China Product
Introduction
Wallpaper groups categorize patterns by their symmetries. Subtle differences may place similar patterns in different groups, while patterns which are very different in style, color, scale or orientation may belong to the same group.
Consider the following examples: hose barb fitting
Example A: Cloth, Tahiti fire hose coupling
Example B: Ornamental painting, Nineveh, Assyria square tube connectors
Example C: Painted porcelain, China
Examples A and B have the same wallpaper group; it is called p4m. Example C has a different wallpaper group, called p4g. The fact that A and B have the same wallpaper group means that they have the same symmetries, regardless of details of the designs, whereas C has a different set of symmetries despite any superficial similarities.
A complete list of all seventeen possible wallpaper groups can be found below.
Symmetries of patterns
A symmetry of a pattern is, loosely speaking, a way of transforming the pattern so that the pattern looks exactly the same after the transformation. For example, translational symmetry is present when the pattern can be translated (shifted) some finite distance and appear unchanged. Think of shifting a set of vertical stripes horizontally by one stripe. The pattern is unchanged. Strictly speaking, a true symmetry only exists in patterns which repeat exactly and continue indefinitely. A set of only, say, five stripes does not have translational symmetry when shifted, the stripe on one end "disappears" and a new stripe is "added" at the other end. In practice, however, classification is applied to finite patterns, and small imperfections may be ignored.
Sometimes two categorizations are meaningful, one based on shapes alone and one also including colors. When colors are ignored there may be more symmetry. In black and white there are also 17 wallpaper groups; e.g., a colored tiling is equivalent with one in black and white with the colors coded radially in a circularly symmetric "bar code" in the centre of mass of each tile.
The types of transformations that are relevant here are called Euclidean plane isometries. For example:
If we shift example B one unit to the right, so that each square covers the square that was originally adjacent to it, then the resulting pattern is exactly the same as the pattern we started with. This type of symmetry is called a translation. Examples A and C are similar, except that the smallest possible shifts are in diagonal directions.
If we turn example B clockwise by 90, around the centre of one of the squares, again we obtain exactly the same pattern. This is called a rotation. Examples A and C also have 90 rotations, although it requires a little more ingenuity to find the correct centre of rotation for C.
We can also flip example B across a horizontal axis that runs across the middle of the image. This is called a reflection. Example B also has reflections across a vertical axis, and across two diagonal axes. The same can be said for A.
However, example C is different. It only has reflections in horizontal and vertical directions, not across diagonal axes. If we flip across a diagonal line, we do not get the same pattern back; what we do get is the original pattern shifted across by a certain distance. This is part of the reason that A and B have a different wallpaper group than C.
History
All 17 groups were used by Egyptian craftsmen, and used extensively in the Muslim world. A proof that there were only 17 possible patterns was first carried out by Evgraf Fedorov in 1891 and then derived independently by George Plya in 1924.
Formal definition and discussion
Mathematically, a wallpaper group or plane crystallographic group is a type of topologically discrete group of isometries of the Euclidean plane which contains two linearly independent translations.
Two such isometry groups are of the same type (of the same wallpaper group) if they are the same up to an affine transformation of the plane. Thus e.g. a translation of the plane (hence a translation of the mirrors and centres of rotation) does not affect the wallpaper group. The same applies for a change of angle between translation vectors, provided that it does not add or remove any symmetry (this is only the case if there are no mirrors and no glide reflections, and rotational symmetry is at most of order 2).
Unlike in the three-dimensional case, we can equivalently restrict the affine transformations to those which preserve orientation.
It follows from the Bieberbach theorem that all wallpaper groups are different even as abstract groups (as opposed to e.g. Frieze groups, of which two are isomorphic with Z).
2D patterns with double translational symmetry can be categorized according to their symmetry group type.
Isometries of the Euclidean plane
Isometries of the Euclidean plane fall into four categories (see the article Euclidean plane isometry for more information).
Translations, denoted by Tv, where v is a vector in R2. This has the effect of shifting the plane applying displacement vector v.
Rotations, denoted by Rc,, where c is a point in the plane (the centre of rotation), and is the angle of rotation.
Reflections, or mirror isometries, denoted by FL, where L is a line in R2. (F is for "flip"). This has the effect of reflecting the plane in the line L, called the reflection axis or the associated mirror.
Glide reflections, denoted by GL,d, where L is a line in R2 and d is a distance. This is a combination of a reflection in the line L and a translation along L by a distance d.
The independent translations condition
The condition on linearly independent translations means that there exist linearly independent vectors v and w (in R2) such that the group contains both Tv and Tw.
The purpose of this condition is to distinguish wallpaper groups from frieze groups, which possess a translation but not two linearly independent ones, and from two-dimensional discrete point groups, which have no translations at all. In other words, wallpaper groups represent patterns that repeat themselves in two distinct directions, in contrast to frieze groups which only repeat along a single axis.
(It is possible to generalise this situation. We could for example study discrete groups of isometries of Rn with m linearly independent translations, where m is any integer in the range 0 m n.)
The discreteness condition
The discreteness condition means that there is some positive real number , such that for every translation Tv in the group, the vector v has length at least (except of course in the case that v is the zero vector).
The purpose of this condition is to ensure that the group has a compact fundamental domain, or in other words, a "cell" of nonzero, finite area, which is repeated through the plane. Without this condition, we might have for example a group containing the translation Tx for every rational number x, which would not correspond to any reasonable wallpaper pattern.
One important and nontrivial consequence of the discreteness condition in combination with the independent translations condition is that the group can only contain rotations of order 2, 3, 4, or 6; that is, every rotation in the group must be a rotation by 180, 120, 90, or 60. This fact is known as the crystallographic restriction theorem, and can be generalised to higher-dimensional cases.
Notations for wallpaper groups
Crystallographic notation
Crystallography has 230 space groups to distinguish, far more than the 17 wallpaper groups, but many of the symmetries in the groups are the same. Thus we can use a similar notation for both kinds of groups, that of Carl Hermann and Charles-Victor Mauguin. An example of a full wallpaper name in Hermann-Mauguin style is p31m, with four letters or digits; more usual is a shortened name like cmm or pg.
For wallpaper groups the full notation begins with either p or c, for a primitive cell or a face-centred cell; these are explained below. This is followed by a digit, n, indicating the highest order of rotational symmetry: 1-fold (none), 2-fold, 3-fold, 4-fold, or 6-fold. The next two symbols indicate symmetries relative to one translation axis of the pattern, referred to as the "main" one; if there is a mirror perpendicular to a translation axis we choose that axis as the main one (or if there are two, one of them). The symbols are either m, g, or 1, for mirror, glide reflection, or none. The axis of the mirror or glide reflection is perpendicular to the main axis for the first letter, and either parallel or tilted 180/n (when n > 2) for the second letter. Many groups include other symmetries implied by the given ones. The short notation drops digits or an m that can be deduced, so long as that leaves no confusion with another group.
A primitive cell is a minimal region repeated by lattice translations. All but two wallpaper symmetry groups are described with respect to primitive cell axes, a coordinate basis using the translation vectors of the lattice. In the remaining two cases symmetry description is with respect to centred cells which are larger than the primitive cell, and hence have internal repetition; the directions of their sides is different from those of the translation vectors spanning a primitive cell. Hermann-Mauguin notation for crystal space groups uses additional cell types.
Examples
p2 (p211): Primitive cell, 2-fold rotation symmetry, no mirrors or glide reflections.
p4g (p4gm): Primitive cell, 4-fold rotation, glide reflection perpendicular to main axis, mirror axis at 45.
cmm (c2mm): Centred cell, 2-fold rotation, mirror axes both perpendicular and parallel to main axis.
p31m (p31m): Primitive cell, 3-fold rotation, mirror axis at 60.
Here are all the names that differ in short and full notation.
