Search results
Results from the WOW.Com Content Network
The golden ratio, the irrational number that is the "most difficult" to approximate rationally (see § A property of the golden ratio φ below). γ = [0;1,1,2,1,2,1,4,3,13,5,1,...] (sequence A002852 in the OEIS). The Euler–Mascheroni constant, which is expected but not known to be irrational, and whose continued fraction has no apparent pattern.
The golden ratio φ and its negative reciprocal −φ −1 are the two roots of the quadratic polynomial x 2 − x − 1. The golden ratio's negative −φ and reciprocal φ −1 are the two roots of the quadratic polynomial x 2 + x − 1. The golden ratio is also an algebraic number and even an algebraic integer.
The distance between the centers along the shortest path namely that straight line will therefore be r 1 + r 2 where r 1 is the radius of the first sphere and r 2 is the radius of the second. In close packing all of the spheres share a common radius, r. Therefore, two centers would simply have a distance 2r.
The silver ratio is a Pisot number, [5] the next quadratic Pisot number after the golden ratio. By definition of these numbers, the absolute value 2 − 1 {\displaystyle {\sqrt {2}}-1} of the algebraic conjugate is smaller than 1, thus powers of σ {\displaystyle \sigma } generate almost integers and the sequence σ n mod 1 ...
Random close packing (RCP) of spheres is an empirical parameter used to characterize the maximum volume fraction of solid objects obtained when they are packed randomly. For example, when a solid container is filled with grain, shaking the container will reduce the volume taken up by the objects, thus allowing more grain to be added to the container.
As for fractions, the simplest form is considered that in which the numbers in the ratio are the smallest possible integers. Thus, the ratio 40:60 is equivalent in meaning to the ratio 2:3, the latter being obtained from the former by dividing both quantities by 20. Mathematically, we write 40:60 = 2:3, or equivalently 40:60∷2:3.
In mathematics, the supersilver ratio is a geometrical proportion close to 75/34. Its true value is the real solution of the equation x 3 = 2x 2 + 1. The name supersilver ratio results from analogy with the silver ratio, the positive solution of the equation x 2 = 2x + 1, and the supergolden ratio.
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is