
Parallelepiped | |
---|---|
Type | Prism Plesiohedron |
Faces | 6 parallelograms |
Edges | 12 |
Vertices | 8 |
Symmetry group | Ci, [2+,2+], (×), order 2 |
Properties | convex, zonohedron |
In geometry, a parallelepiped is a three-dimensional figure formed by six parallelograms (the term rhomboid is also sometimes used with this meaning). By analogy, it relates to a parallelogram just as a cube relates to a square.
Three equivalent definitions of parallelepiped are
- a hexahedron with three pairs of parallel faces,
- a polyhedron with six faces (hexahedron), each of which is a parallelogram, and
- a prism of which the base is a parallelogram.
The rectangular cuboid (six rectangular faces), cube (six square faces), and the rhombohedron (six rhombus faces) are all special cases of parallelepiped.
"Parallelepiped" is now usually pronounced /ˌpærəˌlɛlɪˈpɪpɪd/ or /ˌpærəˌlɛlɪˈpaɪpɪd/; traditionally it was /ˌpærəlɛlˈɛpɪpɛd/ PARR-ə-lel-EP-ih-ped because of its etymology in Greek παραλληλεπίπεδον parallelepipedon (with short -i-), a body "having parallel planes".
Parallelepipeds are a subclass of the prismatoids.
Properties
Any of the three pairs of parallel faces can be viewed as the base planes of the prism. A parallelepiped has three sets of four parallel edges; the edges within each set are of equal length.
Parallelepipeds result from linear transformations of a cube (for the non-degenerate cases: the bijective linear transformations).
Since each face has point symmetry, a parallelepiped is a zonohedron. Also the whole parallelepiped has point symmetry Ci (see also triclinic). Each face is, seen from the outside, the mirror image of the opposite face. The faces are in general chiral, but the parallelepiped is not.
A space-filling tessellation is possible with congruent copies of any parallelepiped.
Volume
A parallelepiped is a prism with a parallelogram as base. Hence the volume of a parallelepiped is the product of the base area
and the height
(see diagram). With
(where
is the angle between vectors
and
), and
(where
is the angle between vector
and the normal to the base), one gets:
The mixed product of three vectors is called triple product. It can be described by a determinant. Hence for
the volume is:
V1 |
Another way to prove (V1) is to use the scalar component in the direction of of vector
:
The result follows.
An alternative representation of the volume uses geometric properties (angles and edge lengths) only:
V2 |
where ,
,
, and
are the edge lengths.
The proof of (V2) uses properties of a determinant and the geometric interpretation of the dot product:
Let be the 3×3-matrix, whose columns are the vectors
(see above). Then the following is true:
(The last steps use , ...,
,
,
, ...)
- Corresponding tetrahedron
The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped (see proof).
Surface area
The surface area of a parallelepiped is the sum of the areas of the bounding parallelograms: (For labeling: see previous section.)
Special cases by symmetry
Octahedral symmetry subgroup relations with inversion center | Special cases of the parallelepiped |
Form | Cube | Square cuboid | Trigonal trapezohedron | Rectangular cuboid | Right rhombic prism | Right parallelogrammic prism | Oblique rhombic prism |
---|---|---|---|---|---|---|---|
Constraints | | | |||||
Symmetry | Oh order 48 | D4h order 16 | D3d order 12 | D2h order 8 | C2h order 4 | ||
Image | |||||||
Faces | 6 squares | 2 squares, 4 rectangles | 6 rhombi | 6 rectangles | 4 rectangles, 2 rhombi | 4 rectangles, 2 parallelograms | 2 rhombi, 4 parallelograms |
- The parallelepiped with Oh symmetry is known as a cube, which has six congruent square faces.
- The parallelepiped with D4h symmetry is known as a square cuboid, which has two square faces and four congruent rectangular faces.
- The parallelepiped with D3d symmetry is known as a trigonal trapezohedron, which has six congruent rhombic faces (also called an isohedral rhombohedron).
- For parallelepipeds with D2h symmetry, there are two cases:
- Rectangular cuboid: it has six rectangular faces (also called a rectangular parallelepiped, or sometimes simply a cuboid).
- Right rhombic prism: it has two rhombic faces and four congruent rectangular faces.
- Note: the fully rhombic special case, with two rhombic faces and four congruent square faces
, has the same name, and the same symmetry group (D2h , order 8).
- Note: the fully rhombic special case, with two rhombic faces and four congruent square faces
- For parallelepipeds with C2h symmetry, there are two cases:
- Right parallelogrammic prism: it has four rectangular faces and two parallelogrammic faces.
