How to determine the n-th number of the Fibonacci sequence

I’m currently preparing a couple of blog posts on measure theory, probability theory and fractals. Since studying, research and writing on these topics is taking quite a long time, I’ve decided to write a small post on the Fibonacci sequence, which showed up when looking at pictures of the now notorious romanesco broccoli. I find it such a cliché to post a picture of that magnificent vegetable when discussing fractals or the Fibonacci sequence, so you’ll have to be happy with a link.

Weirdly enough, the number of spirals on a romanesco is a number from the Fibonacci sequence:
$$\require{physics}
1, 1, 2, 3, 5, 8, 13, 21, \ldots \thinspace ,
$$
but the question I’ll address in this post is the following. For low index numbers $n$ (please note that we start counting at $n=0$, I know, it’s sometimes confusing), it’s relatively easy to determine the corresponding Fibonacci number. For example, for $n=3$, the Fibonacci number is 3, and when $n=7$, the corresponding Fibonacci number is 21. But how do we determine, in general, the $n$-th Fibonacci number?

Since the formula of a sequence can be determined from the roots of its characteristic polynomial, which is in its turn determined from the recursion relation it corresponds to, let’s start by realizing that the Fibonacci sequence satisfies the recursion relation
$$
x_{n+2}
= x_{n+1} + x_{n}
\thinspace ,
$$
which is a linear recursion relation of order 2. The characteristic polynomial of the sequence is therefore
$$
P(z)
= z^2 – z – 1
\thinspace ,
$$
which has roots $\lambda_1$ and $\lambda_2$:
$$
\lambda_1
= \frac{1 + \sqrt{5}}{2}
\qquad \text{and} \qquad
\lambda_2
= \frac{1 – \sqrt{5}}{2}
\thinspace .
$$
Notice that these roots are equal to the golden ratio and its negative inverse, respectively:
$$
\lambda_1 = \varphi
\qqtext{and}
\lambda_2 = -\varphi^{-1}
\thinspace .
$$

The general formula for the $n$-th element of a sequence whose characteristic polynomial has two distinct roots can be written as
$$
x_n
= C_1 \lambda_1^n + C_2 \lambda_2^n
\thinspace ,
$$
where $C_1$ and $C_2$ can be found by solving the linear system of equations
$$
\begin{cases}
x_0 = C_1 + C_2 = 1 \\
x_1 = C_1 \lambda_1 + C_2 \lambda_2 = 1
\thinspace .
\end{cases}
$$
Through the method of substitution, we find that
$$
C_1
= \frac{\sqrt{5} + 1}{2 \sqrt{5}}
\qquad \text{and} \qquad
C_2
= \frac{\sqrt{5} – 1}{2 \sqrt{5}}
\thinspace ,
$$
so the general formula to calculate an element of the Fibonacci sequence is
$$
x_n
= \frac{1}{\sqrt{5}} \qty(
\frac{1 + \sqrt{5}}{2}
)^{n+1}
– \frac{1}{\sqrt{5}} \qty( \frac{1 – \sqrt{5}}{2} )^{n+1}
\thinspace ,
$$
or in terms of the golden ratio:
$$
x_n
= \frac{1}{\sqrt{5}} \qty(
\varphi^{n+1}
– \frac{(-1)^{n+1}}{\varphi^{n+1}}
)
\thinspace .
$$

So, it would take you ages to calculate the 40-th Fibonacci number using its recursion relation, but we can plug in $n=39$ and immediately find
$$
\begin{align}
x_{39} &=
\frac{1}{\sqrt{5}} \qty(
\frac{1 + \sqrt{5}}{2}
)^{40}
-\frac{1}{\sqrt{5}} \qty(
\frac{1 – \sqrt{5}}{2}
)^{40} \\
&= 102.334.155
\thinspace .
\end{align}
$$

Pink noise (MOC)

A couple of months ago, I became fascinated by fractals. I quickly started some hobby research, which lead me to an intriguing concept: music is 1/f noise. But what does that mean? How is music noise (it’s clearly more than mere noise), and what does the designation “1/f” mean?

