Introduction to Mathematical Spirituality – Breathing as a Sine Wave

Mathematical Spirituality is a philosophy that is rooted in two core beliefs:

1. Mathematics is the most fundamental aspect of Reality.
2. Spirituality can achieve its goals more effectively by embracing mathematical principles.

This is not an abstract philosophy, even though sometimes it can also be.

But in this introduction to the topic, you will learn something very practical: how to breathe according to mathematical principles. By doing so, you will acquire a new tool to better handle stress and anxiety and improve your ability to fall asleep.

Breathing is the perfect example of mathematical spirituality because breathing can be:

1. Very spiritual
2. Very mathematical

So, let’s talk about breathing through the lens of Mathematical Spirituality:

Part 1: Breathing as a Spiritual Concept

The connection between breathing and spirituality is not anything new. It is a very ancient connection, in fact. For instance:

• Pranayama is a component of Yoga that focuses on controlling the breath for spiritual purposes.
• In Buddhism, the practice Anapanasati is used to connect the mind and body through breathing mindfulness.
• In some Indigenous practices, breath is believed to have healing powers and is an integral part of certain rituals.

Breathing is also a duality between inspiration and expiration, and dualities are already themselves deeply spiritual concepts, you simply have to consider the Yin Yang symbol as proof of that.

As far as the human being is concerned, the breathing duality is the one that concerns him the most, as his life literally depends on it.

And symbolically, it contains symbolism as profound as the one found in the Yin / Yang.

• Inhalation represents Life, Energy (which is literally provided by the inspired oxygen), Inspiration, Work, and Activity. It is the activation of the sympathetic nervous system. It is the first thing we do when we come into the world.
• Exhalation represents Death, Relaxation, Absence of Energy, Release and Letting Go. It is the activation of the parasympathetic nervous system. It is the last thing we do before we die.

The English word “spirit” derives from the Latin spiritus, meaning “breath”. In fact, even today, the words “respiration” and “spirituality” still share some similarity between themselves.

And in Hebrew, the word “ruach” and in Greek, “pneuma” both mean breath and spirit.

These are the connections between breathing and spirituality.

Now we move on to:

Part 2: Breathing as a Mathematical Idea

In certain fields of Mathematics, it is very common to deal with wave forms like the sine, triangle, the square wave, and others.

Breathing can be connected to Mathematics by the fact that it can be plotted as a waveform onto a graph that shows the lung’s air volume.

In such graph:

• the lowest values correspond to emptied lungs
• and highest values correspond to filled up lungs

If we were to plot our breathing patterns onto this graph, they would draw some sort of wave form. Granted, these shapes would probably be somewhat erratic, because they were the result of unconscious breathing.

But by using our conscious thought, we can control our breath to draw specific shapes, like the ones mentioned before:

• If you inhale and exhale fast at a 1:1 ratio, with abrupt changes in direction, your breathing pattern will be plotted as a triangle wave. There’s a technique known as Shamanic breathing that resembles these characteristics.
• If you inhale quickly and hold your breath, then exhale quickly and hold your breath, your breathing pattern will be plotted as something resembling a square wave.
• You can also breathe like a sine wave. This is a breathing technique that has these two characteristics:
  • It has a symmetrical breathing ratio of 1:1, with inhalation and exhalation times being the same.
  • It has smooth transitions between inhalation and exhalation with a gentle “moment of suspension” during transitions. This can be imagined as if you are jumping on a trampoline, where there is a moment when you reach the highest point, where everything feels weightless. Similarly, there’s a moment of grounded suspension when you hit the lowest point of the trampoline. The breathing transitions should feel somewhat like that.

Why the Sine Wave?

The sine wave shape is more important than the other shapes.

This is because the sine wave has a very important place in Mathematical Spirituality, the same way it has in Mathematics itself.

The mathematical field known as Harmonic Analysis tells us that everything that vibrates can be broken down into, and explained as, a sum of sine waves. In other words, sine waves can be seen as the building blocks of anything that vibrates. And whether you believe in Quantum Field Theory, String Theory, or some other theory, all of them point to something that vibrates at the most fundamental level of Reality (e.g. a quantum field or a string).

This is why the sine wave is the basis of the breathing technique. Because if we breathe following a sinusoidal pattern, we are incorporating the most fundamental of shapes. And I mean incorporating in its literal sense because we’d be doing it quite literally with our bodies.

Part 3: The Sinusoidal Heart

So far, we’ve seen that if we breathe following a sinusoidal pattern, our lungs will draw that pattern onto a graph.

However, that is only the first step.

It is known that:

• Breathing-in makes the heart rate accelerate, by momentarily making the sympathetic nervous system dominate.
• Breathing-out makes the heart rate decelerate, by momentarily making the parasympathetic nervous system dominate.

If we plot these heart rate accelerations and decelerations onto a graph, we end up with something known as the Heart-Rate Variability.

