Parallelepiped
A parallelepiped is a three-dimensional polyhedron formed by six parallelogram faces, where opposite faces are congruent and parallel, analogous to a parallelogram in two dimensions.[1] It can be conceptualized as the set of all points obtained by linear combinations of three linearly independent vectors originating from a common vertex, thus spanning a volume in Euclidean space.[2] This geometric figure generalizes the rectangular box (or cuboid) and appears in classical solid geometry as described in Euclid's Elements, Book XI, where it serves as a fundamental prism-like solid with parallelogram bases.[3] Parallelepipeds are classified into several types based on the shapes of their faces and the angles between edges. A rectangular parallelepiped, also known as a cuboid or rectangular prism, has all six faces as rectangles, with edges meeting at right angles.[4] In contrast, a rhombohedron features six congruent rhombic faces, where all edges are of equal length, and a general parallelepiped (or oblique parallelepiped) has arbitrary parallelogram faces without such symmetries.[5] Special cases include the cube, a rectangular parallelepiped with square faces, and the right parallelepiped, where lateral edges are perpendicular to the base.[6] These variations highlight the parallelepiped's flexibility in modeling skewed or sheared three-dimensional structures. Key properties of a parallelepiped include its volume, which for vectors \vec{u}, \vec{v}, \vec{w} defining its edges from one vertex is the absolute value of the scalar triple product |\vec{u} \cdot (\vec{v} \times \vec{w})|, equivalent to the absolute determinant of the matrix formed by these vectors as columns.[7] This volume formula underscores its role in linear algebra, where the parallelepiped represents the "unit cell" spanned by basis vectors, and in applications such as crystallography for describing lattice structures. Additionally, the surface area consists of the sum of the areas of its six parallelogram faces, each computed as base times height or via vector cross products. Parallelepipeds also exhibit translational symmetry along their edges, making them essential in vector geometry and multivariable calculus for integrals over oriented volumes.[8]Definition and Fundamentals
Definition
A parallelepiped is a three-dimensional geometric figure formed by six parallelogram faces, consisting of three pairs of identical and parallel faces.[9] This structure arises as a prism with parallelogrammatic bases, where opposite faces are congruent and parallel, ensuring the figure maintains translational symmetry along its defining directions.[4] The parallelepiped extends the concept of a two-dimensional parallelogram into three dimensions, much like a cube generalizes a square.[3] In this analogy, just as a parallelogram is bounded by two pairs of parallel sides, the parallelepiped is delimited by three such pairs of faces, creating a solid that can be generated by translating a parallelogram along a third direction not coplanar with the first two. It possesses 6 faces, all parallelograms; 12 edges, with four edges meeting at each vertex in a consistent manner; and 8 vertices, where three edges converge.[10][6] As a fundamental polyhedron, the parallelepiped is inherently convex, meaning that the line segment connecting any two points within the figure lies entirely inside it, and it qualifies as a polyhedral solid with planar faces and straight edges.[11] This convexity underpins its role as a basic building block in three-dimensional geometry, serving as a prerequisite for exploring more advanced properties and constructions.Vector Representation
A parallelepiped can be formally constructed in three-dimensional Euclidean space using vector geometry, starting from a fixed point O and three emanating vectors \mathbf{a}, \mathbf{b}, and \mathbf{c} that are not coplanar. These vectors define the edges originating from O, and the parallelepiped is the convex hull of the eight points formed by their linear combinations with coefficients 0 or 1.[9][12] The vertices of the parallelepiped are precisely: O, \mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{a} + \mathbf{b}, \mathbf{a} + \mathbf{c}, \mathbf{b} + \mathbf{c}, and \mathbf{a} + \mathbf{b} + \mathbf{c}. This set of points ensures that opposite faces are identical parallelograms, with edges parallel to the defining vectors.