Fundamental area

A fundamental area of the area in the graphic below. In this case the fundamental area is a circular sector with an opening angle of 45 °. The same colors mean the same physical properties

Symmetrical two-dimensional area of the rotational symmetry elements and mirror symmetry line has and the type of symmetry of the dihedral group belongs

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A fundamental area (also fundamental region ) is a coherent sub-area of a geometric or physical object with symmetries , which is chosen so that no geometric or physical properties are repeated.

Symmetry means that these properties of a spatial area are present several times in the object. In information theory , information that occurs multiple times in an information source is referred to as redundant . Redundancy also occurs in objects of geometry and physics. If it can be traced back to a symmetry of the object, then a fundamental domain is a suitable means for a description of the object that is free from these redundancies. In such a case one can and should restrict oneself to a fundamental area for pragmatic reasons . As in information theory, redundancy can be used deliberately, for example to find errors in input data and computer programs .

The first two graphics come from the branch of global calculations in reactor physics . The first shows a horizontal cross-section through a fundamental area, the second a horizontal cross-section through the entire reactor (of the EPR series ), which is divided into eight fundamental areas by the four mirror symmetry lines also drawn in.

definition

The fundamental area or fundamental region of a body , a flat geometric figure or a one-dimensional object with symmetries , which are described by a symmetry group , is defined as any connected area that does not contain a pair of equivalent points inside and cannot be further enlarged without this property to lose ( David Hilbert and Stefan Cohn-Vossen , 1932).

The mathematician Felix Klein , to whom we owe significant results in geometry, defined the fundamental domain (restricted to point groups ) in 1884 as follows: We generally designate such a part of space as the fundamental domain of a group of point transformations, the one and only one point from each associated point group contains.

An element of the symmetry group maps a point of the fundamental area to a symmetrically equivalent point in the total area. These two form a pair of equivalent points of the Hilbert and Cohn-Vossen definition. They also emphasize: “In addition to listing the rotations and translations present in a group , each group can also be characterized by a simple geometric figure ”, precisely this fundamental domain . For example, if you know the positions of the atoms in a fundamental area, you know them in the entire crystal .

In physics and chemistry, especially in crystallography , atoms , ions and molecules are considered and they are sometimes abstracted as points. The more general case, however, is that one deals with areas of space and not points. Then the definition by Hilbert and Cohn-Vossen has to be extended to a pair of equivalent spatial areas . In the case of the 2D graphics shown, strictly speaking, no two- dimensional objects are meant, but prismatic 3D objects whose properties do not depend on the third spatial dimension (the “z-axis”) of ( Euclidean ) space.

One can freely choose the fundamental area that one puts into focus from several (or infinitely many). Instead of the fundamental range shown in the first graphic, one of the seven other symmetry sectors of the second graphic could have been selected. In numerical physics , the fundamental domain is often selected according to practical and programming aspects, for example: which fundamental domain is descriptive, which is preferred in a subject? How can the properties of the sub-areas of the fundamental area be clearly stored in a field of a computer program?

Formal definition

A fundamental domain with respect to a transformation group is a special connected subset of a topological space . Be a topological space and a transformation group of . For a point denote the amount of all the images from among the elements of the one track ( English orbit by) . Then the set is called a fundamental domain of if for each it holds that the cut is a one-element set. ${\ displaystyle X}$ ${\ displaystyle G}$ ${\ displaystyle X}$ ${\ displaystyle x \ in X}$ ${\ displaystyle G (x)}$ ${\ displaystyle x}$ ${\ displaystyle G}$ ${\ displaystyle x}$ ${\ displaystyle F \ subset X}$ ${\ displaystyle X}$ ${\ displaystyle x \ in X}$ ${\ displaystyle G (x) \ cap F}$

Formal example

The square is a fundamental domain of with respect to the transformation group of all translations around vectors with integral components. Each point can be written with and . ${\ displaystyle [0,1) \ times [0,1)}$ ${\ displaystyle \ mathbb {R} ^ {2}}$ ${\ displaystyle \ mathbb {Z} ^ {2}}$ ${\ displaystyle (x, y) \ in \ mathbb {R} ^ {2}}$ ${\ displaystyle (u + n, v + m)}$ ${\ displaystyle (u, v) \ in [0,1) \ times [0,1)}$ ${\ displaystyle (n, m) \ in \ mathbb {Z} ^ {2}}$

Point positions

Points can be differentiated according to their location. If the point is not the fixed point of one of the symmetry operations, it has a maximum of many symmetrically equivalent points, in the case of the dihedral group, for example, 8 (see above). However, if the point is a fixed point, for example if it lies on a mirror symmetry line, the points which are symmetrically equivalent with regard to these symmetry operations are identical to the point itself. In the example of the first graphic there is a fixed point, the point at the acute angle of the sector of the circle, which belongs to the fundamental area. All other points of the second (inclined) mirror symmetry line do not belong to the fundamental domain, since they are repetitions of the points on the first mirror symmetry line. ${\ displaystyle D_ {4}}$

Fundamental areas in physics and chemistry

In mathematics, the symmetry of an object and thus the fundamental domain is determined by the geometry of the object alone. In the natural sciences, in addition to the fact that areas of space and not points are compared, there are two additional aspects:

The spatial areas to be compared when using a symmetry operation must have the same material composition and the same physical and chemical properties.
Unless the object is infinitely extended (at least in the model), the external boundary conditions must have the same symmetry elements as the geometry of the object.