During unconscious breathing, negative emotions or stress, this graph would be erratic and unstable. Something like this:

But as soon as we consciously adopt the sine wave breathing technique, the heart-rate variability graph transforms into a sine wave as well.

Spending more time in this harmonious state is more beneficial than spending time in this chaotic state, which is why we should breathe sinusoidally as often as possible.

We’ll discuss how we can do that later, but for now, let’s go deeper and get more mathematical with our bodies.

Part 4: The Frequency of 0.1 Hz

Making your heart-rate variability chart look like a sine wave is an excellent way to improve your well-being.

I already mentioned that some of these benefits include better management of stress, anxiety and sleeping problems, but also better emotional regulation.

However, there is a way we can make this Heart Rate Variability chart look even better by increasing its amplitude (the peaks and valleys of the graph get higher and lower, respectively).

When we increase the amplitude of the heart-rate variability, what that means is that when we breathe-in, our heart beats even faster than before, and when we breathe-out, our heart beats even slower than before. This might sound like a bad thing, but having a heart capable of doing this is actually a sign of health.

And through breathing, there is a way we can achieve this state of increased amplitude.

Our bodies have a mechanism known as the baroreflex response, responsible for regulating our blood pressure, heart-rate and blood vessel constriction and dilation.

By breathing like a sine wave at a frequency that matches the baroreflex’s own natural resonant frequency, we can stimulate it to increase its response, and, as a result, increase the amplitude of the heart-rate variability.

You might ask, what special frequency is that, that will stimulates the baroreflex. For the average person, that frequency is 0.1 Hz, or 6 breaths per second. However, it can have slight variations, based on the person’s height. Personally, I decided to measure mine, just to be sure.

Measuring My Baroreflex Resonant Frequency

To measure my resonant frequency, I bought a special chest strap that not only records my heart-rate but also its heart-rate variability, and then I connected it to a free app that allowed me to conduct some tests at different frequencies.

I found out that 0.11 Hz resulted in the highest heart-rate variability scores, so that’s my resonant frequency.

If you’d like to know what your own resonant frequency is, read this article I wrote: Find Your Heart-Rate Variability Resonant Frequency.

Now I know that if I breathe at 0.11 Hz, I am entering into a state of maximum benefit, both for the body and the mind. I also know that I am in a state of complete mathematical alignment.

Respiratory, Blood Pressure, and Cardiac Phase Alignment

That’s right. When we breathe at our resonant frequency, we enter into a state of complete mathematical alignment.

To explain what this means, let me repeat what I mentioned previously: when we breathe like a sine wave, our breath, our heart rate and our blood pressure all behave like smooth and predictable sine waves as well. This is an excellent state to be in, and it can be achieved at any frequency. whether it’s resonant or not.

However, when we breathe at our specific resonant frequency, all these waves algo align themselves and reinforce each other, which are phenomena that In Mathematics are called phase alignment and phase coherence.

This is the reason that makes this breathing technique, in my opinion, the most relevant for those who appreciate Mathematical Spirituality.

4.1 The Benefits of 0.1 Hz Breathing

As we just saw, breathing like a sine wave at 0.1 Hz results in multiple mathematical convergences in the human body.

This phenomenon has intrigued researchers for some time, leading them to study this specific technique over the last decades. And what they tell us is that the benefits of this breathing technique include things like:

• First, it increases Heart-Rate Variability scores, which by itself is already associated with:
  • Supporting the body in adapting to various daily scenarios, ranging from sleep to intense physical, mental or emotional activity.
  • Resilience to stress.
  • Increased blood cerebral flow
  • Athletic performance
  • Cognitive flexibility
  • Resistance to temptation and addiction
  • Better decision-making skills.
• Anti-Inflammatory effects
• Increased Emotional regulation skills
• Reduces hypertension
• Decreases the number of Alzheimer’s-related proteins in the blood
• Facilitates better pain management
• Reduces stress
• Reduces anxiety

4.1.2 Falling Asleep

Another benefit that I can personally attest to, is this technique’s ability to make oneself fall asleep more easily.

This is a benefit that occurs regardless of whether you do it at 0.1 Hz or at any other frequency (sometimes I enjoy breathing at slower frequencies).

Disclaimer: it won’t work 100% of the time, because there’s no perfect solution for falling asleep, but it’s a pretty good tool to have, and works pretty well most of the time.

The sine wave breathing technique can also calm the “monkey mind”. If you have the discipline to keep doing it regardless of what’s happening inside your mind, most of the time, after 10 or 12 minutes, the body and the monkey mind end up shutting off… whether you want it or not. It’s an intriguing thing to observe – at minute 5 you feel like it will be impossible for your mind to shut down, and yet, 5 minutes later, you’ve fallen asleep.

