Generating Polygons with Triangles - Discrete and Computational Geometry and Graphs - page 112

Generating Polygons with Triangles

B

T. Kuwata 1(

) and H. Maehara 2

1 Tokai University, Kanagawa, Japan
[e-mail]
2 Ryukyu University, Okinawa, Japan

Abstract. A set of triangles F is said to generate a polygon P if a
homothetic transform ʻP of P can be dissected into triangles each con-
gruent to a triangle in F . The simplicial element number of a polygon
P is defined to be the minimum cardinality of a family F of triangles
that can generate P . The simplicial element number of a set of poly-
gons P 1 ,P 2 ,...,P k is defined to be the minimum cardinality of a family
F of triangles that can generate all P 1 ,...,P k . In this paper, we con-
sider simplicial element numbers for several set of regular polygons and
generating relations among triangles.

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Keywords: Original triangle Terminal triangle Intermediate trian-

· ·

gle Simplicial element number Generating chain

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Mathematical subject classification (2010): 52B45 52C20

1 Introduction

A finite set of simplices is said to tile a polytope P ,if P can be represented

as the union of a number of simplices ˄ 1 ,˄ 2 ,...,˄ N such that the interiors of
these ˄ i are mutually disjoint, and each ˄ i is congruent to a member of

F

F .Aset

of simplices F is said to generate a polytope P , written as Fₒ P ,if F tiles

a polytope that is similar to P . For a simplex ˃ , we write ˃ ₒ P instead of

{ P .

The simplicial element number of a polytope P , denoted by e ( P ), is defined
to be the minimum cardinality of a family

˃ }ₒ P . Note that ( ˃ ₒ ˄ ) ∧ ( ˄ ₒ P ) implies ˃ ₒ

of simplices that can generate
P . The simplicial element number of a set of polytopes P 1 ,P 2 ,...,P k , denoted
by e ( P 1 ,P 2 ,...,P k ), is defined to be the minimum cardinality of a family

F

F of

simplices that can generate all P 1 ,...,P k . For example, e ( { 3 } , { 6 } ) = 1, where

{ p } denotes the regular p -gon. It is obvious that the inequality e ( P 1 ,...,P k ) ≤

e ( P 1 )+ + e ( P k ) holds. If equality holds, then P 1 ,...,P k are said to be

independent .
We denote ˃

···

∼ ˄ if two simplices ˃ and ˄ are similar. For a simplex ˃ , define

as follows:

⊧
⊨

original

if ˄

ₒ ˃ implies ˄ ∼ ˃,

˃ is

terminal if ˃ ₒ ˄ implies ˄ ∼ ˃,

⊩

intermediate otherwise.