10 features of the CUBE

As a writer with a background in geometry and design, the cube has been an object of fascination for me since I was a child trying to draw it in my notebooks without making it look like a misshapen square. But beyond its apparent simplicity, this shape hides mathematical secrets, practical applications, and even cultural symbolism. Ready to explore its many facets?

1️⃣ Its definition What geometrically defines a cube?

A cube is a regular polyhedron (a solid with flat faces) whose six faces are identical squares. It belongs to the family of hexahedrons (six-sided polyhedra), but with an additional condition: all its edges are of equal length, and the angles between faces are 90 degrees. This perfection of proportions makes it one of the five Platonic solids , those studied since ancient Greece for their symmetry and balance.

Historical curiosity: Plato associated the cube with the element of earth , perhaps because of its visual stability. Coincidence? Not at all: if you place it on a base, it appears immobile, anchored to the ground.

2️⃣ Faces, edges, and vertices: how are they related?

This is where Euler's formula (V - A + C = 2) comes into play , a mathematical gem that relates vertices (V), edges (A), and faces (C). In the cube:

• 8 vertices (the points where the edges converge)
• 12 edges (each line that joins two vertices)
• 6 faces (the squares that delimit it)

Applying the formula: 8 - 12 + 6 = 2. It works! This numerical balance explains why the cube can tessellate space (fill it without gaps), something we take advantage of when stacking boxes or building with cinder blocks.

3️⃣ Symmetry: Why does it seem so "perfect" to us?

The cube has 24 rotational symmetries (twists that leave it unchanged) and 48 if we include reflections (like looking at it in a mirror). To give you an idea: you can rotate it 90°, 180°, or 270° around axes that pass through the centers of faces, edges, or vertices, and it will always maintain its shape. This property makes it ideal for games like the Rubik's Cube , where symmetry is key to its 43 trillion possible combinations.

4️⃣ Diagonals: the invisible lines that structure it

In a cube there are three types of diagonals:

1. Face diagonals (join opposite vertices on the same face): each face has 2, making a total of 12.
2. Spatial diagonals (cross the cube from opposite vertex to vertex): 4 in total.



3. Edge diagonals (they do not exist here, but they do in other polyhedra).

The length of a space diagonal in a cube of edge a is a3 a 3

. For example, if the edge measures 2 cm, the diagonal measures ≈3.46 cm. This calculation is vital in 3D design for calculating internal distances in structures.

5️⃣ Volume and surface area: how to calculate them?

This is where the cube shines for its simplicity. If the edge measures a :

• Volume (V) : V=a3 V = a 3
• Surface area (A) : A=6a2 A = 6 a 2

Comparison table with examples:

Notice the pattern? The area grows quadratically, and the volume grows cubically. This explains why larger cubes are less efficient in terms of surface area/volume (important in cell biology or thermal storage).

6️⃣ Duality: Does it have a "twin" in the geometric world?

Yes: the octahedron . In geometry, the dual of a polyhedron is obtained by connecting the centers of its faces. If you do this with a cube, you get an octahedron (eight triangular faces). This relationship is reciprocal: the dual of the octahedron is the cube. Together, they form a pair that is repeated in crystallography theories and even in 3D modeling algorithms.

7️⃣ Practical applications: Do we really use cubes in real life?

More than you imagine! Some examples:

• Architecture : Mies van der Rohe 's panels used virtual cubes to create modular spaces.
• Logistics : Standard containers (such as 20-foot containers) are straight prisms, but stacking them creates a network of imaginary cubes.



• Technology : Voxels (3D pixels) are typically cubes in medical rendering software.

Even in nature, pyrite crystals sometimes form almost perfect cubes, although not with the precision of a mathematical model.

8️⃣ The cube in art and culture: symbol or mere form?

From Picasso's Cubism (which fragmented reality into geometric forms) to cinema: in 2001: A Space Odyssey , the monolith is a rectangular prism, but its visual impact evokes the solemnity of the cube. In philosophy , it represents the earthly, the stable, as opposed to the sphere (the divine).

In literature, Borges used it in "The Library of Babel" as a module for an infinite universe. Coincidence? No: the cube, by being able to repeat itself endlessly, suggests both order and claustrophobia.

9️⃣ How does it relate to other polyhedra?

The cube is the basis for building more complex figures:

• Cuboctahedron : Cutting the corners of a cube until the original faces become triangles.
• Rhombicuboctahedron : An "inflation" of the cube that mixes squares and triangles.
• Hypercube : Its 4D version, impossible to fully visualize in our three-dimensional space.

The latter is key in theories of quantum physics and relativity, where time is treated as a "fourth dimension."

🔟 Perspectives: How does it change depending on where you look at it from?

This is where parallel projection and conic perspective come in . In technical drawing, a cube can be represented as:

• Orthogonal views (plan, elevation, profile), typical in engineering drawings.
• Isometric perspective , which maintains proportions but simulates depth.

But in art, we distort its edges to create a sense of volume. A useful exercise: draw a cube rotated 45°. Its faces are deformed into rhombuses, but our brain still recognizes it as a "cube" thanks to learned patterns.

And in case you're wondering: yes, this section is tricky. It's not an intrinsic characteristic of the cube, but rather a function of how our perception interacts with its geometry. But without that interaction, what would be the point of all that math?

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