This is a number series where each member is simply the sum of the previous two numbers. Φ -3 = 2 φ - 3 = φ -4 + φ -5 = φ -1 - φ -2Īnother connection of the Golden Ratio to partial symmetries in nature is through the Fibonacci Numbers ( f n). If the Golden Ratio turns up in examples of five-fold symmetry, it may well be because the number itself is fundamentally related to the number five. More conspicuously, the very irrationality of the Golden Ratio is an artifact of the square root of five. Why would this happen? Well, a complete circular angle (360 o) divided by five is 72 o, which occurred above as one of the angles whose trigonometric function is the Golden Ratio. Now, it just so happens, that, given a large enough area, the ratio of kites to darts is just the Golden Ratio. Multiple decagons, some of which from a distance can look like pentagons, can occur. Within this tiling, however, there can be small areas of five-fold symmetry. However, in the early 70's Roger Penrose discovered that a surface could be completely tiled in an asymetrical and non-periodic way with just two shapes, called "kites" and "darts" - "Penrose tiles" - as seen at right. Periodic five-fold symmetry does not occur in nature. For instance, a surface can be completely and symmetrically tiled with triangles, squares, and hexagons, but not with pentagons. Whether or not the Golden Ratio or the Golden Rectangle are of aesthetic significance, the ratio does turn out to have considerable significance in problems of natural symmetry. We can easily turn this into the previous equation, however, just by muliplying the numbers so as to get integers: φ = 1 / 2 + 2√1.25 / 2 = (1 + 2√1.25) / 2 = (1 + √(4 * 1.25)) / 2 = (1 + √5) / 2. In effect, this gives us an equation for the Golden Ratio: φ =. Thus, if we extend the side of the unit square and draw a circle with a radius of the diagonal and its center at the midpoint on the unit side, the circle will intersect the side at a point that will be 1.6l8033989 units from the corner of the square. 5, this produces the Golden Ratio, 1.6l8033989. The length of the diagonal can be calculated with the Pythagoran Theorem, based on a triangle that is. Beginning with a unit square, first the square is bisected, then a diagonal is drawn in the semi-square. The construction of a Golden Rectangle, however, is an interesting exercise in the geometry of the Golden Ratio, as seen at right. How pleasing the Golden Rectangle is, how often it really does turn up in art, and whether it does really frame the front of the Parthenon, may be largely a matter of interpretation and preference. Thus, for instance, the front of the Parthenon can be comfortably framed with a Golden Rectangle. The theory of the Golden Rectangle is an aesthetic one, that the ratio is an aesthetically pleasing one and so can be found spontaneously or deliberately turning up in a great deal of art. The Golden Ratio seems to get its name from the Golden Rectangle, a rectangle whose sides are in the proportion of the Golden Ratio. The angles in the trigonometric equations in degrees rather than radians are 54 o, 36 o, 18 o, and 72 o, respectively. The Golden Ratio can also be derived from trigonometic functions: φ = 2 sin 3 π/10 = 2 cos π/5 and 1/ φ = 2 sin π/10 = 2 cos 2 π/5. The first number is usually regarded as the Golden Ratio itself, the second as the negative of its reciprocal - or we can use the second in its own right, as the number " τ," for which there will be a use below. The Golden Ratio is an irrational number, but not a transcendental one (like π), since it is the solution to a polynomial equation. Since that equation can be written as φ 2 - φ - 1 = 0, we can derive the value of the Golden Ratio from the quadratic equation,, with a = 1, b = -1, and c = -1. Multiplying both sides of this same equation by the Golden Ratio we derive the interesting property that the square of the Golden Ratio is equal to the simple number itself plus one: φ 2 = φ + 1. It can be defined as that number which is equal to its own reciprocal plus one: φ = 1/ φ + 1. The Golden Ratio ( φ) is an irrational number with several curious properties. The Golden Ratio and The Fibonacci Numbers The Golden Ratio and
0 Comments
Leave a Reply. |
Details
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |