Relation of j to the golden ratio φ of reverse Fibonacci sequence
The relation of j to the golden ratio φ is based on the basic formation of sacred geometry„Vesica piscis„. This shape can be freely translated as“bladder of a fish“and is a pair of intersecting circles with centers on their circumferences. The center therefore has the length of a radius of one of them, see Figure 6.
The ratio j can be expressed by means of the generalized triangle shown in Figure 7. The side a of the newly created triangle (denoted by green on Figure 7) is calculated using the Pythagorean Theorem and simple mathematical adjustments.
If all sides of a general triangle are known and the angle α = π/4 ,the ratio of j can be expressed using the cosine sentence, see derivation (29).
The resulting relation (29) can also be written using simple mathematical operations as follows, see equation (30).
Finally, by calculating the inverse function of the equation (30) we can express the relation of the golden ratio φ and ratio j, see relation (31).
Further information can be find in article „Reverse Fibonacci sequence and its description“ by Jakub Soucek and Ondrej Janicko.