# In the frame of the researches on dendriform

In the frame of the researches on dendriform structures, some researchers have attempted to explore the efficiency of tree׳s fractal-like branching structures with completely different angles. In a research-oriented ‘Fibrous Structures’ workshop hosted by the Istanbul Technical University in Istanbul in 2007, a group of students not only developed a concrete prototype of branching structure on top part to support a roof, but also constructed another but inverted branching structure at the bottom part as a network of foundation to distribute the load evenly to the ground, thus mimicked the concept of structural efficiencies of tree׳s branching stems on top and simultaneously its branching roots at ground for uniform load distribution (Figure 30b). In this workshop, students made a physical model based on the digital model of branching structure, and then after getting the structural feedback from physical model they modified the digital model for getting a final form; thus the interaction between the digital and physical modeling worked as a new approach of form finding process that had produced the most efficient and feasible structure at the end (Pasquero et al., 2007)

Conclusion
In this paper we have discussed how the underlying geometry of trees, possible to be explained by the concepts of fractals and non-Euclidean geometry, was used as an inspiration source by architects and constructors who employed the powerful properties of self-similarity as a way to improve and optimize their structural and architectural solutions. From the analysis of a set of dendriforms from historical to contemporary times, we have attempted to narrate the chemokine receptor antagonist of dendriforms design as a logical consequence of the understanding of intrinsic relations between forms and structures in trees and plants, and as an effect of the advancements in constructional, theoretical, graphical, technological and computational knowledges in different time periods (Table 1).

Acknowledgment
This work was supported by the Politecnico di Torino Ph.D. research grant, XXVII cycle (University Research Scholarship).

Introduction
The rectangular frame in the aforementioned problem is referred to as a rectangular floor plan denoted by . The given spaces represent the different elements of a building, e.g., rooms, offices, kitchens, bathrooms, and toilets.
An algorithm is a step-by-step procedure or formula that is used to solve a problem. The literature describes several algorithms for constructing floor plans. Galle (1981) proposed an algorithm for generating rectangular plans on modular grids to provide a large number of possible solutions. Lai (1988) used graph theory for floor plan design in a study where the rectangular dualization problem was reduced to a matching problem on bipartite graphs. The dual graph of a plane graph G has a vertex corresponding to each face of G and an edge joining two neighboring faces for each edge in G. A plane graph can be embedded in the plane, i.e., it can be drawn in such a way that no edges cross each other. A plane graph is termed rectangular if each of its edges is oriented in either the horizontal or the vertical direction, each of its regions has exactly four sides, and the whole graph can fit a rectangular enclosure. A bipartite graph is a graph whose vertices can be divided into two independent sets, A and B, such that every edge (a, b) connects either a vertex from A to B or a vertex from B to A. Stiny (2013) proposed the construction of floor plans using shape grammar. Shape grammar is a procedure for generating different geometric shapes. Terzidis (2007–08) developed a computer program called autoPLAN that generates architectural plans for the boundary and adjacency matrix of a given site.
Since ancient times, architectural forms composed of mathematical and geometrical relationships have generated great interest (Dabbour, 2012). In the present study, we propose an algorithm for the construction of a rectangular floor plan and use mathematical tools to prove that the obtained floor plan is optimally connected.