Definition:Parallelepiped
Definition
A parallelepiped is a polyhedron formed by three pairs of parallel planes:
In the above example, the pairs of parallel planes are:
- $ABCD$ and $HGFE$
- $ADEH$ and $BCFG$
- $ABGH$ and $DCFE$
Opposite Face of Parallelepiped
The opposite face of the face $F$ of a parallelepiped $P$ is the face of $P$ which is parallel to $F$.
In the above example, the pairs of parallel planes are:
Base of Parallelepiped
One of the faces of the parallelepiped is chosen arbitrarily, distinguished from the others and called a base of the parallelepiped.
The opposite face to that face is also referred to as one of the bases.
It is usual to choose one of the bases to be the one which is conceptually on the bottom.
In the above, $ABCD$ and $EFGH$ would conventionally be identified as being the bases.
Lateral Face of Parallelepiped
The faces of the parallelepiped which are not the bases are called the lateral faces of the parallelepiped.
In the above, the lateral faces are $ABHG$, $BCFG$, $CDEF$ and $ADEH$.
Height of Parallelepiped
The height of a parallelepiped is the length of the perpendicular between the planes of the bases.
In the above diagram, $h$ is the height of the parallelepiped one of whose bases is $AB$.
Types of Parallelepiped
Right Parallelepiped
A right parallelepiped is a parallelepiped whose lateral faces are square or rectangular.
Rectangular Parallelepiped
A rectangular parallelepiped is better known as a cuboid:
A cuboid is a parallelepiped whose faces are all rectangular.
Oblique Parallelepiped
An oblique parallelepiped is a parallelepiped such that the sides of the lateral faces are not perpendicular to the bases.
Also known as
Some sources give this as parallelopiped.
Also see
- Results about parallelepipeds can be found here.
Historical Note
The term parallelepiped is never actually defined by Euclid, although he specifies the concept:
In the words of Euclid:
- If a solid be contained by parallel planes, the opposite planes in it are equal and parallelogrammic.
(The Elements: Book $\text{XI}$: Proposition $24$)
The proposition that follows this one uses the term in a proof:
In the words of Euclid:
- If a parallelepipedal solid be cut by a plane which is parallel to the opposite planes, then, as the base is to the base, so will the solid be to the solid.
(The Elements: Book $\text{XI}$: Proposition $25$)
Linguistic Note
The word parallelepiped is divided into syllables and stressed as par-al-lel-ep-i-ped.
The pronunciation par-al-lel-e-pi-ped is technically incorrect.
The occasionally-seen spelling parallelopiped is also considered by some authorities to be incorrect.
Sources
- 1926: Sir Thomas L. Heath: Euclid: The Thirteen Books of The Elements: Volume 3 (2nd ed.) ... (previous) ... (next): Book $\text{XI}$. Propositions
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): parallelepiped
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): parallelepiped
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): parallelepiped
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): parallelepiped