 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)

17. 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 – =2
a octahedron  6 – = 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

18. 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}

19. 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
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)

20. 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

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

22. 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