Mathematics Interdisciplinary Research
Volume:4 Issue: 1, Spring 2019

The connection between architecture and science and sound based on mathematical relations has continued to develop[ since the rise of the Western classical civilization that originated in Ancient Greece. The mysterious Pythagorean cosmology pursued as secret esoteric knowledge was related to the search of rhythm, proportionality and harmony. Even somewhat earlier, Greek mysteries were based on a concord of music and form. This line of reasoning can be raced as early as when the doctrines of Orphism emerged in early Greece to be followed by the concepts of Pythagoras and his followers and eventually the philosophical school of Neo-Platonists. Early medieval thinkers like St. Aurelius Augustine and Boethius revived and continued this ancient tradition; they sustained and developed further the ideas of dependence between architecture and music (as well as mathematics). Their ideas were further elaborated by later Christian thinkers. Architectural principles practiced by architects belonging to the Western tradition were passed further on. The Pythagorean tradition was still alive during the Renaissance and even baroque. This tradition was gradually marginalized and forgotten with the rise of scientific mentality developed in post-Renaissance era. However, the roots of the application of mathematics and geometry to the design of urban settlements have survived. Such principles can be still observed while studying the early patterns of western as well as non-Western civilizations, and thus one can speak about the universal mathematical geometric character of early urban design.

In this paper, we study Ghiyath al-Din Jamshid al-Kashi's (1380-1429 A.D.) main mathematical achievements. We discuss his al-Risala al-muhitiyya ("The Treatise on the Circumference"), Risala al-watar wa'l-jaib ("The Treatise on the Chord and Sine"), and Miftah al-hisab ("The Key of Arithmetic"). In particular, we look at al-Kashi's fundamental theorem, his calculation of pi, and his calculation of sin1°.

Geometry is a base tool for establishing unity in Iranian architecture and is always considered by architects due to the discipline and rule that gives architecture to architecture. The architecture of the house in terms of its specific functional role, sought to adapt the geometrical principles to the best possible shape and achieve the proper understanding of the proportions and proportions of a harmonious geometry. In shaping the architecture of the Iranian house, the geometrical and functional role of the courtyard is important. This study is conducted through field study of 20 historical houses in Kashan to discover how geometry is used in the architectural design of Kashan historical houses. In this regard, the proportions of the arena and the court in these cells have been compared. The result shows that the yard follows the golden ratio as the house design basis. The 1.414 and 1.618 ratios have the highest frequency in the study samples. In addition, in terms of the level of occupation, the courtyard has occupied 20 to 40 percent of the building.

‎In this work‎, ‎a regular polygon with $n$ sides is described by a periodic (circular) sequence with period $n$‎. ‎Each element of the sequence represents a vertex of the polygon‎. ‎Each symmetry of the polygon is the rotation of the polygon around the center-point and/or flipping around a symmetry axis‎. ‎Here each symmetry is considered as a system that takes an input circular sequence and generates a processed circular output sequence‎. ‎The system can be described by a permutation function‎. ‎Permutation functions can be written in a simple form using circular indexation‎. ‎The operation between the symmetries of the polygon is reduced to the composition of permutation functions‎, ‎which in turn is easily implemented using periodic sequences‎. ‎It is also shown that each symmetry is effectively a pure rotation or a pure flip‎. ‎It is also explained how to synthesize each symmetry using two generating symmetries‎: ‎time-reversal (flipping around a fixed symmetry axis) and unit-delay (rotation around the center-point by $2pi‎ /‎n$ radians clockwise)‎. ‎The group of the symmetries of a polygon is called a dihedral group and it has applications in different engineering fields including image processing‎, ‎error correction codes in telecommunication engineering‎, ‎remote sensing, and radar‎.

‎Aghabozorg Mosque with massive brick dome and the tiled minaret is one of the most magnificent Islamic buildings in Kashan in the Qajar period‎. ‎The unique features of the architecture suggest the architects of this building‎, ‎in terms of the nature of architecture in designing the form and architectural space‎, ‎have considered principles that are based on geometric shapes and proportions between them‎. ‎Regarding the importance of the issue‎, ‎the authors‎, ‎with the approach of geometric proportional analysis‎, ‎seek to answer this question‎: ‎what is the role of geometry and golden proportion in the construction and shaping of the elements of Aghabozorg School-Mosque? For this purpose‎, ‎the descriptive-analytic research method has been used in this study‎. ‎In order to retrieve geometric and proportional data‎, ‎plan‎, ‎elevations‎, ‎and sections of the building were investigated and analyzed accurately‎. ‎The results of this research indicate that the architects had the necessary knowledge about the systems of equations and geometric drawings and used golden proportions and circle divisions to design plan‎, ‎elevations‎, ‎and sections and also applied geometrical knowledge in the direction practical and qualitative for creating the building‎.

Architectural science requires interdisciplinary science interconnection in order to improve this science. Graph theory and geometrical fractal are two examples of branches of mathematics which have applications in architecture and design. In architecture, the vertices are the rooms and the edges are the direct connections between each two rooms. The independence polynomial of a graph G is the polynomial I(G,x)=∑ ikxk, where ik denote the number of independent sets of cardinality k in G. The independence fractal of G is the set I(G)=limk→∞ Roots (I({Gk},x)-1),  where Gk=G[G[...]], and G[H] is the lexicographic product for two graphs G and H. In this paper, we consider graphical presentation of a ground plane as a graph G and use the sequences of limit roots of independence polynomials of Gk to present some animated structures for building.

‎In simple terms‎, ‎architecture and music are two very different things‎, ‎which is the product of one set of materials in the form of one building and the other product is a set of sounds in the form of a song and melody‎. ‎In this sense‎, ‎architecture and music are two separate issues that are not similar‎. ‎But with a little care in the hidden layer of music and architecture we can find amazing similarities‎. ‎Finding the structural‎, ‎quantitative and qualitative similarities between architecture and music is the focus of this article‎. ‎Achieving a qualitative and quantitative correlation between these two arts can be a way to aesthetically improve architecture and achieve its healing principles‎. ‎Therefore‎, ‎the main purpose of this study is the understanding of the immediate beauty of music in mind and the application of its aesthetic elements in architecture‎. ‎The research method is descriptive-analytical study of library documents and case studies‎. ‎In this article‎, ‎first‎, ‎the definitions and the general structure of music and architecture are presented‎, ‎then the results of the studies are introduced in the form of qualitative and quantitative adaptive tables‎. ‎Finally‎, ‎by analyzing data‎, ‎the common structure of music and architecture is determined‎.

The subject of proportion used in architecture -and on a larger scale in any art work- is a debate having a special status in analytical studies on the history of art. “Golden Section” is known as one of the major topics of such debates. It has been given different names during the history of art, and it is generally defined under the ancient and Renaissance art in West. The present paper studies the status of this specific kind of proportion in Islamic civilization. The authors also demonstrate their findings about the manifestation of this proportion in Hasht Behesht Palace, Isfahan. Moreover, they attempt to indicate the roots of familiarity with the usage of respective proportion in Muslim world by referring to some first-hand references of Islamic civilization in the fields of mathematics and geometry. The findings of this paper show that the application of Golden Section in Islamic civilization was independent of the developments of Western Renaissance and Golden Section had practical theorems in Islamic civilization since 10th and 11th centuries AD.