Crystallographic short and full names
Short
p2
pm
pg
cm
pmm
pmg
pgg
cmm
p4m
p4g
p6m
Full
p211
p1m1
p1g1
c1m1
p2mm
p2mg
p2gg
c2mm
p4mm
p4gm
p6mm
The remaining names are p1, p3, p3m1, p31m, p4, and p6.
Orbifold notation
Orbifold notation for wallpaper groups, introduced by John Horton Conway (Conway, 1992), is based not on crystallography, but on topology. We fold the infinite periodic tiling of the plane into its essence, an orbifold, then describe that with a few symbols.
A digit, n, indicates a centre of n-fold rotation. By the crystallographic restriction theorem, n must be 2, 3, 4, or 6.
An asterisk, *, indicates a mirror. It interacts with the digits as follows:
Digits before * are centres of pure rotation (cyclic).
Digits after * are centres of rotation with mirrors through them (dihedral).
A cross, x, indicates a glide reflection. Pure mirrors combine with lattice translation to produce glides, but those are already accounted for so we do not notate them.
The "no symmetry" symbol, o, stands alone, and indicates we have only lattice translations with no other symmetry.
Consider the group denoted in crystallographic notation by cmm; in Conway's notation, this will be 2*22. The 2 before the * says we have a 2-fold rotation centre with no mirror through it. The * itself says we have a mirror. The first 2 after the * says we have a 2-fold rotation centre on a mirror. The final 2 says we have an independent second 2-fold rotation centre on a mirror, one which is not a duplicate of the first one under symmetries.
The group denoted by pgg will be 22x. We have two pure 2-fold rotation centres, and a glide reflection axis. Contrast this with pmg, Conway 22*, where crystallographic notation mentions a glide, but one that is implicit in the other symmetries of the orbifold.
Conway and crystallographic correspondence
Conway
o
xx
*x
**
632
*632
Crystal.
p1
pg
cm
pm
p6
p6m
Conway
333
*333
3*3
442
*442
4*2
Crystal.
p3
p3m1
p31m
p4
p4m
p4g
Conway
2222
22x
22*
*2222
2*22
Crystal.
p2
pgg
pmg
pmm
cmm
Why there are exactly seventeen groups
An orbifold has a face, edges, and vertices; thus we can view it as a polygon. When we unfold it, that polygon tiles the plane, with each feature replicated infinitely by the action of the wallpaper symmetry group. Thus when Conway's orbifold notation mentions a feature, such as the 4-fold rotation centre in 4*2, that feature unfolds into an infinite number of replicas across the plane. Hiding within this description is a key to the enumeration.
Consider a cube, with its corners, edges, and faces. We count 8 corners, 12 edges, and 6 faces. Alternately adding and subtracting, we note that 8 12 + 6 = 2. Now consider a tetrahedron. It has 4 corners, 6 edges, and 4 faces; and we note that 4 6 + 4 = 2. Let us explore further. For generality, use the term vertex instead of corner. Split a face with a new edge, causing one face to become two. Now we have 4 7 + 5 = 2. Next, split an edge with a new vertex, causing the one edge to become two. We have 5 8 + 5 = 2. This is not coincidence; it is a demonstration of the surface Euler characteristic, = V E + F, and the beginning of a proof of its invariance.
When an orbifold replicates by symmetry to fill the plane, its features create a structure of vertices, edges, and polygon faces which must be consistent with the Euler characteristic. Reversing the process, we can assign numbers to the features of the orbifold, but fractions, rather than whole numbers. Because the orbifold itself is a quotient of the full surface by the symmetry group, the orbifold Euler characteristic is a quotient of the surface Euler characteristic by the order of the symmetry group.
The orbifold Euler characteristic is 2 minus the sum of the feature values, assigned as follows:
A digit n before a * counts as (n1)/n.
A digit n after a * counts as (n1)/2n.
Both * and x count as 1.
The "no symmetry" o counts as 2.
For a wallpaper group, the sum for the characteristic must be zero; thus the feature sum must be 2.
Examples
632: 5/6 + 2/3 + 1/2 = 2
3*3: 2/3 + 1 + 1/3 = 2
4*2: 3/4 + 1 + 1/4 = 2
22x: 1/2 + 1/2 + 1 = 2
Now enumeration of all wallpaper groups becomes a matter of arithmetic, of listing all feature strings with values summing to 2.
Incidentally, feature strings with other sums are not nonsense; they imply non-planar tilings, not discussed here. (When the orbifold Euler characteristic is negative, the tiling is hyperbolic; when positive, spherical or bad).
Guide to recognising wallpaper groups
To work out which wallpaper group corresponds to a given design, one may use the following table.
Size of smallest
rotation
Has reflection?
Yes
No
360 / 6
p6m
p6
360 / 4
Has mirrors at 45?
p4
Yes: p4m
No: p4g
360 / 3
Has rot. centre off mirrors?
p3
Yes: p31m
No: p3m1
360 / 2
Has perpendicular reflections?
Has glide reflection?
Yes
No
Has rot. centre off mirrors?
pmg
Yes: pgg
No: p2
Yes: cmm
No: pmm
none
Has glide axis off mirrors?
Has glide reflection?
Yes: cm
No: pm
Yes: pg
No: p1
See also this overview with diagrams.
The seventeen groups
Each of the groups in this section has two cell structure diagrams, which are to be interpreted as follows:
a centre of rotation of order two (180).
a centre of rotation of order three (120).
a centre of rotation of order four (90).
a centre of rotation of order six (60).
an axis of reflection.
an axis of glide reflection.
On the right-hand side diagrams, different equivalence classes of symmetry elements are colored (and rotated) differently.
The brown or yellow area indicates a fundamental domain, i.e. the smallest part of the pattern which is repeated.
The diagrams on the right show the cell of the lattice corresponding to the smallest translations; those on the left sometimes show a larger area.
Group p1
Example and diagram for p1
Cell structure for p1
Cell structure for p1
Orbifold notation: o.
The group p1 contains only translations; there are no rotations, reflections, or glide reflections.
Examples of group p1
Computer generated
Medival wall diapering
The two translations (cell sides) can each have different lengths, and can form any angle.
Group p2
Example and diagram for p2
Cell structure for p2
Cell structure for p2
Orbifold notation: 2222.
The group p2 contains four rotation centres of order two (180), but no reflections or glide reflections.
Examples of group p2
Computer generated
Cloth, Sandwich Islands (Hawaii)
Mat on which Egyptian king stood
Egyptian mat (detail)
Ceiling of Egyptian tomb
Wire fence, U.S.
Group pm
Example and diagram for pm
Cell structure for pm
Cell structure for pm
Orbifold notation: **.
The group pm has no rotations. It has reflection axes, they are all parallel.
Examples of group pm
(The first three have a vertical symmetry axis, and the last two each have a different diagonal one.)
Computer generated
Dress of a figure in a tomb at Biban el Moluk, Egypt
Egyptian tomb, Thebes
Ceiling of a tomb at Gourna, Egypt. Reflection axis is diagonal.
Indian metalwork at the Great Exhibition in 1851. This is almost pm (ignoring short diagonal lines between ovals motifs, which make it p1).
Group pg
Example and diagram for pg
Cell structure for pg
Cell structure for pg
Orbifold notation: xx.
The group pg contains glide reflections only, and their axes are all parallel. There are no rotations or reflections.
Examples of group pg
Computer generated
Mat on which Egyptian king stood
Egyptian mat (detail)
Pavement in Salzburg. Glide reflection axis runs northeast-southwest.
One of the colorings of the snub square tiling; the glide reflection lines are in the direction upper left / lower right; ignoring colors there is much more symmetry than just pg, then it is p4g (see there for this image with equally colored triangles)
Without the details inside the zigzag bands the mat is pmg; with the details but without the distinction between brown and black it is pgg.
Ignoring the wavy borders of the tiles, the pavement is pgg.
Group cm
Cell structure for cm
Cell structure for cm
Orbifold notation: *x.
