Circular Modular Arithmetic: Create Your Own Mandala While You Learn About Universal Mathematics

There are forms that seem to have been drawn by a greater intelligence, patterns that spring from the simple act of multiplying numbers in a circle.

How about experiencing for yourself how mathematics can be a bridge between art, nature and consciousness… and in a way that is practical, visual and fun?

Let me introduce you to the Circular Modular Arithmetic Simulator, created so that you can explore, experiment and be enchanted by the beauty of the natural mathematical patterns that give rise to stunning geometric forms, all of it with an artistic and interactive touch.

You are about to visualize and understand the multiplication table like never before: not in a square grid, but as living geometries that are born from simple rules and create true mathematical mandalas.

But what is Circular Modular Arithmetic?

Imagine a circle full of points, like the minutes on a clock. Now, associate each number with a point and draw a line connecting each point to the one that represents the result of its multiplication table.

For example: if we are using 10 points and the multiplying factor is 2, the line from point 1 will go to point 2 (1×2), the one from point 2 to point 4 (2×2), the one from point 3 to point 6 (3×2), and so on. When the numbers go beyond the total number of points, they “restart” at zero, just like on a clock. And this is what we call modular arithmetic. In other words: instead of growing infinitely, the results are “folded” onto the circle.

An Example: The 2×2 Factor and the Magic of the Cardioid

If you apply the multiplication factor of 2, as you increase the number of modules/points, you will see an incredible pattern emerge: a cardioid, a form that resembles a curved heart, perfectly symmetrical and harmonious.

The word cardioid comes from the Greek kardia, which means “heart”, and eidos, which means “form”, and the resemblance is no accident. Beyond visually evoking the symbol of the heart, this curve appears recurrently in natural phenomena and human technologies. A cardioid is also a type of curve generated by rotation (an epicycloid with circles of equal radius), which also resembles symmetrically a kind of sinusoidal spiral and a polar rose, but we will not delve into these more complex concepts here on this page.

It is observed, for example, in the caustics of reflected light, those bright drawings and gentle curves we see at the bottom of an illuminated coffee cup or in a sunlit swimming pool.

It is also found in acoustic phenomena, as can be illustrated by the sound capture pattern of cardioid microphones, which isolate the sound coming from the front while reducing noise from other directions.

But the most fascinating thing is to realize that the cardioid also manifests within ourselves. This curve traces the line of reflective symmetry of the human body, perceptible in the curves that divide our body down the middle, such as the cleft of the chin, the shape of the skull, the brain seen from above, the natural curve of the breasts, the buttocks, the male and female genitalia and many other structures.

It is as if nature used this pattern as an aesthetic and functional signature of life… in other words, it is not just beautiful, it is naturally recurrent throughout the universe.

This is the kind of pattern that the simulator below will reveal with a few clicks. You set the parameters, and the mathematics of the universe responds with art.

The Harmonic Dance Between Earth and Venus Around the Sun

If the cardioid curve enchants us with its reflections and human symmetries, there is another geometric form that connects us with the heavens: the pentagonal rosette formed by the orbital resonance between Earth and Venus, which evokes an example of the ancient concept defined by the Greeks that postulates the existence of a divine and mathematical harmony between the macrocosm and the microcosm: the Music of the Spheres.

Over the course of 8 Earth years and 13 Venusian years, both planets orbit the Sun tracing a cosmic dance of impressive precision. If we connect the meeting points (conjunctions) between the two, the curve that forms is a five-pointed star, a spiral that reveals a perfect pentagonal symmetry.

This figure reveals a Pentagram, a symbol widely associated with harmony, life and the Golden Ratio. The number 5, associated with the Pentagram, as well as the numbers 8 and 13, which mark the cycle of Earth and Venus, are all numbers from the Fibonacci Sequence, the same one that governs the growth of plants, the proportions of the human body and the patterns of shells and galaxies.

This orbital dance reveals a geometric intelligence that operates on macroscopic, celestial scales, yet repeats itself in the microstructures of life, such as in the shape of flower petals, in the arms of starfish, in the proportions of fruits and even in the arrangements of organic molecules.

What this synchrony suggests is something profound: the universe is coherent at all scales. From the orbit of the planets to the spiral of a pinecone, there is a hidden mathematical language, an invisible music made of symmetry, cycles and natural beauty.

In our simulator, you can experiment with forms that evoke these structures, and allow yourself to touch, even if only symbolically, the same harmony that governs the movement of the stars.

A Fascinating Realization

Perhaps one of the most enchanting realizations when using this simulator is to notice how harmonious curves can emerge from perfectly straight lines. That is right: all the beauty of the cardioid, of the five-pointed star, and of so many other forms that arise, is created merely by connecting points with straight segments.

This invites us to reflect on one of the central principles of Sacred Geometry: the idea that the order, the smoothness and the beauty of the universe can be expressed through simple and precise interactions. Symmetry need not be born of complexity, it can be revealed by patterns that are simple, cyclical and accessible to everyone.

The Simulator

• Choose the number of points on the circle;
• Adjust the multiplication factor;
• Adjust the internal and external rotation speed;
• Observe the formation of dynamic geometric patterns;
• Create mathematical works of art in seconds;
• Turn your art into an image and share it with the world.

Why does this matter?

Because learning Mathematics (and Sacred Geometry) can be something pleasurable, intuitive and transformative. This simulator is more than a little game: it is a playful way to reveal the principles of creation, to connect science and spirituality, to learn about the universal laws of harmony while we have fun.

Get ready to be amazed by what you can create with nothing but numbers and circles. Because, deep down, nature has been doing this for billions of years. The only difference is that now… you can see, understand and take part in the dance.

If, while interacting with the simulator, something inside you vibrated in recognition, perhaps this is your moment to go beyond contemplation… and begin a practical journey of integration with the patterns of universal harmony.

I invite you to discover my eBook “The Initiatic Map of Sacred Geometry”, a guide that does not merely teach, but activates. A progressive and accessible path for those who wish to learn how to apply Sacred Geometry in everyday life, as a true technology of consciousness.

You will learn to work with forms, rhythms and structures that reflect the cosmic order, bringing clarity, balance and purpose to many areas of your life.

This is an invitation to remembrance, to coherence, to conscious creation.

A safe, well-grounded initiation, fully connected with the timeless principles of nature.

May this be only the beginning of a journey that grows ever more beautiful, lucid and coherent with the patterns of your own essence.

A synergistic embrace,

Tiago Braga