Golden ratio in arts
Golden ratio is often used in different kind of arts. Even the architectures and music composers does use the golden ratio. These following things can lead you to understand how the golden ratio is used in geometry and arts.
In this picture I'll show you how to write a golden rectangle:
Construct a square
Then bisect the square
Draw a line from one end of the bisecting line to one of the opposite corners. Extend the baseline of the square.
Using the diagonal line as the radius, drop an arc from the corner of the square down to the baseline.
Draw a line from the point of intersection of the arc and the baseline, perpendicular to the baseline. Extend the top edge of the square to meet this line and form a rectangle.
This rectangle is referred to as the golden rectangle.
This is structure is in Athen, Greece. As you might see the space between the columns form golden rectangle. This structure was made by a Greek sculptor Phidias.
The golden spiral is one thing which is important to know.
To begin constructing the first square, draw an arc from one corner of the rectangle down (or up) until it intersects with the adjacent side. Then draw a line perpendicular to the side that is being intersected, from the point of intersection to the opposite side.
Repeat the process to form the next square...
..and so on.
By drawing connecting arcs within the sequence of squares, you can construct logarithmic curve known as the golden spiral.
Now I'm going to show you how to make a golden section:
Draw a straight line, then draw a perpendicular from one end of the line, to a distance equal to half the length of the first line.
Draw a line connecting the two lines so they form a right triangle.
With a compass, draw an arc, the length of the leg of the triangle, so it cuts the hypotenuse.
Draw an arc with the point of the compass positioned at the other acute angle, from the point at which the first arc cut the hypotenuse down to the baseline. The baseline is now divided into a Golden Section.
Golden triangle:
The Golden Triangle is an isoceles triangle with two angles of 72 degrees and one of 36 degrees. In this triangle, if the baseline is 1.618..., the sides will be 1.618...+1.
Bisect one of the larger angles, and you have formed another Golden Mean.
Keep on bisecting the angles and you have begun to form the structure for another Golden Spiral.
When you've done enough bisecting the angles it looks like this:
Now, draw arcs from the vertices of the obtuse angles of the isoceles triangles from point to point.
Connect the arcs, and you're drawn another Golden Spiral.
You get tired? Just hold on!
This symbol, which is done by the members of the Pythagorean Society used to identify each other, is filled with Golden Sections and Golden Triangles. See how many you can find.
The members of the Pythagorean Society studied many polyhedra; they accorded the dodecahedron special respect. By extending the sides of a face of the dodecahedron, they formed a five-pointed star. When inscribed in a circle, this became known as the pentagram star. The golden ratio in Egypt
The ancient Egyptians used the golden ratio in much of their artwork, including in heiroglyphs, the pyramids, and many statues.
The ancient Egyptians used the golden ratio in much of their artwork, including heiroglyphs, the pyramids, and many statues.
The Ahmes papyrus of Egypt gives an account of the building of the Great Pyramid of Giza in 4700 B.C. with proportions according to a "sacred ratio." Modern measurements show that the ratio of the distance from groud center to base edge, to slant edge of the pyramid is almost exactly 0.618. The sides of the pyramid appear to be golden triangles. A golden triangle is an isosceles triangle with base angles of 72º and a vertex angle of 36º.
Penrose Tilings and the Golden Mean
The British physicist and mathematician, Roger Penrose, has developed an aperiodic tiling which incorporates the golden section and the five-fold symmetry inherent in it. The tiling is comprised of two rhombi, one with angles of 36 and 144 degrees (figure A, which is two Golden Triangles, base to base) and one with angles of 72 and 108 degrees (figure B).
The ratio of tile A to tile B is the Golden Ratio.
By following sets of parallel lines within the tiling, the five-fold symmetry is revealed. I have emphasized two of the five axes in the images below.
In addition to the unusual symmetry, Penrose tilings reveal a pattern of overlapping decagons. Each tile within the pattern is contained within one of two types of decagons, and the ratio of the decagon populations is, of course, the ratio of the Golden Mean. I have highlighted the two types of decagons below.
In the years since Roger Penrose developed the first Penrose tiling, other scientists and Penrose himself have continued to develop new aperiodic tilings. In the February1992 issue of Scientific American Peter W. Stephens and Alan I. Goldman reported that quasicrystals, alloys formed by melting aluminum, copper, and iron together, often revealed the same symmetry present in Penrose Tilings.
This is a modern art. The tiling method is used here too:
It's a beautiful art, isn't it?