The group cm contains no rotations. It has reflection axes, all parallel. There is at least one glide reflection whose axis is not a reflection axis; it is halfway between two adjacent parallel reflection axes.
This group applies for symmetrically staggered rows (i.e. there is a shift per row of half the translation distance inside the rows) of identical objects, which have a symmetry axis perpendicular to the rows.
Examples of group cm
Computer generated
Dress of Amun, from Abu Simbel, Egypt
Dado from Biban el Moluk, Egypt
Bronze vessel in Nimroud, Assyria
Spandrils of arches, the Alhambra, Spain
Soffitt of arch, the Alhambra, Spain
Persian tapestry
Indian metalwork at the Great Exhibition in 1851
Dress of a figure in a tomb at Biban el Moluk, Egypt
Group pmm
Example and diagram for pmm
Cell structure for pmm
Cell structure for pmm
Orbifold notation: *2222.
The group pmm has reflections in two perpendicular directions, and four rotation centres of order two (180) located at the intersections of the reflection axes.
Examples of group pmm
Computer generated
2D image of lattice fence, U.S. (in 3D there is additional symmetry)
Mummy case stored in The Louvre
Ceiling of Egyptian tomb. Ignoring minor asymmetries, this would be cmm.
Mummy case stored in The Louvre. Would be type p4m except for the mismatched coloring.
Group pmg
Example and diagram for pmg
Cell structure for pmg
Cell structure for pmg
Orbifold notation: 22*.
The group pmg has two rotation centres of order two (180), and reflections in only one direction. It has glide reflections whose axes are perpendicular to the reflection axes. The centres of rotation all lie on glide reflection axes.
Examples of group pmg
Computer generated
Cloth, Sandwich Islands (Hawaii)
Ceiling of Egyptian tomb
Floor tiling in Prague, the Czech Republic
Bowl from Kerma
Group pgg
Example and diagram for pgg
Cell structure for pgg
Cell structure for pgg
Orbifold notation: 22x.
The group pgg contains two rotation centres of order two (180), and glide reflections in two perpendicular directions. The centres of rotation are not located on the glide reflection axes. There are no reflections.
Examples of group pgg
Computer generated
Bronze vessel in Nimroud, Assyria
Pavement in Budapest, Hungary. Glide reflection axes are diagonal.
Group cmm
Cell structure for cmm
Cell structure for cmm
Orbifold notation: 2*22.
The group cmm has reflections in two perpendicular directions, and a rotation of order two (180) whose centre is not on a reflection axis. It also has two rotations whose centres are on a reflection axis.
This group is frequently seen in everyday life, since the most common arrangement of bricks in a brick building utilises this group (see example below).
The rotational symmetry of order 2 with centres of rotation at the centres of the sides of the rhombus is a consequence of the other properties.
The pattern corresponds to each of the following:
symmetrically staggered rows of identical doubly symmetric objects
a checkerboard pattern of two alternating rectangular tiles, of which each, by itself, is doubly symmetric
a checkerboard pattern of alternatingly a 2-fold rotationally symmetric rectangular tile and its mirror image
Examples of group cmm
Computer generated
one of the 8 semi-regular tessellations; ignoring color this is this group cmm, otherwise group p1
Suburban brick wall, U.S.
Ceiling of Egyptian tomb. Ignoring colors, this would be p4g.
Egyptian
Persian tapestry
Egyptian tomb
Turkish dish
Group p4
Example and diagram for p4
Cell structure for p4
Cell structure for p4
Orbifold notation: 442.
The group p4 has two rotation centres of order four (90), and one rotation centre of order two (180). It has no reflections or glide reflections.
Examples of group p4
A p4 pattern can be looked upon as a repetition in rows and columns of equal square tiles with 4-fold rotational symmetry. Also it can be looked upon as a checkerboard pattern of two such tiles, a factor smaller and rotated 45.
Computer generated
Ceiling of Egyptian tomb; ignoring colors this is p4, otherwise p2
Ceiling of Egyptian tomb
Frieze, the Alhambra, Spain. Requires close inspection to see why there are no reflections.
Viennese cane
Renaissance earthenware
Group p4m
Example and diagram for p4m
Cell structure for p4m
Cell structure for p4m
Orbifold notation: *442.
The group p4m has two rotation centres of order four (90), and reflections in four distinct directions (horizontal, vertical, and diagonals). It has additional glide reflections whose axes are not reflection axes; rotations of order two (180) are centred at the intersection of the glide reflection axes. All rotation centres lie on reflection axes.
This corresponds to a straightforward grid of rows and columns of equal squares with the four reflection axes. Also it corresponds to a checkerboard pattern of two of such squares.
Examples of group p4m
Examples displayed with the smallest translations horizontal and vertical (like in the diagram):
Computer generated
one of the 3 regular tessellations (in this checkerboard coloring, smallest translations are diagonal)
Demiregular tiling with triangles; ignoring colors, this is p4m, otherwise cmm
one of the 8 semi-regular tessellations (ignoring color also, with smaller translations)
Ornamental painting, Nineveh, Assyria
Storm drain, U.S.
Egyptian mummy case
Persian glazed tile
Examples displayed with the smallest translations diagonal (like on a checkerboard):
Cloth, Otaheite (Tahiti)
Egyptian tomb
Cathedral of Bourges
Dish from Turkey, Ottoman period
Group p4g
Example and diagram for p4g
Cell structure for p4g
Cell structure for p4g
Orbifold notation: 4*2.
The group p4g has two centres of rotation of order four (90), which are each other's mirror image, but it has reflections in only two directions, which are perpendicular. There are rotations of order two (180) whose centres are located at the intersections of reflection axes. It has glide reflections axes parallel to the reflection axes, in between them, and also at an angle of 45 with these.
A p4g pattern can be looked upon as a checkerboard pattern of copies of a square tile with 4-fold rotational symmetry, and its mirror image. Alternatively it can be looked upon (by shifting half a tile) as a checkerboard pattern of copies of a horizontally and vertically symmetric tile and its 90 rotated version. Note that neither applies for a plain checkerboard pattern of black and white tiles, this is group p4m (with diagonal translation cells).
Note that the diagram on the left represents in area twice the smallest square that is repeated by translation.
Examples of group p4g
Computer generated
Bathroom linoleum, U.S.
Painted porcelain, China
Fly screen, U.S.
Painting, China
one of the colorings of the snub square tiling (see also at pg)
Group p3
Cell structure for p3 (the rotation centres at the centres of the triangles are not shown)
Cell structure for p3
Orbifold notation: 333.
The group p3 has three different rotation centres of order three (120), but no reflections or glide reflections.
Imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, but the two are not equal, not each other's mirror image, and not both symmetric (if the two are equal we have p6, if they are each other's mirror image we have p31m, if they are both symmetric we have p3m1; if two of the three apply then the third also, and we have p6m). For a given image, three of these tessellations are possible, each with rotation centres as vertices, i.e. for any tessellation two shifts are possible. In terms of the image: the vertices can be the red, the blue or the green triangles.
Equivalently, imagine a tessellation of the plane with regular hexagons, with sides equal to the smallest translation distance divided by 3. Then this wallpaper group corresponds to the case that all hexagons are equal (and in the same orientation) and have rotational symmetry of order three, while they have no mirror image symmetry (if they have rotational symmetry of order six we have p6, if they are symmetric with respect to the main diagonals we have p31m, if they are symmetric with respect to lines perpendicular to the sides we have p3m1; if two of the three apply then the third also, and we have p6m). For a given image, three of these tessellations are possible, each with one third of the rotation centres as centres of the hexagons. In terms of the image: the centres of the hexagons can be the red, the blue or the green triangles.
Examples of group p3
Computer generated
one of the 8 semi-regular tessellations (ignoring the colors: p6); the translation vectors are rotated a little to the right compared with the directions in the underlying hexagonal lattice of the image
Street pavement in Zakopane, Poland
Wall tiling in the Alhambra, Spain (and the whole wall); ignoring all colors this is p3 (ignoring only star colors it is p1)
Group p3m1
Example and diagram for p3m1
Cell structure for p3m1
Cell structure for p3m1
Orbifold notation: *333.
