Euclid's Geometry - Part II - - The references

I Preliminaries

a/ The historical and scientific context

I propose a completely different interpretation, where what you call a 'point' can also be seen as a very small sphere, with content. This changes the perspective, the structure, and the logic we draw from it."

And yes, the founding fathers of mathematics and geometry could have realized this by making the simple drawing I did, but for now I want to add that for me, it's my observation that acts like a geometric and mathematical microscope. Do you understand? So, if we look for the smallest point, we see an infinite series of central points (because when we enlarge with a microscope, we realize that the infinitely small is actually large, and each point is no longer really a point, but, for example, an atom or a cell).

The founding fathers of ancient geometry (Euclid, Archimedes, etc.) defined the point as a dimensionless entity, a simple abstract "without content." This definition was philosophical and ideal, based on the perception and tools of the time. Today, with microscopes and fine observation instruments, what we call a point can no longer be perceived as a strictly null "point."

In physical reality, this "point" is a small sphere, a very small but non-zero volume, visible due to the resolution limit effect.

What you call a point is a misnomer because to the human eye, what is a sphere actually appears (in my example, with a microscope, for example) as a point, given its smallness, a bit like an atom.

What we call a "point" is often a misnomer or a misnomer, because at a certain scale of perception (human or geometric), a very small sphere looks like a point, but it is not a dimensionless point.

What must be understood is that the fathers of geometry in antiquity did not have access to a microscope, for example, or a magnifying glass, so what they describe as a point can, and I'm being kind, appear today only as a sphere and nothing else.

b/ Geometric and Conceptual Implication

The "mathematical" point is an abstraction.

In my view, this point is best represented by a sphere of very small but non-zero dimension.

This changes the view of nested geometric structures, where a point becomes a sphere, which contains an inner cube, which in turn contains other shapes, and so on. It's a way of reconciling the mathematical abstraction with observable physical reality.

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II The situation :

This not my best picture to formalize a geometric figure. but it’s a good kick off to have a conversation about that. What I wanted to draw, is the following figure (Not easy to code or develop with React and Javascript and personal computer) :

a/ My geometric structure :

1. From the outside to the inside:

The outer cube:

Contains the two pyramids (which form this cube by assembly).

Contains the sphere (inscribed).

The two pyramids:

Form the outer cube.

By contact with the sphere (at the edges/faces/points), they also envelop the sphere (without entirely containing it geometrically).

Serve as mediators between the outer cube and the sphere.

The sphere:

Is contained within the outer cube.

Contains the inner cube (at its center).

Is in contact with the pyramids, which allows for the exchange of perspective.

The inner cube:

Is at the center of the sphere.

Contains the central point.

The central point:

Is at the center of the inner cube.

In your logic, it becomes a sphere, not an abstract point.

b/ In other way:

2. From the inside out:

The "central point":

Becomes a minimal sphere (≠ Euclidean point), therefore a non-zero volume.

This sphere potentially contains another interior cube, and so on → infinite structure (fractal).

This logic induces a hierarchy of centers:

The central point is the center of the interior cube,

The interior cube is the center of the sphere,

The sphere is the center of the pyramids (via contact),

The pyramids are the center of the exterior cube,

And conversely, the exterior cube encompasses everything.

Conclusion of my theory :

The central point is not a point, but a sphere, contained within a cube, itself contained within a sphere, itself encompassed by two pyramids, which form an exterior cube. This reasoning can be extended infinitely, because:

If a "point" is a sphere, then it can contain another interior cube, containing a new point, etc.

I thus propose a complete rewriting of the geometric structure of space, where the centers are never empty, but always full (spherical).

III The other way to see all that :

a/ In classical geometry:

V1 = length (dimension 1)

V2 = area (surface, dimension 2)

V3 = volume (dimension 3)

So V3 = volume, it's the measure of the space occupied by a three-dimensional object.

b/ What I learned in school: V1 = length (1D) V2 = area (2D) V3 = volume (3D);

What I claim is that there is V4 = width (4D), so you're not going to tell me that in a perfect square, length equals width, yes or no? You're going to say that, and I agree. But is that why we mustn’t elude one or the other, or why one or the other mustn’t be erased, as in your 3D system where we have this: V1 = length (1D) V2 = area (2D) V3 = volume (3D) even though the rectangle, which for me is an imperfect square, demonstrates the opposite, right? Well, we must absolutely take into account L and l, i.e., length and width.

IV Coherent and Courageous Reasoning

a/ Classically in geometry:

1/ The square is seen as "the rectangle with equal sides".

2/ So we implicitly say length = width. Therefore, theorists remove one or the other accordingly.

3/ But I say: "No, let's not generalize a particular case."

And I think I'm perfectly right if we want to think of geometry as an open, non-Euclidean, or physical system:

a/ In physics, no measurement guarantees that two lengths are perfectly equal.

b/ So why force width to be equal to length, as in a square? I choose to separate them.

b/ I don't assume : that width is equivalent to length. Why must a perfect square be considered a general definition? In other words:

The perfect square is a special case. The rectangle is the general geometric reality.

For me, all this makes sense:

1/ The square: ideal, abstract form

2/ The rectangle: real, differentiated form

So, in my opinion, the 4th dimension (V4) should serve to distinguish width as a geometric entity independent of length.

Author:

Vidal Bravo - Jandia Miguel

Engineer - Master II in Law

Paris II / Panthéon - Assas

UFR de Montpellier I - Center for Consumer Law