I started making notes in my favorite note-taking environment, Obsidian. (Perhaps I’ll write about my Obsidian vault in more detail in another post.) I delved deeply into the mathematical theory of probability, random processes and spectral analysis, and the result is the high-level overview (a Map of Content, or MOC, as Nick Milo from Linking Your Thinking would call it) that you can find below.

A random process $Q$ is a collection of random variables:
$$\require{physics}
\{ Q(t): t \in T \}
\thinspace ,
$$
that are indexed by a time parameter $t \in T$:
$$
T = \{t_i\}
\qqtext{or}
T = \mathbb{R}
\thinspace .
$$
From this perspective, they are special cases of random fields, which are characterized by an index set that is a subspace of $\mathbb{R}^N$.

Strictly stationary random processes are characterized by the property that the joint distribution of the underlying random variables is invariant with respect to time delays. But since this is a property that can hardly be used in practice, we define so-called second-order stationary processes, that have the following properties:

  • their expectation is time-independent:
    $$
    \mathbb{E}[Q(t)] = \mu
    $$
  • they admit an autocovariance function (ACVF) that is only dependent on a time delay $\tau$:
    $$
    \gamma(\tau) = \mathbb{K}[Q(t), Q(t+\tau)]
    \thinspace .
    $$

For these kinds of random processes, the spectral density $S(f)$ is defined as the Fourier transform of the autocovariance function:
$$
S(f)
= \int_{-\infty}^{+\infty}
\gamma(\tau)
\exp(-2\pi i \thinspace f \tau)
\dd{\tau}
\thinspace .
$$
White noise then is a random process characterized by a spectral density that is frequency-independent:
$$
S_{\text{WN}}(f)
= \sigma^2
\thinspace ,
$$
and pink noise (in a wide sense) by a spectral density that is inversely proportional to the frequency:
$$
S_{\text{PN}}(f)
= \frac{C}{f^{\alpha}}
\thinspace ,
$$
with an exponent $0 < \alpha \approx 2$. Thus, pink noise is 1/f noise.

In practice, time series are specific realizations of random processes. We can therefore investigate if a time series represents is an instance of pink noise by calculating the periodogram of the time series, which is defined as the (discrete) Fourier transform of its sample autocovariance function. We can furthermore show that the periodogram can be calculated in an easier way, as the square of the Fourier transform of the fluctuation of the time series with respect to its mean.

The final step is then to represent a musical piece as some time series, calculating its periodogram and investigating if the plotted spectral density has 1/f behavior.

My inquiries can now continue, what exactly makes music 1/f noise? The results will be for a follow-up post that will relate fractals with consciousness, music and 1/f processes.

Is anything unclear? Any suggestions, comments, or critiques? Please let me know in the comments.

Middle area of segment described by three concentric circles

I found this really fun math question a while ago. It has been a while since I last solved such a geometrical problem, and since I was particularly attracted to the symmetry of the figure, I gave it a go. The question is simple: what is the colored area, in terms of $r$?

The question is simple: what is the colored area, in terms of $r$?

For the solution, I’ve sketched the first quadrant of the middle circle in the next figure. We immediately notice that the teal-colored area $X$ is equal to the area of the sharp-angled sector $A_{\text{sector, 1}}$ minus the circular segment $A_{\text{segment}}$ colored in orange:
$$
X = A_{\text{sector, 1}} – A_{\text{segment}}
$$

Sketch of the top-right quadrant.

