Est-ce que ça va changer quoi si je fais ça à trois dimensions ?
Solid geometry

In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider the cuboid shown in the figure. The length of face diagonal AC is found from Pythagoras' theorem as:

where these three sides form a right triangle. Using diagonal AC and the horizontal edge CD, the length of body diagonal AD then is found by a second application of Pythagoras' theorem as:

or, doing it all in one step:

This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components { v } (the three mutually perpendicular sides):

This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the " n -dimensional Pythagorean theorem":

Let x 1, x 2, …, x n {\displaystyle x_{1},x_{2},\ldots,x_{n}} be orthogonal vectors in R. Consider the n -dimensional simplex S with vertices 0, x 1, …, x n {\displaystyle 0,x_{1},\ldots,x_{n}}. (Think of the (n − 1)-dimensional simplex with vertices x 1, …, x n {\displaystyle x_{1},\ldots,x_{n}} not including the origin as the "hypotenuse" of S and the remaining (n − 1)-dimensional faces of S as its "legs".) Then the square of the volume of the hypotenuse of S is the sum of the squares of the volumes of the n legs.

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:

Given an n -rectangular n -dimensional simplex, the square of the (n − 1)-content of the facet opposing the right vertex will equal the sum of the squares of the (n − 1)-contents of the remaining facets.
Images tridimensionnelles

Outre la sculpture et d'autres activités physiques qui peuvent créer des images tridimensionnelles à partir de matériaux solides, certaines techniques modernes, comme l'holographie, peuvent créer des images tridimensionnelles qui sont reproductibles mais intangibles au toucher. Certains procédés photographiques peuvent désormais rendre l'illusion de profondeur dans une image autrement "plate", mais la "photographie 3D" (stéréoscopie) ou le "film 3D" sont des illusions d'optique qui nécessitent des dispositifs spéciaux, comme des lunettes, pour créer cette illusion de profondeur.

Les copies d'images tridimensionnelles ont traditionnellement dû être fabriquées une par une, généralement par un individu ou une équipe d'artisans. À l'ère moderne, le développement des plastiques et d'autres technologies a permis de créer plusieurs copies d'un objet tridimensionnel avec moins d'efforts ; l'avènement et le développement de l'"impression 3D" ont étendu cette capacité.
### Query ###
Est-ce que ça va changer quoi si je fais ça à trois dimensions ?

### Main Text ###
Solid geometry

In terms of solid geometry, Pythagoras' theorem can be applied to three dimensions as follows. Consider the cuboid shown in the figure. The length of face diagonal AC is found from Pythagoras' theorem as:

where these three sides form a right triangle. Using diagonal AC and the horizontal edge CD, the length of body diagonal AD then is found by a second application of Pythagoras' theorem as:

or, doing it all in one step:

This result is the three-dimensional expression for the magnitude of a vector v (the diagonal AD) in terms of its orthogonal components { v } (the three mutually perpendicular sides):

This one-step formulation may be viewed as a generalization of Pythagoras' theorem to higher dimensions. However, this result is really just the repeated application of the original Pythagoras' theorem to a succession of right triangles in a sequence of orthogonal planes.

A substantial generalization of the Pythagorean theorem to three dimensions is de Gua's theorem, named for Jean Paul de Gua de Malves: If a tetrahedron has a right angle corner (like a corner of a cube), then the square of the area of the face opposite the right angle corner is the sum of the squares of the areas of the other three faces. This result can be generalized as in the " n -dimensional Pythagorean theorem":

Let x 1, x 2, …, x n {\displaystyle x_{1},x_{2},\ldots,x_{n}} be orthogonal vectors in R. Consider the n -dimensional simplex S with vertices 0, x 1, …, x n {\displaystyle 0,x_{1},\ldots,x_{n}}. (Think of the (n − 1)-dimensional simplex with vertices x 1, …, x n {\displaystyle x_{1},\ldots,x_{n}} not including the origin as the "hypotenuse" of S and the remaining (n − 1)-dimensional faces of S as its "legs".) Then the square of the volume of the hypotenuse of S is the sum of the squares of the volumes of the n legs.

This statement is illustrated in three dimensions by the tetrahedron in the figure. The "hypotenuse" is the base of the tetrahedron at the back of the figure, and the "legs" are the three sides emanating from the vertex in the foreground. As the depth of the base from the vertex increases, the area of the "legs" increases, while that of the base is fixed. The theorem suggests that when this depth is at the value creating a right vertex, the generalization of Pythagoras' theorem applies. In a different wording:

Given an n -rectangular n -dimensional simplex, the square of the (n − 1)-content of the facet opposing the right vertex will equal the sum of the squares of the (n − 1)-contents of the remaining facets.

### Supplementary Text ###
Images tridimensionnelles

Outre la sculpture et d'autres activités physiques qui peuvent créer des images tridimensionnelles à partir de matériaux solides, certaines techniques modernes, comme l'holographie, peuvent créer des images tridimensionnelles qui sont reproductibles mais intangibles au toucher. Certains procédés photographiques peuvent désormais rendre l'illusion de profondeur dans une image autrement "plate", mais la "photographie 3D" (stéréoscopie) ou le "film 3D" sont des illusions d'optique qui nécessitent des dispositifs spéciaux, comme des lunettes, pour créer cette illusion de profondeur.

