Roofs and Graph-theoretical Trees

Edwin KOŹNIEWSKI

Białystok University of Technology

Poland

edwikozn@pb.bialystok.pl

Roofs shape the lansdcape

The paper is a continuation of research concerning with geometrical properties of roofs of buildings considered as a special class of polyhedral surfaces from the view point of Graph Theory.

The new concepts (dividing and grafting of trees, primitives of roofs, decomposition and joining of roofs) are introduced and appropriate properties are formulated and proved.

These concepts and properties lead to a description of shapes of regular roofs spread over simply connected v-gons for arbitrary integer v (v 3)

Classification of solutions of shapes of a regular roof generated by a simply connected v-gon for v=3,4,5,6,7,8

Student’s works – 2002-2004.

Civil Engineering Faculty, I semester

Bialystok University of Technology Poland

Fragment „Geometry of Roofs” of Exhibition „Mathematics for artists, mathematics for everyone”.

Bialystok University of Technology 2004., Poland””.

The regular roof generated by a 10-gon. Student’s work – 2004

Civil Engineering Faculty, I semester. Bialystok University of Technology, Poland

Roofs and their skeletons

The part regular roof (with court-yard) of the building of the Białystok University and the block of atrial buildings of the Theological Seminary:

We introduce two new operations in Graph Theory: dividing and grafting of trees.

The least nonelementary tree and roofs

The elementary trees of degree at most 3 and suitable basic (elementary) roofs

(A)

(B)

(C)

In a proof of this theorem we use the geometric operations:

„ccc – chamfering of a convex corner ”

„mcc – making a concave corner ”

Conformal transformations of roofs

scaling

Conformal transformations of roofs

translation

A1) twice used scaling

A2) dividing of the roof into three roofs A3) three elementary roofs

A4) creation of the primitives of roofs

A5) primitives of elementary roofs for the roof A

Thank you very much