Rhombohedron Unveiled

🧊 The Rhombic Hexahedron

In the realm of geometry, a rhombohedron is a specialized type of parallelepiped. Its defining characteristic is that all six of its faces are congruent rhombi.^[4] This geometric solid is also referred to as a rhombic hexahedron. While sometimes inaccurately called a rhomboid, the term 'rhomboid' technically refers to a two-dimensional figure.^[a]

📐 Fundamental Properties

A rhombohedron possesses 6 rhombic faces, 12 edges, and 8 vertices. It is classified as a convex, equilateral, zonohedron, and parallelohedron. Its symmetry group is typically denoted as C_i, with an order of 2.^[1]

🔗 Lattice Systems

The rhombohedron serves as the fundamental cell for the rhombohedral lattice system. This system is crucial in understanding crystal structures and is associated with a specific type of honeycomb structure where the cells themselves are rhombohedra.

⬡ The Cube

As previously noted, a cube is the most symmetrical rhombohedron, occurring when the apex angle ${\displaystyle \theta =90^{\circ }}$. Its faces are squares, and it is one of the five Platonic solids.

√2 The √2 Rhombohedron

This specific form arises when the ratio of the diagonals of the rhombic faces is ${\displaystyle {\sqrt {2}}}$. It is notably found in the dissection of the rhombic dodecahedron.

φ The Golden Rhombohedron

This variant is defined by the ratio of its diagonals being equal to the Golden Ratio (${\displaystyle \phi}$). It appears in the dissection of the rhombic triacontahedron.

📐 Defining Vectors

For a unit rhombohedron (side length ${\displaystyle a}$) with an acute apex angle ${\displaystyle \theta }$, and one vertex at the origin, the three generating vectors originating from this vertex are:

e₁ : ${\displaystyle (1,0,0),}$

e₂ : ${\displaystyle (\cos \theta ,\sin \theta ,0),}$

e₃ : ${\displaystyle (\cos \theta ,{\frac {\cos \theta -\cos ^{2}\theta }{\sin \theta }},{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }).}$

The coordinates of the other vertices are derived by vector addition of these base vectors.^[5]

📏 Volume and Height

The volume (${\displaystyle V}$) of a rhombohedron with side length ${\displaystyle a}$ and acute apex angle ${\displaystyle \theta }$ is given by:

${\displaystyle V=a^{3}(1-\cos \theta ){\sqrt {1+2\cos \theta }}=a^{3}{\sqrt {(1-\cos \theta )^{2}(1+2\cos \theta )}}=a^{3}{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }}~.}$

The height (${\displaystyle h}$) is related to the volume and base area (${\displaystyle a^{2}\sin \theta ~}$):

${\displaystyle h=a~{(1-\cos \theta ){\sqrt {1+2\cos \theta }} \over \sin \theta }=a~{{\sqrt {1-3\cos ^{2}\theta +2\cos ^{3}\theta }} \over \sin \theta }~.}$

🔺 Orthocentric Tetrahedra

A fascinating relationship exists between rhombohedra and orthocentric tetrahedra. Selecting four non-adjacent vertices of a rhombohedron invariably forms an orthocentric tetrahedron. Conversely, every orthocentric tetrahedron can be constructed in this manner.

🧩 Polyhedral Relationships

Rhombohedra are fundamental building blocks in the study of polyhedra. They are related to other important shapes like the rhombic dodecahedron and rhombic triacontahedron, often appearing as components in their dissections. They are also a type of parallelohedron and zonohedron.

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📜 Important Notice

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