The group p3m1 has three different rotation centres of order three (120). It has reflections in the three sides of an equilateral triangle. The centre of every rotation lies on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
Like for p3, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three, and both are symmetric, but the two are not equal, and not each other's mirror image. For a given image, three of these tessellations are possible, each with rotation centres as vertices. In terms of the image: the vertices can be the red, the dark blue or the green triangles.
Examples of group p3m1
one of the 3 regular tessellations (ignoring colors: p6m)
another regular tessellation (ignoring colors: p6m)
one of the 8 semi-regular tessellations (ignoring colors: p6m)
Persian glazed tile (ignoring colors: p6m)
Persian ornament
Floor tiling in Budapest, Hungary (ignoring colors: p6m)
Painting, China (see detailed image)
Computer generated
Group p31m
Example and diagram for p31m
Cell structure for p31m
Cell structure for p31m
Orbifold notation: 3*3.
The group p31m has three different rotation centres of order three (120), of which two are each other's mirror image. It has reflections in three distinct directions. It has at least one rotation whose centre does not lie on a reflection axis. There are additional glide reflections in three distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
Like for p3 and p3m1, imagine a tessellation of the plane with equilateral triangles of equal size, with the sides corresponding to the smallest translations. Then half of the triangles are in one orientation, and the other half upside down. This wallpaper group corresponds to the case that all triangles of the same orientation are equal, while both types have rotational symmetry of order three and are each other's mirror image, but not symmetric themselves, and not equal. For a given image, only one such tessellation is possible. In terms of the image: the vertices can not be dark blue triangles.
Examples of group p31m
Persian glazed tile
Painted porcelain, China
Painting, China
Computer generated
Group p6
Example and diagram for p6
Cell structure for p6
Cell structure for p6
Orbifold notation: 632.
The group p6 has one rotation centre of order six (60); it has also two rotation centres of order three, which only differ by a rotation of 60 (or, equivalently, 180), and three of order two, which only differ by a rotation of 60. It has no reflections or glide reflections.
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with C3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with C6 symmetry (with the edges of the tiles not necessarily part of the pattern).
Examples of group p6
Computer generated
Wall panelling, the Alhambra, Spain
Persian ornament
Group p6m
Example and diagram for p6m
Cell structure for p6m
Cell structure for p6m
Orbifold notation: *632.
The group p6m has one rotation centre of order six (60); it has also two rotation centres of order three, which only differ by a rotation of 60 (or, equivalently, 180), and three of order two, which only differ by a rotation of 60. It has also reflections in six distinct directions. There are additional glide reflections in six distinct directions, whose axes are located halfway between adjacent parallel reflection axes.
A pattern with this symmetry can be looked upon as a tessellation of the plane with equal triangular tiles with D3 symmetry, or equivalently, a tessellation of the plane with equal hexagonal tiles with D6 symmetry (with the edges of the tiles not necessarily part of the pattern). Thus the simplest examples are a triangular lattice with or without connecting lines, and a hexagonal tiling with one color for outlining the hexagons and one for the background.
Examples of group p6m
Computer generated
one of the 8 semi-regular tessellations
another semi-regular tessellation
another semi-regular tessellation
Persian glazed tile
King's dress, Khorsabad, Assyria; this is almost p6m (ignoring inner parts of flowers, which make it cmm)
Bronze vessel in Nimroud, Assyria
Byzantine marble pavement, Rome
Painted porcelain, China
Painted porcelain, China
Lattice types
There are five lattice types, corresponding to the five possible wallpaper groups of the lattice itself. The wallpaper group of a pattern with this lattice of translational symmetry cannot have more, but may have less symmetry than the lattice itself.
In the 5 cases of rotational symmetry of order 3 or 6, the cell consists of two equilateral triangles (hexagonal lattice, itself p6m).
In the 3 cases of rotational symmetry of order 4, the cell is a square (square lattice, itself p4m).
In the 5 cases of reflection or glide reflection, but not both, the cell is a rectangle (rectangular lattice, itself pmm), therefore the diagrams show a rectangle, but a special case is that it actually is a square.
In the 2 cases of reflection combined with glide reflection, the cell is a rhombus (rhombic lattice, itself cmm); a special case is that it actually is a square.
In the case of only rotational symmetry of order 2, and the case of no other symmetry than translational, the cell is in general a parallelogram (parallelogrammatic lattice, itself p2), therefore the diagrams show a parallelogram, but special cases are that it actually is a rectangle, rhombus, or square.
Symmetry groups
The actual symmetry group should be distinguished from the wallpaper group. Wallpaper groups are collections of symmetry groups. There are 17 of these collections, but for each collection there are infinitely many symmetry groups, in the sense of actual groups of isometries. These depend, apart from the wallpaper group, on a number of parameters for the translation vectors, the orientation and position of the reflection axes and rotation centers.
The numbers of degrees of freedom are:
6 for p2
5 for pmm, pmg, pgg, and cmm
4 for the rest.
However, within each wallpaper group, all symmetry groups are algebraically isomorphic.
Some symmetry group isomorphisms:
p1: Z2
pm: Z D
pmm: D D.
Dependence of wallpaper groups on transformations
The wallpaper group of a pattern is invariant under isometries and uniform scaling (similarity transformations).
Translational symmetry is preserved under arbitrary bijective affine transformations.
Rotational symmetry of order two ditto; this means also that 4- and 6-fold rotation centres at least keep 2-fold rotational symmetry.
Reflection in a line and glide reflection are preserved on expansion/contraction along, or perpendicular to, the axis of reflection and glide reflection. It changes p6m, p4g, and p3m1 into cmm, p3m1 into cm, and p4m, depending on direction of expansion/contraction, into pmm or cmm. A pattern of symmetrically staggered rows of points is special in that it can convert by expansion/contraction from p6m to p4m.
Note that when a transformation decreases symmetry, a transformation of the same kind (the inverse) obviously for some patterns increases the symmetry. Such a special property of a pattern (e.g. expansion in one direction produces a pattern with 4-fold symmetry) is not counted as a form of extra symmetry.
Change of colors does not affect the wallpaper group if any two points that have the same color before the change, also have the same color after the change, and any two points that have different colors before the change, also have different colors after the change.
If the former applies, but not the latter, such as when converting a color image to one in black and white, then symmetries are preserved, but they may increase, so that the wallpaper group can change.
Web demo and software
There exist several software graphic tools that will let you create 2D patterns using wallpaper symmetry groups. Usually, you can edit the original tile and its copies in the entire pattern are updated automatically.
Tess, a nagware tessellation program for multiple platforms, supports all wallpaper, frieze, and rosette groups, as well as Heesch tilings.
Kali, free graphical symmetry editor available online and for download.
Inkscape, a free vector graphics editor, supports all 17 groups plus arbitrary scales, shifts, rotates, and color changes per row or per column, optionally randomized to a given degree. (See )
SymmetryWorks is a commercial plugin for Adobe Illustrator, supports all 17 groups.
Arabeske is a free standalone tool, supports a subset of wallpaper groups.
See also
Wikimedia Commons has media related to: Wallpaper group diagrams
List of planar symmetry groups (summary of this page)
Tessellation
Point group
Crystallography
Symmetry groups in one dimension
M. C. Escher
Aperiodic tiling
Notes
^ E. Fedorov (1891) "Simmetrija na ploskosti" [Symmetry in the plane], Zapiski Imperatorskogo Sant-Petersburgskogo Mineralogicheskogo Obshchestva [Proceedings of the Imperial St. Petersburg Mineralogical Society], series 2, vol. 28, pages 345-291 (in Russian).
^ George Plya (1924) "ber die Analogie der Kristallsymmetrie in der Ebene," Zeitschrift fr Kristallographie, vol. 60, pages 278-282.