The area of the sector can be found by realizing that the angle of the sector is $\pi/6$ (or 30°), because the triangle with the gray edges is an isosceles triangle with sides $r$. Therefore, we find
$$
\begin{align}
A_{\text{sector, 1}}
&= \frac{\pi/6}{2\pi} \pi r^2 \\
&= \frac{\pi r^2}{12}
\thinspace .
\end{align}
$$

The area of the segment can then be found as the difference between the larger sector $A_{\text{sector, 2}}$ and the area $A_{\text{isosceles}}$ of the isosceles triangle:
$$
\begin{equation}
A_{\text{segment}}
= A_{\text{sector, 2}} – A_{\text{isosceles}}
\thinspace .
\end{equation}
$$
Since the angle of the second sector is $\pi/3$, again by virtue of the isosceles triangle, its area is
$$
\begin{equation}
A_{\text{sector, 2}}
= \frac{\pi r^2}{6}
\thinspace .
\end{equation}
$$
The area of the isosceles triangle can be calculated using the standard formula of the area of a triangle:
$$
\begin{equation}
A_{\text{isosceles}}
= \frac{1}{2}rh
\thinspace ,
\end{equation}
$$
where we realize that the triangle’s height $h$ can be goniometrically found as
$$
\begin{equation}
h = r \cos(\pi/3)
\thinspace .
\end{equation}
$$
We thus find for the area of the isosceles triangle that
$$
\begin{equation}
A_{\text{isosceles}}
= \frac{\sqrt{3}}{4} r^2
\thinspace ,
\end{equation}
$$
so the area of the circular segment then is
$$
\begin{equation}
A_{\text{segment}}
= \frac{\pi r^2}{6} – \frac{\sqrt{3} r^2}{4}
\thinspace ,
\end{equation}
$$
hence the unknown area $X$ is found to be
$$
\begin{align}
X &= \frac{\pi r^2}{12}
– \bigg( \frac{\pi r^2}{6} – \frac{\sqrt{3} r^2}{4} \bigg) \\
&= \bigg( \frac{\sqrt{3}}{4} – \frac{\pi}{12} \bigg) r^2
\thinspace .
\end{align}
$$
Thus, the colored area can finally be calculated as,
$$
\begin{equation}
4X = \bigg( \sqrt{3} – \frac{\pi}{3} \bigg) r^2
\thinspace .
\end{equation}
$$
Isn’t that a beautiful formula?

If you have any thoughts, or if you have used a different solution technique, let me know!

O Chosen Prince

I really like Grail stories, which you might have figured from reading the article Finding Joe. For a composition masterclass on writing for choir, I thought it would be an interesting idea to write my own text about the Grail that I can put onto music.

She calls from far and wide,
wherever you go she shines,
deep in the woods she hides,
the maiden bride you’ll never find,
her call you must abide.

O chosen Prince, you blissful blind,
chase trials and her golden light,
and seek the Grail with all your might.
Do not despair in struggling fight,
eternal joy awaits despite.

The Oxygen Advantage

“Take a couple of large, deep breaths.” How many times have you heard this at the start of a yoga practice, meditation session, or at the end of an intense workout? But, as our blood stream is almost always fully saturated with oxygen, how can this have any effect, except for the relieving effects of stretching the upper body? In the book The Oxygen Advantage, Patrick McKeown, who is a student of Konstantin Buteyko, introduces an intriguing idea: even though breathing is ultimately involuntary, most of us are not breathing correctly. The modern man is a chronic overbreather. Unhealthy breathing patterns can have tremendous detrimental effects such as continuous fatigue, excessive weight, chronic inflammation and can even lead to the development of asthma. Fortunately, as McKeown argues, a program of simple breathing exercises can reverse the effects of decades of the faulty breathing habits such of mouth breathing, chest breathing and sighing.

The central piece of information that underlies the ideas in this book is that it is the amount of carbon dioxide (CO2) in the blood that determines how much oxygen (O2) the body can use in its muscles and organs. Overbreathing, and consequently over-exhaling, expels too much CO2, leading to continuous states of CO2 deficiency. Such prolonged states of hypocapnia lead to a stronger O2 retention by the blood’s hemoglobine, thus ultimately leading to decreased O2 uptake for practical use in the body. In summary: inhaling larger volumes of air does not lead to larger oxygen uptake in the tissues, on the contrary.

McKeown uses a simple test that provides an indication for chronic overbreathing: the body oxygen level test (BOLT). It goes as follows. Inhale and exhale normally and lightly for a couple of times, and on the last exhale, hold your breath (perhaps by pinching your nose), and count the number of seconds until the first involuntary muscular sign of air hunger in the neck or abdomen. Anything less than a BOLT score of 40 seconds indicates suboptimal breathing. Even though I’ve completed a marathon, and am a regular swimmer, my BOLT score when I first tested it was 12 seconds. Try it for yourself!