- Oblique rhombic prism: it has two rhombic faces, while of the other faces, two adjacent ones are equal and the other two also (the two pairs are each other's mirror image).
Perfect parallelepiped
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and space diagonals. In 2009, dozens of perfect parallelepipeds were shown to exist, answering an open question of Richard Guy. One example has edges 271, 106, and 103, minor face diagonals 101, 266, and 255, major face diagonals 183, 312, and 323, and space diagonals 374, 300, 278, and 272.
Some perfect parallelepipeds having two rectangular faces are known. But it is not known whether there exist any with all faces rectangular; such a case would be called a perfect cuboid.
Parallelotope
Coxeter called the generalization of a parallelepiped in higher dimensions a parallelotope. In modern literature, the term parallelepiped is often used in higher (or arbitrary finite) dimensions as well.
Specifically in n-dimensional space it is called n-dimensional parallelotope, or simply n-parallelotope (or n-parallelepiped). Thus a parallelogram is a 2-parallelotope and a parallelepiped is a 3-parallelotope.
The diagonals of an n-parallelotope intersect at one point and are bisected by this point. Inversion in this point leaves the n-parallelotope unchanged. See also Fixed points of isometry groups in Euclidean space.
The edges radiating from one vertex of a k-parallelotope form a k-frame of the vector space, and the parallelotope can be recovered from these vectors, by taking linear combinations of the vectors, with weights between 0 and 1.
The n-volume of an n-parallelotope embedded in where
can be computed by means of the Gram determinant. Alternatively, the volume is the norm of the exterior product of the vectors:
If m = n, this amounts to the absolute value of the determinant of matrix formed by the components of the n vectors.
A formula to compute the volume of an n-parallelotope P in , whose n + 1 vertices are
, is
where
is the row vector formed by the concatenation of the components of
and 1.
Similarly, the volume of any n-simplex that shares n converging edges of a parallelotope has a volume equal to one 1/n! of the volume of that parallelotope.
Etymology
The term parallelepiped stems from Ancient Greek παραλληλεπίπεδον (parallēlepípedon, "body with parallel plane surfaces"), from parallēl ("parallel") + epípedon ("plane surface"), from epí- ("on") + pedon ("ground"). Thus the faces of a parallelepiped are planar, with opposite faces being parallel.
In English, the term parallelipipedon is attested in a 1570 translation of Euclid's Elements by Henry Billingsley. The spelling parallelepipedum is used in the 1644 edition of Pierre Hérigone's Cursus mathematicus. In 1663, the present-day parallelepiped is attested in Walter Charleton's Chorea gigantum.
Charles Hutton's Dictionary (1795) shows parallelopiped and parallelopipedon, showing the influence of the combining form parallelo-, as if the second element were pipedon rather than epipedon. Noah Webster (1806) includes the spelling parallelopiped. The 1989 edition of the Oxford English Dictionary describes parallelopiped (and parallelipiped) explicitly as incorrect forms, but these are listed without comment in the 2004 edition, and only pronunciations with the emphasis on the fifth syllable pi (/paɪ/) are given.
See also
- Lists of shapes
Notes
- In Euclidean geometry, the four concepts—parallelepiped and cube in three dimensions, parallelogram and square in two dimensions—are defined, but in the context of a more general affine geometry, in which angles are not differentiated, only parallelograms and parallelepipeds exist.
- "parallelepiped". Dictionary.com Unabridged (Online). n.d.
- Oxford English Dictionary 1904; Webster's Second International 1947
- Sawyer, Jorge F.; Reiter, Clifford A. (2011). "Perfect Parallelepipeds Exist". Mathematics of Computation. 80 (274): 1037–1040. arXiv:0907.0220. doi:10.1090/s0025-5718-2010-02400-7. S2CID 206288198..
- Morgan, C. L. (1974). Embedding metric spaces in Euclidean space. Journal of Geometry, 5(1), 101–107. https://doi.org/10.1007/bf01954540
- "parallelepiped". Oxford English Dictionary. 1933.
- parallhlepi/pedon. Liddell, Henry George; Scott, Robert; A Greek–English Lexicon at the Perseus Project.
References
- Coxeter, H. S. M. Regular Polytopes, 3rd ed. New York: Dover, p. 122, 1973. (He defines parallelotope as a generalization of a parallelogram and parallelepiped in n-dimensions.)
External links
- Weisstein, Eric W. "Parallelepiped". MathWorld.
- Weisstein, Eric W. "Parallelotope". MathWorld.
