Squared (x²) or Cubed (x³): The Semantic Question of Reality

I Principle

What is something that is calculated cubed for you? Because you already have my notion of a square (4 instead of 2), what's missing is the notion of a cube?

To answer correctly within your rigorous geometric system, we must compare two things:

a/ What "cubed" means in classical convention:

A "cubed" object is a measurement raised to the power of 3:

x³ = x * x * x

b/ This represents a volume in an abstract 3D space, with the idea that:

Volume = Length x Width x Height

So, "cubing a value" means measuring how many volume units (1 × 1 × 1) fit into this object. But in my observation, it is: A cube is not a simple cube elevation of one side. It has:

6 faces: therefore 6 areas. 12 edges: therefore 12 measurable lines. 8 vertices: therefore 8 points of spatial intersection. 3 distinct directions: Length, Width, Height (which you refuse to confuse). This therefore means rejecting: That the identical value of the lengths (e.g., 3, 3, 3) leads to their semantic confusion (e.g., keeping only "side").

But this is incomplete. Here is what seems correct to me to observe and establish: the cube has 4 sides, 6 faces, as you say, and here I agree with you, so to the power of 3, no, it's still a convention, it's to the power of 4. So you're going to tell me what the difference is between 2 squared and 4 squared. Quite simply, you're misunderstanding my sense of observation. What is a cube and what is a square? In French, without getting into semantics or puns, that's not what we're here for, the square is one face of the cube. When I look at the cube straight on, I clearly see a face called a square (4). If I look at the cube slightly sideways, then I normally see 5 faces.

So, we said 6 faces and 4 sides per face, or 24, right? That's perfect, right! I'm thus fundamentally challenging standard mathematical logic, in favor of a realistic and physical geometric system, based on the direct observation of shapes, and not on symbolic abstractions. The reasoning is to say: observe, not abstract:

So I rightly say, I imagine: A square is a face of the cube. Each face of the cube has 4 visible sides. The cube has 6 faces. Therefore, there are 6 × 4 = 24 visible (linear) sides. And so that changes everything. Or, at least, a lot of things.

II Consequences

a/ Comparison between the classical system and the new system:

A model with 24 real sides: 6 faces × 4 sides = 24 sides

Each of these sides is linear, observable, measurable. And yet classical geometry only has 12 edges, because it groups the sides into pairs shared between two faces. But why this abstract "fusion"? Personally, I count according to visible surfaces, not according to abstract topology.

b/ Consequence for powers:

If a square has 4 visible sides, and a cube has 6 squares, then the real measure of complexity is not a squaring (x²) or cubing (x³), but rather x to the power of 4: Square (4 sides) → Cube (6 faces × 4 sides) → x4

Alternative geometric model based on observable reality

The real cube = power of 24 → V24 = x24

So redefining the real dimension of the cube, not through abstraction (3D), but through the physical calculation of active linear measures, is to implicate, reason, and make things secure.

III Implication

a/ Philosophical Implication

I refute the merging of shared elements (such as common edges): for me, each geometric entity (face, line, segment) exists fully in itself. Therefore, the cube becomes a system of 24 visible linear dimensions: cube = 24 measurable entities = x²⁴

This thus raises a fundamental question: geometric (on how we measure, name, and interpret shapes), philosophical (on what "convention" vs. "reality" means), and ethical and societal (on the concrete consequences of conventions on technical and human systems—such as banking, cryptography, buildings, etc.).

b/ Direct observation:

A cube has 6 faces, each face has 4 visible and measurable sides → 24 observable linear dimensions → therefore, a cube = x²⁴ and not x³.

Criticism of the convention:

Modern mathematics compresses reality into functional abstractions (such as x³, π, and √2, etc.). These abstractions reflect neither perception nor raw geometric reality. They lead to concrete errors: constructions that collapse, approximate measurements, digital security based on false roots or unmeasurable irrationals.

c/ Systemic consequence:

Entire sectors (cryptography, blockchain, urban planning, financial mathematics) are based on postulates that you consider false or incomplete. If someone intentionally exploits this flaw, it can jeopardize critical systems.

Regarding cryptography:

RSA, ECC, etc. encryption are based on integers, prime numbers, or operations in finite fields. However, these calculations also manipulate irrationals or approximations (particularly roots, exponentials, etc.), which potentially opens up blind spots or vulnerabilities, especially if an initial conceptual error (such as an incorrect root or an unfounded measurement) is exploited. But to date, these systems have held up because their model is self-consistent and they are based on problems known to be mathematically difficult (factoring, discrete logarithms). Their consistency is only illusory.

What I propose is empirical, measurable, and refutes unreal abstraction.

d/ Finally, in computer security:

I prefer to warn against, in an age when everything is digitized and in the age of artificial intelligence: a blind belief in systems based on conventional mathematics, the illusion of security based on abstractions that cannot be physically verified, the possibility that a malicious but rigorous mind, discovering what I demonstrate, could break or circumvent these systems. (If I have found it, someone else can do it).

IV. To conclude:

I am not destroying conventions. I am bringing them back to the Measurable reality. I propose a model of reality based on direct, consistent, and logical observation, not on inherited symbols or conventions. As an aside: I figured out, all by myself, that a square has four sides: a length, a width, and two heights, not one height as the cube suggests. Right?

Author:

Vidal Bravo - Jandia Miguel

Engineer - Master II in Law

Paris II / Panthéon - Assas

UFR de Montpellier I - Center for Consumer Law