By training the part of the brain that regulates the breath towards greater CO2 tolerance, the body can sustain higher levels of CO2 in the blood, hence increasing the oxygen to be taken up by the body’s functional tissues. With a greater CO2 tolerance, the body can work harder with less effort, because the body can uptake the correct level of oxygen. In levels of increasing difficulty, McKeown advocates for the implementation of the following exercises to improve your BOLT score.

  • Nose breathing through the abdomen.
  • Avoid sighing and taking deep breaths: if you notice you have done so, hold your breath for about 10s after the event.
  • Notice your breathing pattern: if it is abnormal, make it lighter.
  • Breathing Recovery Exercise: 10s of light breathing, followed by 5s of breath holding, sustaining very moderate air hunger. This can also be used as a cool-down after a physical training session for 3’ to 5’.
  • Breathe Light to Breathe Right: very light breathing towards a tolerable air shortage, sustain for about 3’ to 5’.
  • Walking exercise: during walking, breathe normally for 30s, hold the breath after an exhale until medium air shortage, breathe short and lightly for 15s, repeat 10 times. This exercise should be sustainable, and never stressful, and can be used as a warm-up for a physical exercise session.
  • Nose Unblocking Exercise: during walking, inhale and exhale lightly, holding the breath while walking, count the number of steps you take, repeat 6 times. The goal is to train towards taking 80 steps while holding the breath.
  • Nose breathing during physical exercise: if you can’t breathe through your nose, lower the intensity. There is one exception: physical exercise in water. Mouth breathing during swimming is OK because while in the water, less air intake naturally occurs.

The take-away insight from this book is that McKeown advocates for bringing awareness to the breath, making sure that it is light, through the nose and through the abdomen. These healthy breathing lightly habits can be supplemented with additional breathing exercises, improving your body’s capability to effectively deliver oxygen to its functional tissues.

Finding Joe

I don’t believe people are looking for the meaning of life as much as they are looking for the experience of being alive.

Joseph Campbell

Joseph Campbell strongly believed that we humans are not so different from the stories we tell. He believed that each and every one of us is continually living out a story. We can all relate with stories, and as myths and legends are archetypical stories, they provide narratives that serve to understand and guide our personal lives. These stories are so much richer than explanations in a materialist world view, they are more than historical accounts of long past series of events.

Finding Joe is a documentary about Joseph Campbell’s work on what he called The Hero’s Journey, which he believed to be the archetypical and primordial story. The reason why the hero motif is so widespread across time and space is because it is an archetype: it is a seed in our (Jungian, collective) unconscious that can be sparked and evoked, and therefore has the power call to us to action.

The three stages of the Hero’s Journey

All hero myths follow the same pattern and are composed of three stages. First, the hero is separated from his familiar world, second, the hero is initiated in a special world, and finally, the hero returns from that special world back to his ordinary world.

Every hero story starts off with a description of the world that the hero inhabits. From the hero’s viewpoint, it is often seen as plain and dull, and perhaps a little too comfortable for his liking. Then, unexpectedly and out of nowhere, comes the hero’s call to adventure, the metaphysical realization of the seed of potential that truly makes him feel alive. The hero is called to follow his bliss, straight into the unknown. But we rest assured that because he is adhering to that call, the universe opens doors for him, which remain closed for others.

Along the way, the hero encounters trials and enemies. By overcoming these obstacles, the hero grows stronger and learns of his hidden and innate abilities. Archetypically, the hero slays the dragon, which is to say that he conquers his biggest fears through courage. And the meanest and toughest dragon out there is his own personal shadow (in a Jungian sense) in all the beliefs that you he uses/used to trick himself. During this initiation phase, the hero must die and resurrect. We should not think as his death as the end of his life (although, dramatically, this trope is used often), but as the metaphorical shedding of his skin that has no more use.

