Polar reciprocation Dual polyhedron

the duality of polyhedra defined in terms of polar reciprocation concentric sphere. here, each vertex (pole) associated face plane (polar plane or polar) ray center vertex perpendicular plane, , product of distances center each equal square of radius. in coordinates, reciprocation sphere

x

2


+

y

2


+

z

2


=

r

2


,


{\displaystyle x^{2}+y^{2}+z^{2}=r^{2},}

the vertex

(

x

0


,

y

0


,

z

0


)


{\displaystyle (x_{0},y_{0},z_{0})}

is associated plane

x

0


x
+

y

0


y
+

z

0


z
=

r

2




{\displaystyle x_{0}x+y_{0}y+z_{0}z=r^{2}}

.

the vertices of dual poles reciprocal face planes of original, , faces of dual lie in polars reciprocal vertices of original. also, 2 adjacent vertices define edge, , these reciprocate 2 adjacent faces intersect define edge of dual. dual pair of edges orthogonal (at right angles) each other.

if




r

0




{\displaystyle r_{0}}

radius of sphere, ,




r

1




{\displaystyle r_{1}}

,




r

2




{\displaystyle r_{2}}

respectively distances centre pole , polar, then:

r

1


.

r

2


=

r

0


2




{\displaystyle r_{1}.r_{2}=r_{0}^{2}}

for more symmetrical polyhedra having obvious centroid, common make polyhedron , sphere concentric, in dorman luke construction described below.

however, possible reciprocate polyhedron sphere, , resulting form of dual depend on size , position of sphere; sphere varied, dual form. choice of center sphere sufficient define dual similarity. if multiple symmetry axes present, intersect @ single point, , taken centroid. failing that, circumscribed sphere, inscribed sphere, or midsphere (one edges tangents) commonly used.

if polyhedron in euclidean space has element passing through center of sphere, corresponding element of dual go infinity. since euclidean space never reaches infinity, projective equivalent, called extended euclidean space, may formed adding required plane @ infinity . theorists prefer stick euclidean space , there no dual. meanwhile, wenninger (1983) found way represent these infinite duals, in manner suitable making models (of finite portion!).

the concept of duality here closely related duality in projective geometry, lines , edges interchanged. projective polarity works enough convex polyhedra. non-convex figures such star polyhedra, when seek rigorously define form of polyhedral duality in terms of projective polarity, various problems appear. because of definitional issues geometric duality of non-convex polyhedra, grünbaum (2007) argues proper definition of non-convex polyhedron should include notion of dual polyhedron.

canonical duals

any convex polyhedron can distorted canonical form, in unit midsphere (or intersphere) exists tangent every edge, , such average position of points of tangency center of sphere. form unique congruences.

if reciprocate such canonical polyhedron midsphere, dual polyhedron share same edge-tangency points , must canonical. canonical dual, , 2 form canonical dual pair.