Les copies d'images tridimensionnelles ont traditionnellement dû être fabriquées une par une, généralement par un individu ou une équipe d'artisans. À l'ère moderne, le développement des plastiques et d'autres technologies a permis de créer plusieurs copies d'un objet tridimensionnel avec moins d'efforts ; l'avènement et le développement de l'"impression 3D" ont étendu cette capacité.

### Draft ###

### 1. Query Decomposition

"Est-ce que ça va changer quoi si je fais ça à trois dimensions ?"

Breaking down:
- "ça" = "this/that" → ambiguous reference
- "fais ça" = informal imperative, suggests familiar procedure  
- "trois dimensions" = 3D space vs 2D plane

? What procedure? Pythagorean theorem application most probable given geometric context.

### 2. Contextual Triangulation

Response claims "théorème de Pythagore" → solid geometry extension.

Standard 2D Pythagorean: a² + b² = c²
3D generalization: a² + b² + c² = d²

But phrasing suggests user knows this. Query implies something beyond basic extension.

### 3. Dimensional Scaling Analysis

2D → 3D transition:
- Independent dimensions increase: 2 → 3
- Variables: (a,b) → (a,b,c) 
- Operations: addition → 3D distance formula

Simple extension: YES
But "fais ça" suggests user wants practical implications, not just formula.

### 4. De Gua's Theorem Connection

⟨H≈0.8⟩ Response mentions "théorème de De Gua" → Jean Paul de Gua de Malves, 18th century. Special case: right tetrahedron. Area relationships preserve Pythagorean scaling but add geometric constraints. Not just formula expansion but structural implications...

Back to concrete: ◐ De Gua's theorem states:
For right-angled tetrahedron with edges a, b, c, d forming right angle:
S²opp = S²a + S²b + S²c

Where S = area of corresponding face.

### 5. Practical Implications Assessment

"Qu'est-ce que ça change concrètement ?"

Key differences 2D→3D:

**Computational**:
- Operations: 2 summands → 3 summands
- Distance formula: √(a²+b²) → √(a²+b²+c²)

**Geometric constraints**:
- 2D: infinite orientations possible
- 3D: rotational degrees of freedom increase
- Right triangle → right parallelepiped/tetrahedron

**Physical applications**:
- Navigation: 2D maps → 3D spatial reasoning
- Construction: planar framing → structural analysis
- Physics: 2D forces → 3D field calculations

### 6. Knowledge Domain Requirements

2D Pythagorean: Basic algebra, geometry
3D Pythagorean: Vector algebra, coordinate geometry
De Gua's theorem: Advanced geometry, specialized knowledge

∴ Response correctly identifies need for specialized geometric understanding.

### 7. Answer Validation

Response structure:
☑ Acknowledges ambiguity in "ça"
☑ Correctly identifies Pythagorean theorem connection  
☑ Provides 3D formula extension
☑ Mentions De Gua's theorem (appropriate complexity)
☑ Addresses practical implications
☑ Recognizes complexity limits

**Confidence assessment**: ● High for geometric claims, ◐ moderate for practical applications interpretation

### 8. Final Synthesis

User query: informal, suggests familiar mathematical procedure
Response: provides geometric scaling analysis, appropriate complexity level, addresses practical implications while acknowledging formula limitations

**Translation challenge**: French informal phrasing ("fais ça") vs formal mathematical exposition → response strikes appropriate balance

**Key insight**: 3D extension isn't just computational - it fundamentally changes geometric relationships and practical applications. Response captures this distinction effectively.

### Answer###
Votre question touche à une extension naturelle du théorème de Pythagore en géométrie tridimensionnelle. Lorsque vous appliquez ce concept en trois dimensions, vous devez considérer que le théorème s'étend de manière relativement simple : au lieu d'avoir deux côtés formant un triangle rectangle, vous travaillez avec trois composantes orthogonales.

Dans l'espace tridimensionnel, le théorème peut s'exprimer comme : d² = a² + b² + c², où a, b et c sont les trois côtés du triangle rectangle dans le plan, et d représente la diagonale spatiale. Cette formule peut se visualiser facilement dans un cube ou un parallélépipède rectangle.

Il existe même une généralisation plus avancée connue sous le nom de théorème de De Gua, qui établit une relation entre les aires des faces d'un tétraèdre ayant un angle droit. Cette extension montre que le carré de l'aire de la face opposée à l'angle droit est égal à la somme des carrés des aires des trois autres faces.

Concernant ce que cela change concrètement, passer de deux à trois dimensions implique essentiellement d'ajouter une troisième dimension orthogonale aux calculs et de considérer les relations géométriques dans l'espace tridimensionnel plutôt que dans le plan. Les applications pratiques incluent le calcul de distances dans l'espace, l'analyse structurelle en architecture ou l'ingénierie, et la résolution de problèmes géométriques complexes.

Cependant, les applications précises dépendent largement du contexte spécifique de votre travail ou de vos études.