When choosing a physical fundamental area, one will first start from the geometric one and then include “fillings” of the spatial area and boundary conditions at its outer boundary.

External boundary conditions

External boundary conditions are boundary conditions for the "outside space" and are determined by the surroundings of the object. For example, if you want to numerically calculate the temperature distribution when a homogeneous and homogeneously heated cube cools down (at a given point in time), the geometric fundamental area of the cube (see below) can only be used if the boundary conditions are right. This is the case if the temperature of the space around the cube is kept constant. If one insulates a side surface of the cube, another, a larger fundamental area must be chosen.

If you use an appropriate computer program, the boundary conditions are usually known before the start of the invoice. They belong to the input data.

Inner boundary conditions

External boundary conditions are to be distinguished from further boundary conditions. If only a fundamental domain is specified, it is not always clear from this alone whether it is a fundamental domain of the symmetry type point symmetry , mirror symmetry , rotational symmetry or translational symmetry . This is determined by specifications on the inner delimitation lines of the fundamental area through boundary conditions, which are called symmetry boundary conditions (even if they are not uniform across disciplines) .

Inner boundary conditions are usually specified as parameters in the computer program and are also part of the input data.

Examples from crystallography and chemistry

Grid points , grid lines, unit cells and associated Wigner-Seitz cells (red) of a parallelogram grid at different angles. One of these cells can be chosen as the fundamental domain

The fundamental domain is named differently in different branches of physics and chemistry. In crystallography, a unit cell is a fundamental domain in the form of a parallelepiped that belongs to the subgroup of translational symmetries of a crystal. A Wigner-Seitz cell is also a fundamental domain in some cases. The magazine article "On the Constitution of Metallic Sodium" by Wigner and Seitz was the starting point and model for many subsequent work to solve the Schrödinger equation (approximately) using symmetries and fundamental domains in order to use the resulting wave function to solve physical and chemical properties of chemical elements , chemical compounds and to calculate crystals. These include lattice constants , binding energies , heat of vaporization , compressibilities, etc.

Examples from reactor physics

In reactor physics, physical quantities , primarily neutron fluxes and neutron flux spectra , are calculated for Wigner-Seitz cells or cells of other types. Existing symmetries are used or symmetries are even introduced artificially, for example a square is replaced by a circle of the same area in order to save storage space and computing time, as far as this is physically justifiable. This is a procedure that goes back directly to Wigner and Seitz, who replaced a polyhedron with a sphere of equal volume. In the branches of cell calculations and the global calculations of reactor physics mentioned at the beginning, fundamental domains play a major role without the explicit use of the name fundamental domain, which was coined by mathematicians . Many types of nuclear reactors are deliberately designed symmetrically, also in order to be able to calculate them at all, because symmetries and fundamental areas (drastically) reduce the storage space requirements and the computing times of the computer programs required for construction and operation.

Examples of fundamental domains in three-dimensional Euclidean space

Hilbert and Cohn-Vossen stated in 1932:

“Such fundamental areas play an important role in all discontinuous mapping groups, not just in the movement groups. In general, it is not an easy task to determine a fundamental domain for a given group, or even to prove the existence of a fundamental domain for a species of groups. Fundamental domains can easily be constructed for the flat, discontinuous groups of movements in any case. "

In his eighteenth problem , Hilbert asked in 1900 whether there are polyhedra in three-dimensional space that do not appear as the fundamental area of a movement group, but with which the entire space can still be tiled without gaps. Karl Reinhardt was able to show that this is the case for the first time in 1928 by giving a case. In 1932 Heinrich Heesch found such a solution for the plain as well. The area is an active research area, for example in quasicrystals according to Roger Penrose and self-similar fractal tiling according to William Thurston .

Cases of easy to construct fundamental domains in three-dimensional Euclidean space are the following:

Rotation by 180 ° around an axis: The trajectory is either a set of two points that are opposite each other in relation to the axis, or a single point on the axis. The fundamental domain is a half-space that is bounded by an arbitrary plane. Of this plane itself, only a half plane delimited by the axis belongs to the fundamental domain.

n-fold rotation around an axis: The path is either a set of points around the axis or a single point on the axis. The fundamental area is a sector. ${\ displaystyle n}$

Reflection on a plane: The trajectory is either a set of two points, one on each side of the plane, or a single point on the plane. The fundamental area is a half-space that is bounded by this plane.

Point Symmetry : The orbit is a set of two points, one on each side of the center, with the exception of an orbit that consists only of the center. The fundamental domain is a half-space bounded by an arbitrary plane through the center. Again, only one half plane belongs to the fundamental domain.

Discrete translation symmetry in one direction: The paths are translations of a 1D lattice in the direction of the translation vector. The fundamental area is an infinite plate.