^ Weyl, Hermann (1952), Symmetry, Princeton University Press, ISBN 0-691-02374-3
^ It helps to consider the squares as the background, then we see a simple patterns of rows of rhombuses.
References
The Grammar of Ornament (1856), by Owen Jones. Many of the images in this article are from this book; it contains many more.
J. H. Conway (1992). "The Orbifold Notation for Surface Groups". In: M. W. Liebeck and J. Saxl (eds.), Groups, Combinatorics and Geometry, Proceedings of the L.M.S. Durham Symposium, July 515, Durham, UK, 1990; London Math. Soc. Lecture Notes Series 165. Cambridge University Press, Cambridge. pp. 438447
Grnbaum, Branko; Shephard, G. C. (1987): Tilings and Patterns. New York: Freeman. ISBN 0-7167-1193-1.
Pattern Design, Lewis F. Day
External links
The 17 plane symmetry groups by David E. Joyce
Introduction to wallpaper patterns by Chaim Goodman-Strauss and Heidi Burgiel
Description by Silvio Levy
Example tiling for each group, with dynamic demos of properties
Overview with example tiling for each group
Tiling plane and fancy by Steve Edwards
Escher Web Sketch, a java applet with interactive tools for drawing in all 17 plane symmetry groups
Burak, a Java applet for drawing symmetry groups.
Beobachtungen zum geometrischen Motiv der Pelta
Categories: Crystallography | Discrete groups | Euclidean symmetries | OrnamentsHidden categories: Articles to be merged from February 2010 | All articles to be merged
Monday, April 26, 2010
Wallpaper group
Bicycle infantry
China Product
History
Origins
Bicycle Corps at Fort Missoula in 1897. mesh truck tarps
Numerous experiments were carried out to determine the possible role of bicycles and cycling within military establishments until in 1894 a turning point occurred due to improved resilience of pneumatics and the shorter sturdier construction of the frame. To some extent, bicyclists took over the functions of dragoons, especially as messengers and scouts, substituting for horses in warfare. Bicycle units or detachments were formed at the end of the 19th century by all European armies and the US armed forces. rv tire pressure
The United Kingdom employed bicycle troops in militia or territorial units, but not in regular units. In France, several experimental units were created, starting in 1886. They attempted to adopt folding bicycles early on. In the United States, the most extensive experimentation on bicycle units was carried out by a 1st Lieutenant Moss, of the 25th United States Infantry (Colored) (an African American infantry regiment with white officers). Using a variety of cycle models, Lt. Moss and his troops carried out extensive bicycle journeys covering between 500 and 1,000 miles (800 to 1,600 km). Late in the 19th century, the United States Army tested the bicycle's suitability for cross-country troop transport. Buffalo Soldiers stationed in Montana rode bicycles across roadless landscapes for hundreds of miles at high speed. chrome grab handle
The first known use of the bicycle in combat occurred during the Jameson Raid, in which cyclists carried messages. In the Second Boer War, military cyclists were used primarily as scouts and messengers. One unit patrolled railroad lines on specially constructed tandem bicycles that were fixed to the rails. Several raids were conducted by cycle-mounted infantry on both sides; the most famous unit was the Theron se Verkenningskorps (Theron Reconnaissance Corps) or TVK, a Boer unit led by the scout Daniel Theron, whom British commander Lord Roberts described as "the hardest thorn in the flesh of the British advance." Roberts placed a reward of 1,000 on Theron's headead or alivend dispatched 4,000 soldiers to find and eliminate the TVK.
World Wars
Danish soldiers cycling to the front to fight the Germans during the Battle of Denmark in 1940
Photo showing Italian Bersaglieri during World War I with bicycles strapped to their backs. 1917.
German bicycle infantry on the northern Russian front in 1941
During World War I, cycle-mounted infantry, scouts, messengers and ambulance carriers were extensively used by all combatants. Italy used bicycles with the Bersaglieri (light infantry units) until the end of the war. German Army Jger (light infantry) battalions each had a bicycle company (Radfahr-Kompanie) at the outbreak of the war, and additional companies were raised during the war bringing the total to 80 companies, a number of which were formed into eight Radfahr-Bataillonen (bicycle battalions). In its aftermath, the German Army conducted a study on the use of the cycle and published its findings in a report entitled Die Radfahrertruppe[citation needed].
In its 1937 invasion of China, Japan employed some 50,000 bicycle troops. Early in World War II their southern campaign through Malaya en route to capturing Singapore in 1941 was largely dependent on bicycle-riding soldiers. In both efforts bicycles allowed quiet and flexible transport of thousands of troops who were then able to surprise and confuse the defenders. Bicycles also made few demands on the Japanese war machine, needing neither trucks, nor ships to transport them, nor precious petroleum. Using bicycles, the Japanese troops were able to move faster than the withdrawing Allied Forces, often successfully cutting off their retreat. The speed of Japanese advance have also caught Allied Forces defending the main roads by surprise by attacking them from the rear.
The Finnish Army utilized bicycles extensively during the Continuation War and Lapland War. Bicycles were used as a means of transportation in Jaeger Battalions, divisional Light Detachments and regimental organic Jaeger Companies. Bicycle units spearheaded the advances of 1941 against Soviet Union. Especially successful was the 1st Jaeger Brigade which was reinforced with a tank battalion and an anti-tank battalion, providing rapid movement through limited road network. During winter time these units, like the rest of the infantry, switched to skis.
Within 1942-1944 bicycles were also added to regimental equipment pools. During the Summer 1944 battles against the Soviet Union, bicycles provided quick mobility for reserves and counter-attacks. In Autumn 1944 bicycle troops of the Jaeger Brigade spearheaded the Finnish advance through Lapland against the Germans; tanks had to be left behind due to the German destruction of the Finnish road network.
The hastily assembled German Volksgrenadier divisions had a battalion of bicycle infantry, to have some mobile reserve.
Allied use of the bicycle in World War II was limited, but included supplying folding bicycles to paratroopers and to messengers behind friendly lines. The term, "bomber bikes" came into use during this period, as US forces dropped bicycles out of planes to reach troops behind enemy lines.
By 1939, the Swedish army operated six bicycle infantry regiments. They were equipped with domestically produced Swedish military bicycles. Most common was the m/42, an upright, one-speed roadster produced by several large Swedish bicycle manufacturers. These regiments were decommissioned between 1948 and 1952, and the bicycles remained for general use in the Army, or transferred to the Home Guard. Beginning in the 1970s, the Army began to sell these as military surplus. They became very popular as cheap and low-maintenance transportation, especially among students. Responding to its popularity and limited supply, an unrelated company, Kronan, began to produce a modernized version of the m/42 in 1997.
Later uses
Although much used in World War I, bicycles were largely superseded by motorized transport in more modern armies. In the past few decades, however, they have taken on a new life as a "weapon of the people" in guerrilla conflicts and unconventional warfare, where the cycle's ability to carry large, about 400 lb (180 kg), loads of supplies at the speed of a pedestrian make it vastly useful for lightly-equipped forces. For many years the Viet Cong and North Vietnamese Army used bicycles to ferry supplies down the "Ho Chi Minh trail", avoiding the repeated attacks of United States and Allied bombing raids. When heavily loaded with supplies such as sacks of rice, these bicycles were seldom rideable, but were pushed by a tender walking alongside. With especially bulky cargo, tenders sometimes attached bamboo poles to the bike for tiller-like steering (this method can still be seen practiced in China today). Vietnamese "cargo bikes" were rebuilt in jungle workshops with reinforced frames to carry heavy loads over all terrain.
Modern times
LTTE bicycle infantry platoon north of Killinochi in 2004
Bicycles continue in military use today, primarily as an easy alternative for transport on long flightlines. The use of the cycle as an infantry transport tool continued into the 21st century with the Swiss Army's Bicycle Regiment, which maintained drills for infantry movement and attack until 2001, when the decision was made to phase the unit out.