- Paper model parallelepiped (net)
ParallelepipedType Prism PlesiohedronFaces 6 parallelogramsEdges 12Vertices 8Symmetry group Ci 2 2 order 2Properties convex zonohedron In geometry a parallelepiped is a three dimensional figure formed by six parallelograms the term rhomboid is also sometimes used with this meaning By analogy it relates to a parallelogram just as a cube relates to a square Three equivalent definitions of parallelepiped are a hexahedron with three pairs of parallel faces a polyhedron with six faces hexahedron each of which is a parallelogram and a prism of which the base is a parallelogram The rectangular cuboid six rectangular faces cube six square faces and the rhombohedron six rhombus faces are all special cases of parallelepiped Parallelepiped is now usually pronounced ˌ p aer e ˌ l ɛ l ɪ ˈ p ɪ p ɪ d or ˌ p aer e ˌ l ɛ l ɪ ˈ p aɪ p ɪ d traditionally it was ˌ p aer e l ɛ l ˈ ɛ p ɪ p ɛ d PARR e lel EP ih ped because of its etymology in Greek parallhlepipedon parallelepipedon with short i a body having parallel planes Parallelepipeds are a subclass of the prismatoids PropertiesAny of the three pairs of parallel faces can be viewed as the base planes of the prism A parallelepiped has three sets of four parallel edges the edges within each set are of equal length Parallelepipeds result from linear transformations of a cube for the non degenerate cases the bijective linear transformations Since each face has point symmetry a parallelepiped is a zonohedron Also the whole parallelepiped has point symmetry Ci see also triclinic Each face is seen from the outside the mirror image of the opposite face The faces are in general chiral but the parallelepiped is not A space filling tessellation is possible with congruent copies of any parallelepiped VolumeParallelepiped generated by three vectors A parallelepiped is a prism with a parallelogram as base Hence the volume V displaystyle V of a parallelepiped is the product of the base area B displaystyle B and the height h displaystyle h see diagram With B a b sin g a b displaystyle B left mathbf a right cdot left mathbf b right cdot sin gamma left mathbf a times mathbf b right where g displaystyle gamma is the angle between vectors a displaystyle mathbf a and b displaystyle mathbf b and h c cos 8 displaystyle h left mathbf c right cdot left cos theta right where 8 displaystyle theta is the angle between vector c displaystyle mathbf c and the normal to the base one gets V B h a b sin g c cos 8 a b c cos 8 a b c displaystyle V B cdot h left left mathbf a right left mathbf b right sin gamma right cdot left mathbf c right left cos theta right left mathbf a times mathbf b right left mathbf c right left cos theta right left left mathbf a times mathbf b right cdot mathbf c right The mixed product of three vectors is called triple product It can be described by a determinant Hence for a a1 a2 a3 T b b1 b2 b3 T c c1 c2 c3 T displaystyle mathbf a a 1 a 2 a 3 mathsf T mathbf b b 1 b 2 b 3 mathsf T mathbf c c 1 c 2 c 3 mathsf T the volume is V det a1b1c1a2b2c2a3b3c3 displaystyle V left det begin bmatrix a 1 amp b 1 amp c 1 a 2 amp b 2 amp c 2 a 3 amp b 3 amp c 3 end bmatrix right V1 Another way to prove V1 is to use the scalar component in the direction of a b displaystyle mathbf a times mathbf b of vector c displaystyle mathbf c V a b scala b c a b a b c a b a b c displaystyle begin aligned V left mathbf a times mathbf b right left operatorname scal mathbf a times mathbf b mathbf c right left mathbf a times mathbf b right frac left left mathbf a times mathbf b right cdot mathbf c right left mathbf a times mathbf b right left left mathbf a times mathbf b right cdot mathbf c right end aligned The result follows An alternative representation of the volume uses geometric properties angles and edge lengths only V abc1 2cos a cos b cos g cos2 a cos2 b cos2 g displaystyle V abc sqrt 1 2 cos alpha cos beta cos gamma cos 2 alpha cos 2 beta cos 2 gamma V2 where a b c displaystyle alpha angle mathbf b mathbf c b a c displaystyle beta angle mathbf a mathbf c g a b displaystyle gamma angle mathbf a mathbf b and a b c displaystyle a b c are the edge lengths Proof of V2 The proof of V2 uses properties of a determinant and the geometric interpretation of the dot product Let M displaystyle M be the 3 3 matrix whose columns are the vectors a b c displaystyle mathbf a mathbf b mathbf c see above Then the following is true V2 detM 2 detMdetM detMTdetM det MTM