Discrete translation symmetry in two directions: The paths are displacements of a 2D grid in the plane spanned by the translation vectors. The fundamental domain is an infinite bar with the cross-section of a parallelogram.

Discrete translation symmetry in three directions: The paths are translations of the lattice. The fundamental domain is a unit cell.

Fundamental domains of a homogeneous cube. The faces of 24 fundamental areas (the 48 in total) are visible in the graph

In the case of translational symmetry in combination with other symmetries, the fundamental domain is part of the unit cell. For example, for wall pattern groups, the fundamental range is a factor of 2, 3, 4, 6, 8 or 12 smaller than the unit cell.

Fundamental areas of the Platonic solids

Fundamental areas of cube or octahedron illustrated by central projection from the fixed point onto an enveloping sphere

It is a little more complicated to find the geometric shape of the fundamental domain of a homogeneous cube . A homogeneous cube has 48 elements of symmetry, the neutral element , 23 elements of rotational symmetry and reflections on 24 planes of symmetry. The cube can be broken down into its 48 (equivalent) fundamental domains by making cuts along the 24 planes of mirror symmetry. The result shows the figure fundamental areas of a homogeneous cube . There are two types of fundamentals that are mirror symmetric. Although colored differently, the (physical) properties of the two types are the same. A fundamental domain has the shape of a (irregular) tetrahedron . Its edges, which are not visible in the graphic, run from the corner points of the visible right-angled triangle to the fixed point of the symmetry operations, the center of the cube.

The Platonic Solids Cube and Regular Octahedron are dual solids . Therefore the cube group and the octahedral group are isomorphic , since dual bodies have the same type of symmetry. Consequently the octahedron also has 48 fundamental regions. Together, the fundamental areas of the cube or octahedron can be visualized by central projection from the fixed point onto an enveloping sphere, as shown in the figure. The planes of mirror symmetry cut the sphere in great circles . This projection of the regular bodies onto a sphere goes back to Felix Klein, who introduced it in the first section of his famous monograph.

The regular homogeneous tetrahedron has 24 symmetry elements that form the tetrahedron group. It is a subgroup of the cube group (octahedron group). The tetrahedron therefore has 24 fundamental domains. The dual body of the tetrahedron is again a tetrahedron.

The Platonic solids regular pentagonal dodecahedron and regular icosahedron are dual and have 120 symmetry elements (icosahedron group) and 120 fundamental areas. Analogous to the figure fundamental areas of cube or octahedron , the projection of the fundamental areas of dodecahedron or icosahedron onto a sphere is shown in the article icosahedron group .

Spektrum has put an interesting and well-illustrated introduction to the subject of the fundamental domains of polyhedra on the web .

Individual evidence

↑ ^a ^b ^c David Hilbert, Stefan Cohn-Vossen: Illustrative Geometry . Springer, Berlin 1932, p. 56–61 (VIII, 310, limited preview in Google Book search).
↑ ^a ^b Felix Klein: Lectures on the icosahedron and the solution of the equations of the fifth degree . Teubner, Leipzig 1884, p. 22 and 3 (VIII, 260, online ).
↑ Fundamental area . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
^ Eugene Wigner, Frederick Seitz: On the constitution of metallic sodium . In: Physical Review . tape 43 , no. 10 , 1933, pp. 804 ( online [PDF]).
^ Samuel Glasstone, Milton C. Edlund: The elements of nuclear reactor theory . MacMillan, London 1952, pp. 265 f . (VII, 416, online ).
^ Karl Reinhardt: On the decomposition of the Euclidean spaces into congruent polytopes , session reports of the Prussian Academy of Sciences, 1928, pp. 150–155
↑ Christoph Pöppe: Fundamental areas on the sphere and the family register of the polyhedra. Spectrum, March 28, 2004, accessed August 24, 2019 .

[Hilbert_1932-1] David Hilbert, Stefan Cohn-Vossen: Illustrative Geometry . Springer, Berlin 1932, p. 56–61 (VIII, 310, limited preview in Google Book search).

[Klein_1884-2] Felix Klein: Lectures on the icosahedron and the solution of the equations of the fifth degree . Teubner, Leipzig 1884, p. 22 and 3 (VIII, 260, online ).

[Walz_2000-3] Fundamental area . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .

[Wigner_1933-4] Eugene Wigner, Frederick Seitz: On the constitution of metallic sodium . In: Physical Review . tape 43 , no. 10 , 1933, pp. 804 ( online [PDF]).

[Glasstone_1952-5] Samuel Glasstone, Milton C. Edlund: The elements of nuclear reactor theory . MacMillan, London 1952, pp. 265 f . (VII, 416, online ).

[Reinhard_1928-6] Karl Reinhardt: On the decomposition of the Euclidean spaces into congruent polytopes , session reports of the Prussian Academy of Sciences, 1928, pp. 150–155

[Poeppe_2004-7] Christoph Pöppe: Fundamental areas on the sphere and the family register of the polyhedra. Spectrum, March 28, 2004, accessed August 24, 2019 .