Fibonacci Range
Vinay Pettyjohn
The Fibonacci numbers were first described in Indian mathematics as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths.[2][3][4] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci.[5]
Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci Quarterly. Applications of Fibonacci numbers include computer algorithms such as the Fibonacci search technique and the Fibonacci heap data structure, and graphs called Fibonacci cubes used for interconnecting parallel and distributed systems. They also appear in biological settings, such as branching in trees, the arrangement of leaves on a stem, the fruit sprouts of a pineapple, the flowering of an artichoke, and the arrangement of a pine cone's bracts, though they do not occur in all species.
Fibonacci numbers are also strongly related to the golden ratio: Binet's formula expresses the n-th Fibonacci number in terms of n and the golden ratio, and implies that the ratio of two consecutive Fibonacci numbers tends to the golden ratio as n increases. Fibonacci numbers are also closely related to Lucas numbers, which obey the same recurrence relation and with the Fibonacci numbers form a complementary pair of Lucas sequences.
The Fibonacci sequence appears in Indian mathematics, in connection with Sanskrit prosody.[3][9][10] In the Sanskrit poetic tradition, there was interest in enumerating all patterns of long (L) syllables of 2 units duration, juxtaposed with short (S) syllables of 1 unit duration. Counting the different patterns of successive L and S with a given total duration results in the Fibonacci numbers: the number of patterns of duration m units is Fm+1.[4]
Variations of two earlier meters [is the variation] ... For example, for [a meter of length] four, variations of meters of two [and] three being mixed, five happens. [works out examples 8, 13, 21] ... In this way, the process should be followed in all mātrā-vṛttas [prosodic combinations].[a]
The Fibonacci sequence first appears in the book Liber Abaci (The Book of Calculation, 1202) by Fibonacci[16][17] where it is used to calculate the growth of rabbit populations.[18][19] Fibonacci considers the growth of an idealized (biologically unrealistic) rabbit population, assuming that: a newly born breeding pair of rabbits are put in a field; each breeding pair mates at the age of one month, and at the end of their second month they always produce another pair of rabbits; and rabbits never die, but continue breeding forever. Fibonacci posed the puzzle: how many pairs will there be in one year?
Like every sequence defined by a linear recurrence with constant coefficients, the Fibonacci numbers have a closed-form expression. It has become known as Binet's formula, named after French mathematician Jacques Philippe Marie Binet, though it was already known by Abraham de Moivre and Daniel Bernoulli:[22]
These last two identities provide a way to compute Fibonacci numbers recursively in O(log n) arithmetic operations. This matches the time for computing the n-th Fibonacci number from the closed-form matrix formula, but with fewer redundant steps if one avoids recomputing an already computed Fibonacci number (recursion with memoization).[31]
The only nontrivial square Fibonacci number is 144.[48] Attila Pethő proved in 2001 that there is only a finite number of perfect power Fibonacci numbers.[49] In 2006, Y. Bugeaud, M. Mignotte, and S. Siksek proved that 8 and 144 are the only such non-trivial perfect powers.[50]
If the members of the Fibonacci sequence are taken mod n, the resulting sequence is periodic with period at most 6n.[63] The lengths of the periods for various n form the so-called Pisano periods.[64] Determining a general formula for the Pisano periods is an open problem, which includes as a subproblem a special instance of the problem of finding the multiplicative order of a modular integer or of an element in a finite field. However, for any particular n, the Pisano period may be found as an instance of cycle detection.
The Fibonacci sequence is one of the simplest and earliest known sequences defined by a recurrence relation, and specifically by a linear difference equation. All these sequences may be viewed as generalizations of the Fibonacci sequence. In particular, Binet's formula may be generalized to any sequence that is a solution of a homogeneous linear difference equation with constant coefficients.
The figure shows that 8 can be decomposed into 5 (the number of ways to climb 4 steps, followed by a single-step) plus 3 (the number of ways to climb 3 steps, followed by a double-step). The same reasoning is applied recursively until a single step, of which there is only one way to climb.