det a aa ba cb ab bb cc ac bc c a2 b2c2 b2c2cos2 a abcos g abcos g c2 accos b bccos a accos b abcos g bccos a accos b b2 a2b2c2 a2b2c2cos2 a a2b2c2cos2 g a2b2c2cos a cos b cos g a2b2c2cos a cos b cos g a2b2c2cos2 b a2b2c2 1 cos2 a cos2 g cos a cos b cos g cos a cos b cos g cos2 b a2b2c2 1 2cos a cos b cos g cos2 a cos2 b cos2 g displaystyle begin aligned V 2 amp left det M right 2 det M det M det M mathsf T det M det M mathsf T M amp det begin bmatrix mathbf a cdot mathbf a amp mathbf a cdot mathbf b amp mathbf a cdot mathbf c mathbf b cdot mathbf a amp mathbf b cdot mathbf b amp mathbf b cdot mathbf c mathbf c cdot mathbf a amp mathbf c cdot mathbf b amp mathbf c cdot mathbf c end bmatrix amp a 2 left b 2 c 2 b 2 c 2 cos 2 alpha right amp quad ab cos gamma left ab cos gamma c 2 ac cos beta bc cos alpha right amp quad ac cos beta left ab cos gamma bc cos alpha ac cos beta b 2 right amp a 2 b 2 c 2 a 2 b 2 c 2 cos 2 alpha amp quad a 2 b 2 c 2 cos 2 gamma a 2 b 2 c 2 cos alpha cos beta cos gamma amp quad a 2 b 2 c 2 cos alpha cos beta cos gamma a 2 b 2 c 2 cos 2 beta amp a 2 b 2 c 2 left 1 cos 2 alpha cos 2 gamma cos alpha cos beta cos gamma cos alpha cos beta cos gamma cos 2 beta right amp a 2 b 2 c 2 left 1 2 cos alpha cos beta cos gamma cos 2 alpha cos 2 beta cos 2 gamma right end aligned The last steps use a a a2 displaystyle mathbf a cdot mathbf a a 2 a b abcos g displaystyle mathbf a cdot mathbf b ab cos gamma a c accos b displaystyle mathbf a cdot mathbf c ac cos beta b c bccos a displaystyle mathbf b cdot mathbf c bc cos alpha Corresponding tetrahedron The volume of any tetrahedron that shares three converging edges of a parallelepiped is equal to one sixth of the volume of that parallelepiped see proof Surface areaThe surface area of a parallelepiped is the sum of the areas of the bounding parallelograms A 2 a b a c b c 2 absin g bcsin a casin b displaystyle begin aligned A amp 2 cdot left mathbf a times mathbf b mathbf a times mathbf c mathbf b times mathbf c right amp 2 left ab sin gamma bc sin alpha ca sin beta right end aligned For labeling see previous section Special cases by symmetryOctahedral symmetry subgroup relations with inversion center Special cases of the parallelepipedForm Cube Square cuboid Trigonal trapezohedron Rectangular cuboid Right rhombic prism Right parallelogrammic prism Oblique rhombic prismConstraints a b c displaystyle a b c a b g 90 displaystyle alpha beta gamma 90 circ a b displaystyle a b a b g 90 displaystyle alpha beta gamma 90 circ a b c displaystyle a b c a b g displaystyle alpha beta gamma a b g 90 displaystyle alpha beta gamma 90 circ a b displaystyle a b a b 90 displaystyle alpha beta 90 circ a b 90 displaystyle alpha beta 90 circ a b displaystyle a b a b displaystyle alpha beta Symmetry Oh order 48 D4h order 16 D3d order 12 D2h order 8 C2h order 4ImageFaces 6 squares 2 squares 4 rectangles 6 rhombi 6 rectangles 4 rectangles 2 rhombi 4 rectangles 2 parallelograms 2 rhombi 4 parallelogramsThe parallelepiped with Oh symmetry is known as a cube which has six congruent square faces The parallelepiped with D4h symmetry is known as a square cuboid which has two square faces and four congruent rectangular faces The parallelepiped with D3d symmetry is known as a trigonal trapezohedron which has six congruent rhombic faces also called an isohedral rhombohedron For parallelepipeds with D2h symmetry there are two cases Rectangular cuboid it has six rectangular faces also called a rectangular parallelepiped or sometimes simply a cuboid Right rhombic prism it has two rhombic faces and four congruent rectangular faces Note the fully rhombic special case with two rhombic faces and four congruent square faces a b c displaystyle a b c has the same name and the same symmetry group D2h order 8 For parallelepipeds with C2h symmetry there are two cases Right parallelogrammic prism it has four rectangular faces and two parallelogrammic faces Oblique rhombic prism it has two rhombic faces while of the other faces two adjacent ones are equal and the other two also the two pairs are each other s mirror image Perfect parallelepipedA perfect parallelepiped is a parallelepiped with integer length edges face diagonals and space diagonals In 2009 dozens of perfect parallelepipeds were shown to exist answering an open question of Richard Guy One example has edges 271 106 and 103 minor face diagonals 101 266 and 255 major face diagonals 183 312 and 323 and space diagonals 