Golden Ratio, the Phi, and Its Geometrical Substantiation A study on the Golden Ratio, Dynamic Rectangles and Equation of Phi Md. Akhtaruzzaman 1* and Amir A. Shafie 2 Dept. of Mechatronics Engineering International Islamic University Malaysia (IIUM) 53100 Kuala Lumpur, Malaysia akhter900@yahoo.com 1 ; aashafie@iium.edu.my 2 Sharif M. Raihan 3 and M. K. Hasan 4 Dept. of Electrical and Communication Engineering International Islamic University Malaysia (IIUM) 53100 Kuala Lumpur, Malaysia drraihan@iium.edu.my 3 ; hasankamrul@msn.com 4 Tanveer Ahsan 5 and M. S. Alam 6 Department of Computer Science and Engineering International Islamic University Chittagong (IIUC) 154/A College Road, Chittagong-4203, Bangladesh tanveer.ahsan@gmail.com 5 ; alam_cse@yahoo.com 6 Md. Bahlul Haider 7 Department of Computer Science University of Western Ontario Ontario N6A 3K7, Canada mhaider5@uwo.ca 7 Abstract— Golden ratio is often denoted by the Greek letter, usually in lower case, Phi (φ) which is an irrational mathematical constant, approximately 1.6180339887. Because of its unique and interesting properties, many mathematicians as well as renaissance artists and architects studied, documented and employed golden section proportions in remarkable works of sculpture, painting and architecture. Robot sizing especially for the Humanoid Robot, Phi is considered as the key to achieve the human friendly look. The ratio also plays an enigmatic role in the geometry and mathematics. The basic concept of golden ratio and its relation with the geometry are represented and described in this paper. The paper also explains about the structure and construction strategies of various dynamic rectangles by establishing some relations and dependencies with each other. The main contribution of the paper is to study about the validation and substantiation of the Equation of Phi based on classical geometric relations. The technique can be considered as an interesting strategy to prove the Equation of Phi. Keywords- Golden Ratio; Golden Section; Golden Mean; Golden Spiral; Equation of Phi; Dynamic rectangle. I. INTRODUCTION The golden section proportions are found in natural world through human proportions and the growth patterns of many living plants, animals, and insects and also in the universe [1] [5]. Basically golden ratio is always considered as the most pleasing proportion to human eyes and many work has been done, also going on where the ratio concept are analyzed. Human detection, human face detection and recognition, emotion detection, beauty detection of an image, biological inspired robot structure design, locomotion analysis of human and animals are some of the fields where the golden section proportion are being used [5]. The interrelation between proportion and good looks has made a lot of discussion in science because of the accidental occurrence of the shapes in various designs of objects like books, paintings, edifices and so on. The designs are approximated by a rectangle shaped such that the ratio of its length and height is equal to the Golden Ratio, φ = (1+5 1/2 )/2 = 1.6180339887 (approx.). The ‘φ’ also called the divine proportion, golden section, golden cut, golden ratio, golden mean etc. which is the result of dividing a segment into two segments (A+B) such that A/B=(A+B)/A=1.6180339887 (approx.) [1] [2] [4] [5] [6]. II. GOLDEN RATIO AND ITS CHRONICLE The natural proportioning system provides the foundation of the work of many artists and designers. The proportion known as the Golden Mean has always existed in mathematics and in the physical universe and it has been of interest to mathematicians, physicists, philosophers, architects, artists and even musicians since antiquity. According to the great German mathematician Johannes Kepler (December 27, 1571 - November 15, 1630), geometry has two great treasures, theorem of Pythagoras and the division of a line into extreme and mean ratio [2]. In the early days of the 19 th century it was suggested that the Greek letter ‘φ’ (Phi), the initial letters of Phidias’s name, should be adopted to designate the golden ratio [2]. In the medieval age and during the Renaissance, the ubiquity of ‘φ’ in mathematics aroused the involvement of many mathematicians. It is unknown exactly when the idea was first discovered and applied by mankind. The presence of both Pi and Phi in the design of the Pyramids represents that the Egyptians were aware of the ratio. The Greeks based the design of the Parthenon (example of Doric architecture, the main temple of the goddess Athena built more than 400 years BC) on this proportion. Phidias, a Greek sculptor and mathematician (490 - 430 BC), studied Phi and applied it to the design of sculptures for the Parthenon [2]. Plato (427 - 347 BC), in his views on natural science and cosmology presented 2011 IEEE Student Conference on Research and Development (SCOReD) 978-1-4673-0102-2/11/$26.00 ©2011 IEEE 326