Fibonacci sequences appear in biological settings,[77] such as branching in trees, arrangement of leaves on a stem, the fruitlets of a pineapple,[78] the flowering of artichoke, the arrangement of a pine cone,[79] and the family tree of honeybees.[80][81] Kepler pointed out the presence of the Fibonacci sequence in nature, using it to explain the (golden ratio-related) pentagonal form of some flowers.[82] Field daisies most often have petals in counts of Fibonacci numbers.[83] In 1830, Karl Friedrich Schimper and Alexander Braun discovered that the parastichies (spiral phyllotaxis) of plants were frequently expressed as fractions involving Fibonacci numbers.[84]
Przemysław Prusinkiewicz advanced the idea that real instances can in part be understood as the expression of certain algebraic constraints on free groups, specifically as certain Lindenmayer grammars.[85]
Fibonacci analysis can be applied to financial markets in an attempt to discover potential future price action, and some may even consider it as a leading indicator. This article touches on range trading and explores how traders can utilize Fibonacci retracements when looking for potential areas of support and resistance that may appear in ranging markets.
A ranging market environment develops when price trades between two well-established areas called support and resistance. Price tends to rise and often touches the area of resistance before failing to break higher and eventually turns lower. Likewise, price tends to drop toward support before failing to break lower and subsequently reverses higher. Witnessing successive instances of price action bouncing off support and resistance is crucial to establish a trading range.
For range traders, the possibility of a break above or below the range should never be discounted. Traders are encouraged to utilize the risk management tools available to them and adopt various risk management techniques.
When an asset advances or declines significantly, creating a major move, there is a tendency for the market to consolidate as it partially retraces or fully retraces the initial move. Fibonacci levels can provide clues around areas of potential support or resistance where such consolidation may take place.
The daily GBP/USD chart below presents an area where price had a tendency to range between two Fibonacci levels. The major move produced in the month of March, 2020 presented a major move from which a Fibonacci retracement could be drawn; and for approximately four months after, price action showed multiple inflections off of these retracement levels while prices were mean reverting/ranging. Also added here is the 76.4% retracement level, which is a commonly included level to be used with Fibonacci retracements (1-.236 = .764).
Returning to the same GBP/USD chart, from left to right, it is clear to see an extended period of lower highs and lower lows, presenting us with the initial downward trend. This was followed by a strong pullback that erased more than 50% of that prior major move, after which price action moved into a range (shown in grey below).
This can allow for relatively tight stop placement, particularly if support is defined by a Fibonacci retracement level. This can allow the trader to focus on risk minimization when plotting for range continuation; whether that support is coming from a 38.2, 50 or 61.8% retracement levels.
Fibonacci Ranges are used to specify vertical lines at date/time levels. An initial range is selected and that range is then multiplied by the Fibonacci number sequence to generate where the date/time vertical lines will be plotted.
To add the tool to your chart, select the tool from the Fibonacci tool group, and left-click on the first bar you want the range to start. Next, left-click on the bar where you want the range to end. Optuma will then draw the tool using the default settings.
Apply Settings to All: When multiple Fibonacci Range tools have been applied to a chart, page or workbook, this action can be used to apply the settings of the one selected to other instances of the tool. This is a great time saver if an adjustment is made to the tool - such as line colour - as this allows all the other Fibonacci Range tools in the chart, page or entire workbook to be updated instantly.
Line Over All Views: Check this box to ensure the that the Fibonacci Range lines extend over all tools or indicators that are placed on the chart, for example, the lines will run through the volume indicator if this box is checked.
Line Style: The Line Style property allows you to adjust the type of the range lines displayed. There are 8 options available: Solid, Dots, Dash, Dash Dots, Long Dash, Long Dash Dot, Long Dash Dot Dot, Stippled.
Line Colour: Allows you to select the colour of the range lines. Clicking on the drop down arrow will display a colour swatch. Locate the desired colour and left-click it once to select it.
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