374 300 278 and 272 Some perfect parallelepipeds having two rectangular faces are known But it is not known whether there exist any with all faces rectangular such a case would be called a perfect cuboid ParallelotopeCoxeter called the generalization of a parallelepiped in higher dimensions a parallelotope In modern literature the term parallelepiped is often used in higher or arbitrary finite dimensions as well Specifically in n dimensional space it is called n dimensional parallelotope or simply n parallelotope or n parallelepiped Thus a parallelogram is a 2 parallelotope and a parallelepiped is a 3 parallelotope The diagonals of an n parallelotope intersect at one point and are bisected by this point Inversion in this point leaves the n parallelotope unchanged See also Fixed points of isometry groups in Euclidean space The edges radiating from one vertex of a k parallelotope form a k frame v1 vn displaystyle v 1 ldots v n of the vector space and the parallelotope can be recovered from these vectors by taking linear combinations of the vectors with weights between 0 and 1 The n volume of an n parallelotope embedded in Rm displaystyle mathbb R m where m n displaystyle m geq n can be computed by means of the Gram determinant Alternatively the volume is the norm of the exterior product of the vectors V v1 vn displaystyle V left v 1 wedge cdots wedge v n right If m n this amounts to the absolute value of the determinant of matrix formed by the components of the n vectors A formula to compute the volume of an n parallelotope P in Rn displaystyle mathbb R n whose n 1 vertices are V0 V1 Vn displaystyle V 0 V 1 ldots V n is Vol P det V0 1 T V1 1 T Vn 1 T displaystyle mathrm Vol P left det left left V 0 1 right mathsf T left V 1 1 right mathsf T ldots left V n 1 right mathsf T right right where Vi 1 displaystyle V i 1 is the row vector formed by the concatenation of the components of Vi displaystyle V i and 1 Similarly the volume of any n simplex that shares n converging edges of a parallelotope has a volume equal to one 1 n of the volume of that parallelotope EtymologyThe term parallelepiped stems from Ancient Greek parallhlepipedon parallelepipedon body with parallel plane surfaces from parallel parallel epipedon plane surface from epi on pedon ground Thus the faces of a parallelepiped are planar with opposite faces being parallel In English the term parallelipipedon is attested in a 1570 translation of Euclid s Elements by Henry Billingsley The spelling parallelepipedum is used in the 1644 edition of Pierre Herigone s Cursus mathematicus In 1663 the present day parallelepiped is attested in Walter Charleton s Chorea gigantum Charles Hutton s Dictionary 1795 shows parallelopiped and parallelopipedon showing the influence of the combining form parallelo as if the second element were pipedon rather than epipedon Noah Webster 1806 includes the spelling parallelopiped The 1989 edition of the Oxford English Dictionary describes parallelopiped and parallelipiped explicitly as incorrect forms but these are listed without comment in the 2004 edition and only pronunciations with the emphasis on the fifth syllable pi paɪ are given See alsoLists of shapesNotesIn Euclidean geometry the four concepts parallelepiped and cube in three dimensions parallelogram and square in two dimensions are defined but in the context of a more general affine geometry in which angles are not differentiated only parallelograms and parallelepipeds exist parallelepiped Dictionary com Unabridged Online n d Oxford English Dictionary 1904 Webster s Second International 1947 Sawyer Jorge F Reiter Clifford A 2011 Perfect Parallelepipeds Exist Mathematics of Computation 80 274 1037 1040 arXiv 0907 0220 doi 10 1090 s0025 5718 2010 02400 7 S2CID 206288198 Morgan C L 1974 Embedding metric spaces in Euclidean space Journal of Geometry 5 1 101 107 https doi org 10 1007 bf01954540 parallelepiped Oxford English Dictionary 1933 parallhlepi pedon Liddell Henry George Scott Robert A Greek English Lexicon at the Perseus Project ReferencesCoxeter H S M Regular Polytopes 3rd ed New York Dover p 122 1973 He defines parallelotope as a generalization of a parallelogram and parallelepiped in n dimensions External linksLook up parallelepiped in Wiktionary the free dictionary Wikimedia Commons has media related to Parallelepipeds Weisstein Eric W Parallelepiped MathWorld Weisstein Eric W Parallelotope MathWorld Paper model parallelepiped net