Historical context. This page records an earlier stage of GRG. Personal motivations and earlier terminology are distinct from the current documented formulation.

From the very beginning, the problem of the circle left me dissatisfied: its area depends on π, an irrational number that makes the calculation incomplete and always “open.” I could not accept that the most perfect figure in geometry was based on an infinite and never-ending value.
It may sound strange, but I spent entire nights trying to find a possible solution, constantly reasoning about all of geometry and about the fateful circle.

The area of a rectangle is easy to understand: base times height. We simply repeat the base line along the height, like a raft made of logs placed side by side. Figuratively and geometrically, everything is simple.
With the triangle, things are already more complicated: in theory we should do the same, but with a point and a line—or alternatively, with two lines meeting at a point. In the first case we try to merge two incompatible entities (the point has no dimension, the line has one); in the second, we end up with an “accumulation of matter” at the vertex and a stretching on the opposite side. This may not make sense in an analogical way, but in my mind a digital vision of geometry was already emerging.

One night I asked myself: how did they really solve the triangle? They could not project a point onto a line, so they used a trick: they projected the base along the normal towards the opposite vertex and closed the open sides. In this way, they actually duplicated it. Try cutting a triangle out of paper and see. Now the solution was acceptable: we had returned to the rectangle, base × height, and since it was doubled, the formula became (base × height) ÷ 2. An artifice gave the solution.

But what about the circle? Why does it have that formula?
The circle should be conceived as a concentric extrusion of circumferences towards the center or vice versa… but then we face the original problem: from a point it becomes a line. At first I thought: maybe π arises precisely from this unnatural fusion—something impossible.

I also tried imagining a radius revolving around a pivot to form the circle. But again, there was too much “material” at the sides and compression at the center.

Then I realized I needed a trick, like with the triangle. I thought: let’s imagine a rubber circle, cut it along one radius, and stretch the circumference onto a straight base. At that point I get a triangle with base = circumference (2πr) and height = r. The area of this triangle is (2πr × r) ÷ 2, which simplifies to πr².
At first I was happy with this discovery, but then I told myself: if the solution is like this, it has no true geometric foundation! I can really duplicate a triangle, but not a circle.

And here came the turning point: I cannot duplicate it, but I can halve it.

I looked at the semicircle. What do I have? I know the radius, of course, but I also know the diameter!
And the area? Excluding the previous artifices, only one logical path remained: projecting the progressive heights of the arc onto the diameter. That was the only coherent development.
Do I have those positions? Looking at the circle… yes, I have them all! Because I can apply triangulation. By drawing several radii, I realized that I could determine each point with absolute precision, without using π.

I noticed that I could use Pythagoras or trigonometry to compute the values. But soon I realized that I wanted to solve not only the area but also the circumference. For the circumference, I could divide the angle into N equal parts and calculate the chords subtended by the radii: the more divisions, the closer the chords matched the arc, until for N → ∞ they perfectly overlapped.

For the area I needed equally spaced heights along the diameter; for the circumference I needed equally spaced chords along the angle. For this reason, I decided: Pythagoras for the area, trigonometry for the circumference.

As my reflection progressed, I realized I did not even need the semicircle, but only a quarter of a circle. And from there my mind started to wander further: to the sphere, and eventually to geometry as a whole.

Before, we solved the circle with simpler figures. Now, in my mind, the opposite was happening: the circle itself was generating all the others.
This is where Generative Radial Geometry begins.