Ancient Greek architects in the 5th century BC already knew the influence of this coordination. The Parthenon is an example of an early building that used a golden rectangle. At that time, the ancient Greeks already had the knowledge of the golden mean and how to make it, and they also knew how to approximate it and how to use it to construct a golden rectangle. It is not just a coincidence that the pronunciation of the golden section φ(phi) and the first three letters of the name are the same as those of the famous ancient Greek sculptor Pheidias. I believe that Pheidias used the golden section and the golden rectangle in his works. Since Pythagoras society can choose the five-pointed star as a symbol of rank, it is hard to say that it has nothing to do with Pheidias to use φ to represent the golden mean.
The golden rectangle not only affects architecture, but also appears in art. In the book Magic Proportion by L Pacilio in A.D. 1509, Leonardo da Vinci illustrated the golden mean in human body structure. The application of the golden mean in art is marked by vivid symmetry skills. A. Diu Lei, G. Cirette, P. Manzhoulian and Da Vinci.
A rectangle with the ratio of short side to long side of 0.6 18 is called golden section rectangle, also called golden section rectangle. The aesthetic feeling of the golden section rectangle lies in: firstly, the golden section rectangle has the aesthetic feeling of affirming the shape. Secondly, the golden section rectangle can produce unique rhythmic beauty visually.
The rest of a golden rectangle except the square is still a reduced golden rectangle, which can be divided into infinite golden rectangles with the same proportion according to the same proportion (the property that a golden rectangle can be divided into squares and rectangles indefinitely is the expression of "certainty" in appearance and the strict restriction of "dynamic balance"). For example, if the vertices of each square are connected by arcs with the radius of the corresponding square side length, these arcs will become a special vortex line-golden vortex line after being connected. The vortex line forms a rectangular vortex eye at the infinite vanishing point. This golden vortex line has the characteristics of "endless life" and creates a unique rhythmic beauty visually. This aesthetic feeling has its psychological and physiological reasons.
When people's eyes are at a fixed viewpoint, due to the parallel distance between them, theoretically we can form an overlapping visual circle with the focus of one eye as the center and the focus of the other eye as the radius. If the overlapping field of vision of these eyes is simplified to a rectangle, then this rectangle is similar to a golden rectangle. Therefore, taking the two vortex eyes of the golden rectangle (we can make one vortex eye on the left in the figure) as the stopping point of the head-up condensation of human eyes can produce the greatest visual comfort (human eyes have the visual habit of high center of gravity). This physiological mechanism of human eyes coincides with the changing law of objective images, which makes the endless and varied phenomenon of the golden rectangle conform to the principle of unified change and produces the best visual interest and universal aesthetic feeling.
The golden ratio and the aesthetic feeling of the golden rectangle make it have high aesthetic value in plastic arts from ancient Greece to19th century. Many beautiful shapes in the world are created according to this ratio. For example, the statues of Venus and Apollo, the ancient Greek architecture of the goddess Athens and Notre Dame de Paris, and the Eiffel Tower in Paris are all closely related to the golden ratio. The proportion of China ancient Qin bricks and Han tiles is also similar to the golden ratio. Windows, desktops, newspapers, books and magazines, and film bases in modern life are also related to the golden ratio. Moreover, the structure of many natural forms in nature is also related to the golden ratio.
Geometrically, a golden rectangle can be easily made by the following steps:
1) Given any line segment AC, divide the line segment AC into a golden section with point B to make a square ABED. ..
2) See ⊥ Communication.
3) Extend the DE ray so that the DE line and CF line intersect at point F. 。
ADN is a golden rectangle.
The golden rectangle can also be made without the existing golden section:
1) is an arbitrary square ABCD ..
2) Divide the square into two halves with line segment MN.
3) Draw an arc with a compass with n as the center and |CN| as the radius.
4) Extend ray AB until it intersects the above arc at point E. 。
5) expanded ray DC.
6) Make a line segment EF⊥AE, so that the rays DC and EF intersect at point F. 。
Then ADFE is a golden rectangle.
The golden rectangle can also generate itself: starting with the golden rectangle ABCD below, it is easy to get the golden rectangle ECDF. By drawing a square ABEF, it is easy to form a golden rectangle DGHF. By drawing a square. This process can continue indefinitely.
Another kind of equiangular spiral (also called logarithmic spiral) can be made of infinite golden rectangles finally obtained. As shown in the figure below, draw a quarter arc on each square of a series of golden rectangles with a compass. These circular arcs form the outline of an equiangular helix.
Nautilus's shell, trunk, horn, parrot's claw, etc. They are all isometric snails. If you carefully observe the arrangement of stamens in daisies, you will find that they are also equilateral spiral shapes. There are two views on this arrangement: left-handed and right-handed. The left-handed and right-handed numbers of most daisies are 2 1 and 34, which are two adjacent terms of Fibonacci sequence. The scales of pinecones and pineapples have a similar arrangement, and the arrangement numbers are 5 and 8 and 8 and 13 respectively, which are also two adjacent terms of Fibonacci series. So is the sunflower. Usually the number of left hands and right hands are 34 and 55 respectively. The larger sunflowers are 89 and 144, and even 144 and 233, which are two adjacent terms in Fibonacci series. Some people say that Virgil and many Roman poets at that time often used Fibonacci series in their works; Even the keys of the piano have five black keys and eight white keys between an octave! Fibonacci sequence can be seen everywhere.
The ratio of two adjacent terms in Fibonacci sequence is close to the golden section law, and an equiangular spiral can be drawn from the golden rectangle, which appears in pinecones, pineapples, daisies, sunflowers and so on. And their left and right spiral numbers are just two adjacent terms of Fibonacci sequence, so the natural creation is amazing!