4.1.3 Save Time

Using this technique to fall asleep also has the benefit of not requiring taking extra time out of the day to achieve an harmonious sinusoidal state. So even those with no extra time in their days to spare can do and benefit from this technique.

Part 5: The Sine Wave Breathing Guide Tracks

Breathing like a sine wave is fairly easy:

• Inhale counting to 4
• Brief moment of suspension during the transition
• Exhale counting to 4
• Brief moment of suspension during the transition

As long as you follow these 4 steps, both your lungs and heart will follow a sinusoidal shape, and that’s good!

However!

If you want to maximize the health benefits, you must breathe at your resonant frequency, which for the average person is 0.1 Hz.

To breathe at the rate of 0.1 Hz, you can’t simply do it on your own, you need to have some sort of reference to guide you.

Something like this (listen at low volume)

This sound track is clicking at the exact frequency of 0.1 Hz, and so you simply need to synchronize your breath to it.

If you’re not quite sure how to do it or what all the clicks mean, take a look at this video. It’ll help!

Using an audio guiding track allows you to know for sure you are breathing at your resonant frequency.

If your resonant frequency is 0.11 Hz and not 0.1 Hz that’s not a problem—simply use a 0.11 Hz guiding track instead.

I have dozens of these tracks for you to use for free. Tracks like these:

You can get these mp3 tracks for free on my Gumroad page.

Simply use the code ZERO-DOWNLOAD at checkout 😉

These are the guiding tracks I have been using regularly since 2019 (so they have a proven track record).

For me, these mp3 files are the most practical method of breathing at my resonant frequency. Here’s why I love using them:

They are mp3 files:
✅ I can use them anytime, anywhere.
✅ I frequently sync my breath to my resonant frequency to fall asleep more easily. Since mp3 files can be downloaded to my phone, I can play them without an internet connection, reducing my exposure to artificial wireless radiation during the night.
✅ Audio player apps let me play mp3 files with the phone screen off, allowing me to sleep in a dark environment.
✅ I can set the audio player to stop playing after a while, letting me transition into a silent bedroom, and without draining the phone’s battery from continuous playback throughout the night.

They are audio guiding tracks, which are superior to visual guides:
✅ You can synchronize your breath with sound as you fall asleep—something that’s impossible with videos.
✅ You can engage in breathing exercises while multitasking (like driving or washing dishes) using sound, but images require full attention.
✅ The process of syncing your breath to audio creates a deeper focus and connection with the breathing practice, something that visuals simply can’t match.

Some other benefits specific to the Sinewave Breathing tracks are:
✅ They offer multiple frequencies ranging from 0.05 Hz to 0.12 Hz in precise increments of 0.005 Hz. Plus, I welcome community requests for even more finely-tuned tracks!
✅ They have been designed to have a soft clicking sound that is pleasant and musical to listen to (for the best experience, you should listen to them at low volume, almost at the edge of your hearing).
✅ I’ve personally been using these tracks since 2019 and never get tired of listening to them.
✅ This is a brand-new product I just launched—I’m giving it away completely free to the first 500 users! Simply use code ZERO-DOWNLOAD at checkout for a 100% discount.

Part 6: The Mandala

If you do decide to try the sine wave breathing technique, you now have all the theoretical and practical information to get started.

You have learned that breathing is spiritual, and breathing is mathematical. And the more mathematical, the more spiritual.

However, this technique can be supplemented with another extra layer of mathematical spirituality.

Many people don’t know this, but the work of psychologist Carl Jung is deeply rooted in mathematical spirituality, even if Jung himself never used that term. Still, his work is full of dualities, opposites, singularities and circles, which are all mathematical notions. He also expressed, more than once, a longing for a mathematical basis to the Psyche.

Jung regarded the mandala as a symbol of extreme importance in the human psyche, and helpful in helping individuals confront inner conflicts, identifying unconscious patterns, and moving towards psychological integration.

He suggested people draw or coloring mandalas, or even contemplating them during meditation.

How is this related to the sinewave breathing technique?

Because a mandala is a circle.
And a circle is a generator of sine waves.

Due to the close relation between these two geometrical shapes, it makes sense, intuitively, to connect Jung’s visualization technique to the sinewave breathing technique.

So, as you breathe, you can imagine a mandala glowing, rotating, changing in complexity and color, and pulsing in harmony with your breath.

Let your imagination dictate the transformations of the mandala, allowing it to evolve organically as you breathe.

And let this practice not only improve your focus, but ultimately also guide you toward an inner equilibrium.

This was the introduction to Mathematical Spirituality. I hope the theory has left you intrigued, and the practice has left you convinced.

Talk to you later.

Stay Synced!

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▪ The Science: What science has been saying about brain waves and brainwave entrainment, specific frequencies like 432 Hz and 0.1 Hz, and more.
▪ The Esoteric: My ideas on geometrical and mathematical spirituality.
▪ The Practical: How to transform these ideas into practical tools.