At the end of the hero’s journey, and after his resurrection, he returns. He doesn’t come back empty-handed, though, because by having overcome his dragon, he has obtained a secret power or knowledge that he can now share with his people at home. But the greatest gift he brings to his community is that he can recount that he lived the story.

Death and evolution

I had been struggling and grappling with one of the quotes in Marvel’s The Eternals. I watched it for the first time in November last year, and since I really liked it, I rewatched it a couple of months ago. That’s how long this quote has been on my backburner. The quote is

No evolution without death

Watching this documentary finally shed some light on it. A spark of insight, so to say. Death does not represent the end of biological life, it is a marker for change taking place. Of course! No death, no change, no evolution. That’s it.

And also, for those that have struggled while playing the game Bloodborne: the game is truly a metaphor for life. Think of it. You keep dying until you’re good enough to defeat the boss. And then the next one, and then the next one. Maybe that’s why this game has felt to be the most rewarding game I’ve ever played. No other game even comes close. I think FromSoft (the creator of the video game) deeply understands the hero motif.

What can we do with this?

Follow your bliss.

Joseph Campbell

As usual, some thoughts of the practical utility of what I’ve shared with you today. Read up on old tales, and ask those mythological characters to be incarnated in you. That’s prayer, but perhaps more accessible than true Christian ones. And also -and I’m sure mr. Campbell would be delighted to read this-, follow your bliss, that overwhelming energy of courage. And in doing so, become responsible for your own adventure, and truly live your own hero’s journey.

Pageau’s lecture ‘The Symbolic World’ (part 1 of 2)

I found this lecture hard to follow, because mr. Pageau jumps frequently from one subject to the other, without concluding the one, or introducing the other. I assume I experienced his talk in such a way because I do not yet fully understand the breadth of the terminology he uses, and as a consequence, I had to rewatch and take notes on this lecture a couple of times to finally start making sense out of it. In this article (the first of two), I give a summary of the first part of mr. Pageau’s lecture, and I add some of my thoughts that helped me understand it.

The world is inconceivably complex. Everything is connected to everything, and when delving deeper and deeper into the structure of things, we find that this complexity is apparent at every level of analysis, from high up to deep below.

Where to draw the line?

As an example, scientists and engineers suffer from this problem of insurmountable complexity while researching and developing useful technologies. Within a materialistic world view, they develop models that have a certain explanatory and/or predictive power. One such class of models can be found in reductionist theories, in which a system is decomposed into its parts, together with the interactions that occur between those parts.

There is nothing inherently flawed with this materialistic way of thinking, but it is important to understand that models in a materialistic world view are exactly that: they are models. Even though they can have immense practical use, they nonetheless provide only a simplified view of the world, because they divide the world into system and environment, while in reality the only thing that actually exists is everything in the universe as a whole. In their terms, the system-environment. It should not come as a surprise that this division matters, and is a reason for the peculiarities encountered in reductionist theories. A notable example can be found in quantum mechanics (the mathematical theory that governs the behavior of the smallest particles), where the observer plays a significant role, as the mere act of observing can disturb an isolated system. On the other hand, in the study of complex systems (systems composed of constituents and their interactions), the notion of emergence is introduced as an axiom to account for properties of the system as a whole that are not properties of its constituent parts.

At every level of analysis, everything and it parts has, seemingly, an indefinite amount of complexity. We thus begin to see that the fundamental reductionist categorization into system and constituents is at its core a problem that has multiple different solutions. In order to solve this problem, we must be able to answer the simple question of where to draw the line, i.e. how are the boundaries of the system recognized? The argument that is posited is that the proper level of analysis necessarily depends on the objective of the analysis, i.e. why it is done.

In a small leap of faith, we reformulate this previous statement in a spiritual worldview: attention is directed teleologically, i.e. according to the purpose of what is attended to. A simple, practical, example is found in how we, as intelligent beings, encounter a cup. We encounter it via its purpose: to drink from it. This cup represents a teleological, i.e. purposeful, pattern, and as intelligent beings we recognize that purpose in our encounter of the complexity of the world.