[9][13] Any point within or on the boundary of the parallelepiped can be represented parametrically as a position vector \mathbf{r} = u \mathbf{a} + v \mathbf{b} + w \mathbf{c}, where the parameters satisfy $0 \leq u, v, w \leq 1. This parameterization describes the solid as the image of the unit cube under the linear transformation with columns \mathbf{a}, \mathbf{b}, and \mathbf{c}.[12][9] The edges of the parallelepiped consist of three sets of four parallel edges each, with lengths given by the magnitudes |\mathbf{a}|, |\mathbf{b}|, and |\mathbf{c}|, corresponding to the directions of the defining vectors.[13][9] The six faces are pairwise parallel parallelograms, each spanned by a pair of the defining vectors: one pair by \mathbf{a} and \mathbf{b}, another by \mathbf{a} and \mathbf{c}, and the third by \mathbf{b} and \mathbf{c}. These faces meet at the vertices, forming the closed surface of the parallelepiped.[12][9]Properties
Geometric Properties
A parallelepiped has six faces, each a parallelogram, twelve edges, and eight vertices.[9] Opposite faces are congruent and parallel, forming three pairs of identical faces. It is a type of zonohedron, generated as the Minkowski sum of three line segments in different directions.[14]Symmetry and Tessellation
A general parallelepiped possesses point group symmetry C_i, which consists solely of the identity and inversion operations through its center.[15] This inversion symmetry arises because the parallelepiped's opposite faces are equal parallelograms, ensuring that the figure maps onto itself under central inversion.[16] In special cases, such as when edges are orthogonal or faces are regular polygons, the symmetry elevates to higher point groups, but the generic form retains only this minimal symmetry.[15] The parallelepiped admits translations along its edge directions, allowing identical copies to be shifted uniformly along the three directions defined by its edge vectors to form a periodic structure.[17] These translations form the basis of periodic repetition in three dimensions, where each shift corresponds to one of the primitive lattice vectors spanning the figure. This translational property enables the parallelepiped to tessellate three-dimensional Euclidean space completely, filling it without gaps or overlaps through successive translations along its edge directions.[18] Such space-filling behavior is fundamental to periodic arrangements, as congruent parallelepipeds can replicate indefinitely to cover the entire volume. In crystallography, the parallelepiped serves as the unit cell for crystal lattices, particularly in the triclinic system, where it encapsulates the minimal repeating volume that generates the full lattice via translations.[18] All 14 Bravais lattices can be described using parallelepiped unit cells, with the general form corresponding to the lowest-symmetry triclinic lattice.[16]Geometric Quantities
Volume
The volume of a parallelepiped, defined by three vectors \mathbf{a}, \mathbf{b}, and \mathbf{c} emanating from a common vertex, is given by the absolute value of the scalar triple product:V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|.
This formula arises from the geometric interpretation of the parallelepiped as a prism with a parallelogram base spanned by \mathbf{b} and \mathbf{c}, extruded along \mathbf{a}. The area of the base parallelogram is the magnitude of the cross product \|\mathbf{b} \times \mathbf{c}\|, which provides a vector normal to the base plane. The height is then the projection of \mathbf{a} onto this normal direction, computed as the absolute value of the dot product \mathbf{a} \cdot (\mathbf{b} \times \mathbf{c}) / \|\mathbf{b} \times \mathbf{c}\|. Multiplying the base area by this height yields V = |\mathbf{a} \cdot (\mathbf{b} \times \mathbf{c})|, where the absolute value ensures a positive volume regardless of the vectors' orientation.[7][19] An equivalent expression uses the determinant of the matrix M formed by \mathbf{a}, \mathbf{b}, and \mathbf{c} as its columns:
V = |\det(M)|.
The scalar triple product equals the determinant \det(M), linking the volume directly to linear algebra properties; the absolute value again accounts for signed orientation, with the magnitude representing the unsigned volume. This formulation generalizes to higher dimensions via the determinant's role in computing the volume of parallelotopes.[12][20] If the vectors are measured in units of length (e.g., meters), the volume V has units of cubic length (e.g., cubic meters). Furthermore, under uniform scaling of all linear dimensions by a factor k > 0, the volume scales by k^3, as the determinant of the scaled matrix is k^3 \det(M).[21][22]