The LTTE Tamil Tigers made use of bicycle mobility in the fighting in Sri Lanka. The Sri Lankan army also has a bicycle unit. They are mainly stationed and deployed in high security zones in the capital city Colombo. The theory and the basis of their usage is still not well known.
In media
In the expansion pack for EA game Command & Conquer: Generals Zero Hour has a unit called a ombat Cycle which are dirt bike mounted infantry used for reconnaissance and light combat.
See also
Paratrooper folding Tactical Mountain Bicycle
Army Cyclist Corps
References
^ Leiser 10
^ Leiser 11-16
^ Leiser 11
^ "Danie Theron" (html). http://www.instinsky.de/Boer_War/Danie_Theron/danie_theron.html. Retrieved 2007-10-07.
^ Doole, Claire, End of road for Swiss army cyclists, BBC News, http://news.bbc.co.uk/1/hi/world/europe/1325485.stm, retrieved 2008-02-05
Bibliography
Leiser Rolf (sup.) (1991). Hundert Jahre Radfahrer-Truppe (100 Years of Bicycle Troops). Bern, Switzerland: Bundesamt fr Mechanisierte u. Leichte Truppen (Federal Office for Mechanized and Light Troops).
Fitzpatrick, Jim (1998). The Bicycle In Wartime: An Illustrated History. Washington, DC: Brassey's Inc.. ISBN 1-57488-157-4.
Ekstrm, Gert; Husberg, Ola (2001). lskade cykel (1st ed.). Bokfrlaget Prisma. ISBN 91-518-3906-7.
Categories: Combat occupations | Cycling | History of cycling | InfantryHidden categories: All articles with unsourced statements | Articles with unsourced statements from January 2008
Jianzi
China Product
The game
The shuttlecock, called a jianzi in the Chinese game and also known in English as a 'Chinese hacky sack' or 'kinja', typically has four feathers fixed into a rubber sole or plastic discs. Some handmade jianzis make use of a washer or a coin with a hole in the center.
During play, various parts of the body, but not the hands, are used to keep the shuttlecock from touching the ground. It is primarily balanced and propelled upwards using parts of the leg, especially the feet. Skilled players may employ powerful and spectacular overhead kicks. wintec western saddle
The formal game horse lead rope
Competitively, the game is played on a rectangular court 6.10 by 11.88 metres, divided by a net (much like badminton) at a height of 1.60 metres (1.50 metres for women). fleece horse blanket
The informal game
There are unlimited variations of the game, such as trying to keep the feathercock in the air until an agreed target of kicks (eg. 100) is reached, either alone or in a pair. In circle play, the aim may be simply to keep play going. In all but the most competitive formats, a skillful display is a key component of play.
History
The first known version of jianzi was in the 5th century BC in China. The name ti jian zi, means simply 'kick shuttlecock' ('ti' = kick, 'jian zi' = little shuttlecock). The game is believed to have evolved from cuju, a game similar to football that was used as military training. Over the next 1000 years, this shuttlecock game spread throughout Asia, acquiring a variety names along the way.
Jianzi has been played since the Han Dynasty (206 BC220 AD), and was popular during the Six Dynasties period and the Sui and Tang dynasties. Thus the game has a history of two thousand years. Several ancient books attest to its being played.
Shuttlecock player
Modern history
Jianzi came to Europe in 1936, when a Chinese athlete from the province of Jiangsu performed a demonstration at the 1936 Summer Olympics in Berlin. In Germany and other countries people began to learn and play the sport, now called 'shuttlecock'.
In June 1961, a film about the sport called The Flying Feather was made by the Chinese central news movie company, winning a gold medal at an international movie festival. In 1963, jianzi was taught by teachers in elementary school so that it became even more popular.
Well known in Asia, the game has been gaining popularity in Europe. The World Shuttlecock Championship is an annual event held since the founding of the International Shuttlecock Federation (ISF) in 1999. Until then, various countries took turns organising championships.
The sport continues to receive greater recognition, and was included as a sport in the 2003 Southeast Asian Games and in the Chinese National Peasants' Games. Among the members of ISF are China, Taiwan, Finland, Germany, the Netherlands, Hungary, Laos, Vietnam, Greece, France, Romania, and Serbia. Vietnam and China are generally considered best, while in Europe, Hungary and Germany are strongest. On August 11, 2003, delegates from Finland, France, Germany, Greece, Hungary, Romania, and Serbia founded the Shuttlecock Federation of Europe (S.F.E.) in Ujszasz (Hungary).
Health benefits
Playing shuttlecock is vigorous aerobic exercise, and provides the health benefits of any active sport. Building hand-eye coordination is also good for the health of the brain.
Shuttlecock sport Jianzi
Official jianzi for competitions
The official featherball used in the sport of shuttlecock consists of four equal-length goose or duck feathers conjoint at a plastic base. It weighs approximately 15 grams. The total length is 15 to 18 cm. The feathers vary in colour, usually dyed red, yellow, blue and/or green. However, in competitions a white featherball is preferred. The Official Jianzi for Competitions
Other names
Vietnam - cu
Malaysia - sepak bulu ayam
Singapore (and SE Asia) - chapteh or capteh or chatek
Korea - jegichagi or jeigi (to most Koreans know as sports only for children)
Indonesia - bola bulu tangkis or sepak kenchi
Philippines - larong sipa
Macau - chiquia
India - poona (forerunner of badminton) (unknown to most Indians)
Greece - Podopterisi
France - da cu or plumfoot or [pili]http://www.pili-aep.com
Related games, derivatives and variants
Sepak takraw is popular in Malaysia, using a light rattan ball about five inches in diameter. (Sepak means "kick" in Malay, and takraw means "ball" in Thai.)
Indiaca or featherball is played with the same shuttlecock as jianzi but on a court, similar to a badminton court, and played over the net using the hands.
Kemari was played in Japan (Heian Period). It means trike the ball with the foot.
Chinlone is a non-competitive Burmese game that uses a rattan ball and is played only in the circle form, not on a court.
Cuju or tsu chu, the possible forerunner of both football and jianzi
Myachi
Sipa
Ebon
Footbag and footbag net
Hacky Sack
Footvolley
Bossaball
List of circle kick variants
References
^ "History of Shuttlecock Sport". Iordanis Stavridis. 2002-02-14. http://www.shuttlecock-europe.org/sport_history.php. Retrieved 2008-08-20.
^ Rules
^ Melakcom.net
^ "History of Shuttlecock Sport". Iordanis Stavridis. 2002-02-14. http://www.shuttlecock-europe.org/sport_history.php. Retrieved 2008-10-27.
^ "History of Shuttlecock Sport". Iordanis Stavridis. 2002-02-14. http://www.shuttlecock-europe.org/sport_history.php. Retrieved 2008-08-20.
^ "History of Shuttlecock Sport". Iordanis Stavridis. 2002-02-14. http://www.shuttlecock-europe.org/sport_history.php. Retrieved 2008-08-20.