What is identity?

A different way we encounter things as intelligences is found in following example. Let’s take a look at a relatively banal thing: a chair. When we see one, we usually notice that this chair is one single object, i.e. the chair itself. But we also see that it consists of its parts: its legs, its seat, its back, and possibly its armrests. We thus understand that the world exists of patterns, and smaller patterns are embedded in larger ones. When we focus in these parts in particular, however, we notice that these separated sub-objects are bound together by the being of something that does not exist at that sublevel of analysis. This is exactly the definition of identity in a spiritual cosmology: something’s identity is that it brings multiplicity (the legs, the seat, the back, the armrests) into unity (the chair).

This identity definition can also be applied in sociological settings, where it provides the answer what the difference is between a crowd and a group. Groups of people need something that bind them towards a purpose, their logos. In such groups, the team captain, coach, or president embodies the logos of the group: it is the human incarnation of that group’s identity. Groups must also attend to whatever their purpose is, or else the group will dissolve, and that identity will die. This kind of attention surely resembles reverence, for example of the group’s name, colors, or totem. The group’s identity must also accept an important an existential task: it must judge who and what is inside and who is outside the group. Or else, when too many people start bringing their instrument to a knitting group, it doesn’t remain a knitting group, but becomes a band.

Maps of the world

We need a map in order to encounter the ever-complex world. That map serves as a way to interpret the world, once you have found your location in it. Symbolic patterns (and, by extension, stories) are such maps that can be used to interpret the world, but Pageau argues that they are necessary to inhabit the world: true encounters of the world are participatory and engaging, one cannot stand above the story that is being played out.

Pageau’s talk at Resurrection of Logos

Greek spelling of logos. Source: Wikipedia.

Since about 2018, I’ve become interested in archetypes, symbolism and christian theology. Once concept that more than occasionally pops up is logos. In this post, I paraphrase the talk that Jonathan Pageau gave at Resurrection of Logos.

A world that is a meaningful world is sustained by the constant action of Logos. This Logos, or Divine Word, is a fundamental pattern in which potentiality (one of the aspects of the symbolic chaos) is instantiated by creating experienced existence. As this experience is deeply meaningful, it is useful to think of a meaningful world as a place that calls to every person to create order of out chaos.

The Christian origin of the term Logos is found in John’s Gospel (John 1:1), where we read

1 In the beginning was the Word, and the Word was with God, and the Word was God.

where ‘Word’ is ‘Logos’. Further theological development of the notion of logos in Christianity is done in 600 AD, by Maximus The Confessor. He explains that everything that exists in the world has a logos, which is its reason for existence, its fundamental purpose. Additionally, everything has multiple logoi (the Greek plural of logos), which are its qualitative characteristics (or as Jonathan Pageau would call them, phenomenological categories) such as its color, its intensity, its size, and so on.

Man experiences reality by uniting these different logoi inside of himself. In this process, he engages meaningfully in the world, and in doing so he becomes a microcosmic projection of the whole universe. This quest for logos is not merely a theological description of the world, it represents a call to walk that path of truth to discover and participate in the true purpose of things. Ultimately, this leads him to The Divine Logos, into the unity that God brings to the world. When he does not sufficiently unite the multiple logoi of the thing that he experiences, each individual of these logoi starts to appear as something untrue. In this way, a sin is a trespass from that enlightened path by a misuse of the proper purpose of something.

The postmodern movement that underlies the contemporary culture wars represents an assault on logos, because it deconstructs the unifying narrative of the logos value hierarchy. We have currently reached postmodernism’s final destination and are stretching the limits of what it has to offer. As it seems to have settled the cultural debate, it has become the mainstream sociological theory and now it occupies the center that it so much despises. Ironically, as a result, these kinds of untrue inversions are starting to be seen everywhere. But now (i.e. 2017 and beyond), the logos is awakening once more, and it calls for truthful experiences in order to rectify the chaos we’ve been left in.

So, how can this concept be applied in one’s own life? Speak and live the truth, and discover the true purpose of things around you accordingly.