Why I Started with Pi

From Pi to Pattern – A Personal Geometry Shift (1/4)

I started with something simple. A curiosity, really. About how the radius relates to the circle it creates. When you take the radius and try to wrap it around half the circle, it almost fits three times, but not quite. There’s always a sliver left. A remainder that refuses to settle. And that remainder isn’t just hard to express. It’s fundamentally irrational.
Which means: we can never describe the circle’s perimeter or area in exact terms. Only approximations. But here’s the contradiction that stayed with me.
When we draw a circle (whether by hand, in code, or on a screen) the result has a very real perimeter. A very real area. We can see it. Count it. Fill it.

So my question became:
If the shape is real, why can’t our numbers be?

I wasn’t trying to fight π. In classical geometry, it fits beautifully. It’s elegant. It holds a powerful place in theory. But when it comes to visual systems (to images, to pixels, to actual form) it leaves a gap.

A gap between what we calculate and what we can see.

I wanted to understand that gap. Not to dismiss classical geometry, but to complement it.
To find a way to describe shape in terms of what it occupies, not what it suggests.

At some point, I started mixing things up. Not to be clever, but just to make sense of it all.
Circles were hard to calculate. So I turned to other shapes. How do we measure them?

It didn’t take long to realize that squares were different.
They’re simple. Reliable. Their sides match, their diagonals behave, and their angles are always exactly 90°.
Mathematically, they’re honest. And with the help of Pythagoras, I could trace the logic of triangles within those squares, especially right-angled ones. That 90° curve started to feel like more than just a property.
It felt like a signature. That’s when something clicked.
What if the square isn’t just a shape, but a container?
A reference frame that other forms grow from, or fold into?

Many geometric shapes relate to the square in some way.
But the circle, the one I’d started with, fits it perfectly. This couldn’t be coincidence, there had to be more to it. You can inscribe a circle inside a square so precisely that the square’s base becomes the circle’s diameter. Half of that, the radius, is no longer abstract. It’s anchored. And suddenly, I wasn’t just looking at shapes. I was starting to see ratios.

From the idea of ratio I went onwards. If the circle fits its “bouding box” the exact perimeter and area should be smaller than the bounding box, but how much? This is we classic geometry helped me a lot. It turned out that the perimeter and area of the perfectly inscribed circle are always 78.45% of its “bounding box”. This doesn’t replace Pi because Pi is used to calculate the 78.45% but it gives another perspective. I didn’t need to perform difficult calculations anymore to get to know how much is “space is occupied” by a circle. I can now very simply scale it. When I know the diameter or radius, I easily calculate the perimeter or area of the bounding box and the circle perimeter and area are always 78.54% This has an error marging < 0,0001 which is scientificly acceptable.
The most beautiful part is the scalability making the error marging more handeleble.
If the boundingbox is 0,1cm2 the circle is 78,54%, but is the bounding box is 100km2 the circle area is also 78,54%. So if you know the exact value of the bounding box you know the circle value with an 0.0001 certainty. That’s what I found very interesting.

With this idea of ratio in mind, I kept going. If a circle fits exactly inside its bounding box (a perfect square) then its perimeter and area must be smaller than that square. But by how much?

That’s where classical geometry helped me. It turns out that a perfectly inscribed circle always takes up about 78.54% of its square. Both in area and in perimeter, relative to the square’s structure.

This doesn’t replace π, far from it. In fact, it’s π that defines the 78.54% (π/4). But what this gave me was a new kind of handle. A way to understand space without relying on continuous, irrational numbers.

Instead of calculating endlessly, I could now scale. If I knew the side of the square, or the diameter of the circle, I instantly knew both shapes. No approximations needed. The circle will always be 78.54% of the square. And the margin of error, even in rounded use, stays below 0.0001. Which means scientifically negligible for most real-world applications.

What made this powerful wasn’t just the ratio. It was the scalability.
Whether my square was 0.1 cm² or 100 km², the proportion held.
If I knew the bounding box, I knew the circle, with near absolute certainty.
And that’s what truly fascinated me.

That’s when it became clear to me:
The square isn’t just a frame, it is a measure.
A consistent, scalable, rational unit from which other shapes could be understood.
Not just in form, but in proportion.

But if that was true, then the way we measure needed to shift too.
Not in rejection of traditional units, but in addition to them.

I started thinking in ratios.
If the square was my reference, then the shapes inside it had their own proportions.
A perfectly inscribed circle?
It needs a unit that expresses its ratio to the square.

That’s how I came to define two new terms:

• SPU – Square Perimeter Unit
• SAU – Square Area Unit

Instead of measuring a circle’s area in abstract units, I could now say:
“It occupies 0.7854 SAU.”
No rounding errors, no irrational leftovers, just proportion.
And proportion, it turns out, is something visual systems can work with directly.

From Pi to Pattern – Insights into the Digital Geometry Shift
Series introduction

What started as a mathematical curiosity became something else.
I thought I was solving a classical geometry problem, something about π and curves and how we calculate them. But after further exploration, drawing, comparing and building, I now realize this wasn’t just a mathematical issue.

It is a digital issue.

This blog series is not about formulas or proofs.
It’s about a shift, from timeless theory to systems that need precision, patterns, and proportion.
From abstract constants to visual logic. From calculation… to comprehension.

Over the next few days, I’ll share the path that led me there.
Not the clean version, but the real one.