^ The Featherball - a handy game around the world
^ German Wikipedia
External links
Basic Rules of Shuttlecock Sport
Official organisations
International Shuttlecock Federation
European Shuttlecock Federation
German Shuttlecock Federation
Hong Kong Shuttlecock Association Limited (A member agency of Sports Federation and Olympic Committee of Hong Kong, China) (Chinese)
USA Shuttlecock Association
Greek Shuttlecock Federation
Hungarian Shuttlecock Federation
French Shuttlecock Federation
Finnish Shuttlecock Federation
Holland Shuttlecock Federation
v d e
Interdependent team sports
Sport Governing bodies Sportspeople National sport
Basket codes
Basketball (Wheelchair) Korfball (Beach) Netball (Fastnet, Indoor) Slamball
Football codes
Association football (Beach, Futsal, Indoor, Street, Paralympic) Australian football (9-a-side, Rec footy, Metro footy) Gaelic football (Ladies') Powerchair football
Gridiron codes
American football (Eight-man, Flag, Indoor, Nine-man, Six-man, Sprint, Touch) Arena football Canadian football Iceball (Sport)
Hybrid codes
Austus International rules football Samoa Rules Universal football Volata
Medieval football codes
Ba game Caid Calcio Fiorentino Camping Cnapan Cornish hurling Cuju Harpastum Kemari La soule Mob football Royal Shrovetide Uppies and Downies
Rugby codes
Beach Threes Rugby league (Masters, Mini, Mod, Nines, Sevens, Tag, Touch, Wheelchair) Rugby union (American Flag, Mini, Sevens, Tag, Touch, Tens)
Handball codes
Goalball Handball (Beach, Outside) Torball
Safe haven codes
Baseball Brnnboll British baseball Cricket (Indoor, Limited Overs, Test, Twenty20) Danish longball Kickball Lapta Oin Over-the-line Pespallo Rounders Softball Stoolball Town ball Vigoro
Stick and ball codes
Composite rules shinty-hurling Hurling (Camogie) Lacrosse (Box, Field, Indoor, Women's) Polocrosse Shinty
Hockey codes
Bandy Broomball Field hockey (Indoor) Floorball Ice hockey Sledge hockey Ringette Rinkball Roller hockey (Inline, Quad) Rossall Hockey Street hockey Underwater hockey Unicycle hockey
Polo codes
Canoe polo Cowboy polo Cycle polo Elephant polo Horseball Polo Segway polo Yak polo
Ball over a net codes
Biribol Bossaball Fistball Football tennis Footvolley Jianzi Net footbag Peteca Sepak takraw (Beach) Volleyball (Beach, Paralympic)
Other codes
Airsoft Basque pelota (Frontenis, Jai alai, Xare) Buzkashi Curling Cycle ball Gateball Kabaddi Mesoamerican ball game Paintball Ptanque Roller derby Tchoukball (Beach) Ultimate Underwater rugby Water polo Wheelchair rugby
Categories: Individual sports | Chinese words and phrases | Team sports
Gibson Firebird
China Product
History
The Gibson Guitar Corporation released several new styles during the 1950s to compete with Fender's instruments, such as the Telecaster and Stratocaster. After success with the Les Paul in the 1950s, Gibson's popularity began to ebb in the 1960s. Fender's colors, shapes and multiple pickups were endorsed by notable guitarists. Gibson guitars seemed old-fashioned, and this with higher prices contributed to a decline in sales.
Gibson had made forays into radical body shapes - the Flying V and Explorer in the 1950s - but they failed. The president of Gibson, Ted McCarty, hired car designer Ray Dietrich to design a guitar that would have popular appeal. Under Dietrich, the Firebird took on the lines of mid-50s car tailfins. Dietrich took the Explorer design and rounded the edges. The most unusual aspect is that the guitar is "backward" in that the right-hand (treble) horn of the body is longer than the other. Thus, the original Firebirds were unofficially referred to as "reverse". best compound bow
The Firebird is the first Gibson solid-body to use neck-through construction, wherein the neck extended to the tail end of the body. The neck itself is made up of five plies of mahogany interspersed with four narrow strips of walnut for added strength. Other features were reverse headstock (with the tuners on the treble side) and "banjo" tuning keys, as well as mini-humbucking pickups. These were not identical to Epiphone mini-humbuckers, having an alnico bar magnet polepiece in each coil. In contrast, the Epiphone design is a mini-PAF, with a single alnico bar magnet at the pickup's base and six iron polepieces per coil. Some Firebirds from 1965 featured Gibson's single-coil P-90 pickup. discount paintball supplies
A Gibson Firebird V played by Johnny Winter onstage in 1990 sniper paintball markers
The Firebird line went on sale in mid-1963 with four models distinguished by pickup and tailpiece configurations (see below). Unlike the Les Paul and SG line, which used the terms "Junior", "Special", "Standard" and "Custom", the Firebird used the Roman numerals "I", "III" , "V" and "VII". Gibson's line of Thunderbird basses is rooted in the design of the Firebird, and uses even Roman numerals ("II" and "IV") to distinguish it. From 1965 to 1969, Gibson introduced "non-reverse" models after failing to achieve marketing success with the unusual reverse-body design. Gibson had also received complaints from Fender that the Firebird headstock mirrored the Stratocaster and that the body violated Fender's design patents, with Fender threatening a lawsuit. The "non-reverse" body is a more standard double-cutaway design, with the bass horn being longer than the treble horn and the headstock having the tuners mounted on the bass side. It also had a standard glued-in ("set") neck rather than neck-through construction, as well as other, less noticeable changes in design and build. Pickup and tailpiece configuration for the V and VII were the same as the earlier "reverse" models, although the I- and III-models were now shipped with two or three P-90 pickups and plain vibratos. After a few years of disappointing sales, the "non-reverse" line was dropped. "Reverse" body Firebirds were first reissued from 1972-1979, with a commemorative "Bicentennial" model released in 1976.
Reissues
The "reissue" Firebirds are usually based on the original reverse body design; however Gibson reintroduced the non-reverse Firebird in 2002 as a Custom Shop guitar. While prices for just about all vintage guitars have rocketed, the reverse body Firebirds are the more popular design, and command more money than the non-reverse. Many types have been released. Epiphone, owned by Gibson, who also issued Firebirds.
Notable Firebird players
Main article: List of Gibson players
Configurations
The distinctive "banjo" tuners on a Firebird V
Firebird I - One pickup. Combination stud bridge/tailpiece. Chrome hardware. Dot inlays, no neck binding.
Firebird II Artist CMT - A limited production instrument from the early 1980s; features a maple top, mahogany body, set mahogany neck, two standard-sized humbuckers, and toggle switches to control active Moog electronic tone controls.
Firebird III - Two pickups, stud bridge/tailpiece and Gibson Vibrola (player could choose to use the stud or Vibrola). Chrome hardware. Dot inlays and neck binding.
Firebird V - Two pickups, Tune-o-matic bridge with Maestro "Lyre" Vibrola; reissues feature a stop-bar tailpiece. Chrome hardware. Trapazoid inlays and neck binding.
Firebird VII - Three pickups, Tune-o-matic bridge and Maestro "Lyre" Vibrola tailpiece. Gold hardware. Block inlays and neck binding.
Firebird Studio - Two standard-sized Alnico humbuckers, Tune-o-matic bridge with stop-bar tailpiece. Chrome or gold hardware. Set neck. Dot inlays and no neck binding.
Firebird XII - Two pickups, twelve string non-reversed version on the Firebird.
Non-Reverse Firebird - Collectors' term for a Firebird I, III, V or VII featuring a headstock with the tuners on the bass side, and a body having the bass side horn slightly longer than the treble side horn.
See also
Gibson Explorer
Gibson Firebird
Gibson Thunderbird
External links
Gibson Firebird
Categories: 1963 introductions | Gibson electric guitars
Sunday, April 25, 2010
HPS
China Product
Look up hps in Wiktionary, the free dictionary. shatterproof light bulbs
HPS may stand for: dimming compact fluorescent
Hantavirus pulmonary syndrome, a deadly disease transmitted by infected rodents through urine, droppings, or saliva colored glass lamp
Heart Protection Study, a large randomized controlled trial run in the United Kingdom
Hermansky-Pudlak syndrome, a rare autosomal recessive disorder which results in oculocutaneous albinism
High Performance Switch (used in some IBM Scalable POWERparallel and IBM System p models)
High Pressure Sodium, a type of light bulb
History and philosophy of science, an academic discipline that encompasses the philosophy of science and the history of science
Croatian Mountaineering Association (from Hrvatski planinarski savez), an association covering both mountaineering and a broad range of related activities
Hyderabad Public School, one of the oldest and most prestigious public schools in Hyderabad in India
Ilford HPS, a cubic-grain black-and-white film
HP Services, part of the Technology Solutions Group business segment of Hewlett-Packard
Helsingin Palloseura (HPS), a sports club from Helsinki, Finland
Hydraulic power steering, a power steering system in automobiles
This disambiguation page lists articles associated with the same title.
