Some Research On Convex Polygon Tiling And Self-similar Tiling

These thesis includes two parts.In the first part we study such a tiling problem:can we dissect a square into some congruent convex polygons?In other words,we consider the possibility of the equation ?=?j=1N,where ??R2 is a square,P is a polygon with n vertex and such n sides,Pj is congruent with P for each j ? {1,2,…,N},n?4 is a positive integer,and the union in the right hand is measure-disjoint,i.e.,?(Pj?Pi)=0,where ? is the Lebesgue measure.We first find a necessary condition from the tiling equation,which can be used to deny the case q? 6 directly,and also can be used to find the necessary conditions of t he angles of P when q=5.This result tell us that to make sure the tiling equation holding,the angles of P just have five possibilities.which should be considered their side relations further more.We talk about the situation that P is a right angle trapezoid when q=4.The first part is an answer to the generalized Danzer's conjecture.In the second part we discuss the topological properties about self-similar tiles.We mainly talk about the topological properties,including connecting and the number of connected component,of self-similar tiles with product form digits.The main work of this thesis is spread in the following chapters:The second chapter includes some basic knowledges which we need.In the third chapter,we classify all the vertex of Pj's in the tiling equation according their degree.As a result,we find a necessary condition about the tiling equation.Especially,this condition cannot hold when q?6.In the fourth chapter,we first use the result of such a condition about q=5 to talking about the angles of P included in the tiling equation.The result,which obtained just through some combination discussions,tell us there are only five angles of P should be considered further.Next,we design a program to find the side relations of P when the angles of P(in clockwise or counter-clockwise)known.Finally when the angles and sides of P are known,it is easy to determine whether or not a square can be tiled by such a P.In the fifth chapter,we talk about the case when P is a right angle trapezoid.By talking about the hypotenuse graph about {Pj}j=1N,we show that N must be even when the tiling equation hold according the hypotenuse graph is Euler or not.This result shows the L.Danzer conjecture is true when P is a right angle trapezoid.In the sixth chapter,some necessary about the convex quadrilateral which satisfying the tiling equation are discuss.In the seventh chapter,we first introduce the basic properties of self similar tiles.And then we focus on the topological properties of self similar tiles with product form digits,mainly including the connecting and the number of connected components in R2.At last we simply talk about the properties of self similar tiles in R3,which is also the problem that the author want to study next.