If an internal link led you here, you may wish to change the link to point directly to the intended article.
Categories: Disambiguation pagesHidden categories: All article disambiguation pages | All disambiguation pages
Geothermal gradient
China Product
Heat sources
Temperature within the Earth increases with greater depth. Highly viscous or partially molten rock at temperatures between 650 to 1,200 C (1,202 to 2,192 F) is postulated to exist everywhere beneath the Earth's surface at depths of 50 to 60 miles (80 to 100 kilometers), and the temperature at the Earth's center, nearly 4,000 miles (6,400 km) deep, is estimated to be 5650 600 kelvins. The heat content of the earth is 1031 Joules.
Much of the heat is believed to be created by decay of naturally radioactive elements. An estimated 45 to 90 percent of the heat escaping from the Earth originates from radioactive decay of elements within the mantle. flat head rivets
Heat of impact and compression released during the original formation of the Earth by accretion of in-falling meteorites. plastic snap rivets
Heat released as abundant heavy metals (iron, nickel, copper) descended to the Earth's core. brass rivet
Some heat may be created by electromagnetic effects of the magnetic fields involved in Earth's magnetic field.
10 to 25% of the heat flowing to the surface may be produced by a sustained nuclear fission reaction in Earth's inner core, the "georeactor" hypothesis.
Heat may be generated by tidal force on the Earth as it rotates; since land cannot flow like water it compresses and distorts, generating heat.
Present-day major heat-producing isotopes
Isotope
Heat release [W/kg isotope]
Half-life [years]
Mean mantle concentration [kg isotope/kg mantle]
Heat release [W/kg mantle]
238U
9.46 10-5
4.47 109
30.8 10-9
2.91 10-12
235U
5.69 10-4
7.04 108
0.22 10-9
1.25 10-13
232Th
2.64 10-5
1.40 1010
124 10-9
3.27 10-12
40K
2.92 10-5
1.25 109
36.9 10-9
1.08 10-12
Heat flow
Sequence of the burning of a shrub by geothermal heat.
Heat flows constantly from its sources within the Earth to the surface. Total heat loss from the earth is 42 TW (4.2 1013 Watts). This is approximately 1/10 watt/square meter on average, (about 1/10,000 of solar irradiation,) but is much more concentrated in areas where thermal energy is transported toward the crust by Mantle plumes; a form of convection consisting of upwellings of higher-temperature rock. These plumes can produce hotspots and flood basalts. The Earth's crust effectively acts as a thick insulating blanket which must be pierced by fluid conduits (of magma, water or other) in order to release the heat underneath. More of the heat in the Earth is lost through plate tectonics, by mantle upwelling associated with mid-ocean ridges. The final major mode of heat loss is by conduction through the lithosphere, the majority of which occurs in the oceans due to the crust there being much thinner than under the continents.
The heat of the earth is replenished by radioactive decay at a rate of 30 TW. The global geothermal flow rates are more than twice the rate of human energy consumption from all primary sources.
The geothermal gradient has been exploited for space heating and bathing since ancient roman times, and more recently for generating electricity. About 10 GW of geothermal electric capacity is installed around the world as of 2007, generating 0.3% of global electricity demand. An additional 28 GW of direct geothermal heating capacity is installed for district heating, space heating, spas, industrial processes, desalination and agricultural applications.
Variations
The geothermal gradient varies with location and is typically measured by determining the bottom open-hole temperature after borehole drilling. To achieve accuracy the drilling fluid needs time to reach the ambient temperature. This is not always achievable for practical reasons.
In stable tectonic areas in the tropics a temperature-depth plot will converge to the annual average surface temperature. However, in areas where deep permafrost developed during the Pleistocene a low temperature anomaly can be observed that persists down to several hundred metres. The Suwaki cold anomaly in Poland has led to the recognition that similar thermal disturbances related to Pleistocene-Holocene climatic changes are recorded in boreholes throughout Poland, as well as in Alaska, northern Canada, and Siberia.
In areas of Holocene uplift and erosion (Fig. 1) the initial gradient will be higher than the average until it reaches an inflection point where it reaches the stabilized heat-flow regime. If the gradient of the stabilized regime is projected above the inflection point to its intersect with present-day annual average temperature, the height of this intersect above present-day surface level gives a measure of the extent of Holocene uplift and erosion. In areas of Holocene subsidence and deposition (Fig. 2) the initial gradient will be lower than the average until it reaches an inflection point where it joins the stabilized heat-flow regime.
In deep boreholes, the temperature of the rock below the inflection point generally increases with depth at rates of the order of 20 K/km or more.[citation needed] Fourier's law of heat flow applied to the Earth gives q = Mg where q is the heat flux at a point on the Earth's surface, M the thermal conductivity of the rocks there, and g the measured geothermal gradient. A representative value for the thermal conductivity of granitic rocks is M = 3.0 W/mK. Hence, using the global average geothermal conducting gradient of 0.02 K/m we get that q = 0.06 W/m. This estimate, corroborated by thousands of observations of heat flow in boreholes all over the world, gives a global average of 6102 W/m. Thus, if the geothermal heat flow rising through an acre of granite terrain could be efficiently captured, it would light four 60 watt light bulbs.
A variation in surface temperature induced by climate changes and the Milankovitch cycle can penetrate below the Earth's surface and produce an oscillation in the geothermal gradient with periods varying from daily to tens of thousands of years and an amplitude which decreases with depth and having a scale depth of several kilometers. Melt water from the polar ice caps flowing along ocean bottoms tends to maintain a constant geothermal gradient throughout the Earth's surface.
If that rate of temperature change were constant, temperatures deep in the Earth would soon reach the point where all known rocks would melt. We know, however, that the Earth's mantle is solid because it transmits S-waves. The temperature gradient dramatically decreases with depth for two reasons. First, radioactive heat production is concentrated within the crust of the Earth, and particularly within the upper part of the crust, as concentrations of uranium, thorium, and potassium are highest there: these three elements are the main producers of radioactive heat within the Earth. Second, the mechanism of thermal transport changes from conduction, as within the rigid tectonic plates, to convection, in the portion of Earth's mantle that convects. Despite its solidity, most of the Earth's mantle behaves over long time-scales as a fluid, and heat is transported by advection, or material transport. Thus, the geothermal gradient within the bulk of Earth's mantle is of the order of 0.3 kelvin per kilometer, and is determined by the adiabatic gradient associated with mantle material (peridotite in the upper mantle).
This heating up can be both beneficial or detrimental in terms of engineering: Geothermal energy can be used as a means for generating electricity, by using the heat of the surrounding layers of rock underground to heat water and then routing the steam from this process through a turbine connected to a generator.
On the other hand, drill bits have to be cooled not only because of the friction created by the process of drilling itself but also because of the heat of the surrounding rock at great depth. Very deep mines, like some gold mines in South Africa, need the air inside to be cooled and circulated to allow miners to work at such great depth.
See also
Sustainable development portal
Geothermal power
Hydrothermal circulation
References
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^ Rybach, Ladislaus (September 2007). "Geothermal Sustainability". Geo-Heat Centre Quarterly Bulletin (Klamath Falls, Oregon: Oregon Institute of Technology) 28 (3): pp. 27. ISSN 0276-1084. http://geoheat.oit.edu/bulletin/bull28-3/art2.pdf. Retrieved 2009-05-09.
^ The Frozen Time, from the Polish Geological Institute
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"Geothermal Resources". DOE/EIA-0603(95) Background Information and 1990 Baseline Data Initially Published in the Renewable Energy Annual 1995. http://www.eia.doe.gov/cneaf/solar.renewables/renewable.energy.annual/backgrnd/chap9b.htm. Retrieved May 4, 2005.
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