Auguste Bravais
On the Distribution of Regular Points
In order to obtain a system of regular spacing in the space of divided panels, we take two randomly chosen points and connect them with a straight free line, which we extend in both directions to a dead end. We occupy this line with a row of other points, all separated by a constant interval between the two points. The linear system of these equidistant points will be referred to as P-series in the course of this discussion. The fundamental distance between two consecutive points will be referred to as the parameter of the point series.
We take a wide point series of the same parameter, arrange them parallel to each other over an arbitrarily chosen length, and combine these rows to form a geometric plane — the height of which, by its nature, is not limited in any direction. We occupy this plane with a sequence of such panel rows that are parallel and equidistant from each other.
Now, from the position of these point rows — each of them being linear — we slide them in their longitudinal direction so that the points which have served as initial points in each row now all lie at one end of the same straight line, which is inclined more or less with respect to the direction of the point rows. We will designate a system of points distributed in this way on a plane with the general name net.
We take a second net of the same standard and size as the previous one and place it on a plane parallel to the first, separated by a fixed distance — taking care that all homologous lines in both nets run in the same direction. This alignment can be achieved by parallel translation of all parts of the corresponding net.
We now distribute an infinite number of such identical, equally directed nets on an infinite number of planes, which are all parallel and equidistant to one another. We allow each net to slide along its plane until all the points used as origin markers lie along one and the same line, necessarily outside the original plane of the net. The system of points thus obtained will be referred to in this treatise as a lattice. It is unlimited in all three dimensions.
Figure 1 illustrates the result of the operations described. The sequence of points N, N′... lie on the first row; the sequence B, B′... on the second. From the points B′, B″, etc., other equal and parallel rows of points begin. The origin points O, B, B′, J′... of the rows lie necessarily on a straight line. Since all rows are equidistant, the sequence OB, BB′, B′B″... also forms a point series of the system; but it differs from ON, N′, N″... by its direction and also by the size of its parameter, which apparently equals OB.
A second net, similar to the one formed by OB, N′... and B, P, P′... starts at point D and lies in a plane parallel to the one containing O, A, B. A first point row lies along the line D, D′, D″, etc., and the homologous points from the previous net are denoted Z, Z′, Z″...
The other nets of the system are named according to their starting points: D′, F″, etc. All points O, D, Z, D′... lie on a straight line, and due to the regular spacing of the planes — each of which contains a net — the sequence OD, DZ, D′... also forms a point series of the system. But this one differs from ON or OB not only in direction but also in the magnitude of its parameter.
The resulting lattice shows a regular distribution, characterized by the following properties:
No individual point can be distinguished from another by any peculiarity of its position.
The configuration surrounding any arbitrarily chosen point is the same as the configuration surrounding any other point.
For example, if a point is taken as the origin of a system of rectangular or orthogonal coordinates, every subsequently chosen point in the lattice will have the same coordinates, provided the directions of the axes are preserved in the shift of origin.
Before I proceed further, I want to assign a special designation to the points that form the lattice. It is, in fact, necessary to distinguish them from purely mathematical points, which are considered abstract and placeless.
I will therefore call them lattice points (Gitterpunkte). There is no need for a seam or boundary in this naming. These lattice points can be thought of as small dimensional physical objects — real particles, especially the centers of cells whose polyhedral shape remains otherwise undetermined.
I will assume that these lattice points are connected by such forces that the entire lattice maintains a constant structure, with fixed distances between the points. The body cannot be moved inside this lattice — except when the entire structure is moved together by translation or rotation.
If we shift the entire lattice parallel with itself so that a lattice point (e.g., point N in Figure 1) arrives at the place previously occupied by another lattice point O, then all other lattice points also move to positions previously occupied by their equivalents. Thus, the entire structure remains unchanged in form. This kind of general motion, which preserves the locations of all lattice points, is called a symmetry operation of the lattice.
As long as the lines OA, OB, etc., and the planes that contain the lattice remain fixed, we retain the standard construction process. However, in our minds, we can relax the rigidity of these lines and planes and consider a more general approach, which we will describe next.
Task I — Construction of the Lattice from Given Points and Planes
Let us begin by selecting two lattice points, such as O and A (Figure 1), which are situated within a single plane of the lattice. We connect them with the line segment OA. If there are other lattice points a, b, c, ... on this line that belong to the same system, then point a, as the one closest to O, must be considered a simple subdivision of OA.
We can always assume that no other lattice point lies between two chosen points O and A. We extend OA in both directions and obtain the line segments OA′, ON, N′, N″...
All these points N′, A″, etc., must repeat themselves in accordance with the general laws that characterize a regular distribution of lattice points.
Let us now take another point B and connect it with point O. If any other lattice points lie between O and B, we only retain the one nearest to O. We can then assume again that no other point from the system lies between O and B.
Having fixed this, we construct the parallelogram OAPB. Point P is given by the intersection of the lines from B and A, within the plane OAB. One can, in general, find a finite number of lattice points inside this parallelogram, such as m, n, etc., which belong to the plane. In such a case, point B must be discarded and replaced with the point whose distance to O is smallest.
Let us denote this point as P′. We draw a new line OP′ parallel to OA and define the parallelogram OPP′A. Then we may assume that no point of the general lattice lies within this parallelogram — neither in the interior nor on any of its sides — apart from the four lattice points O, N, P, P′.
To avoid cluttering Figure 1, we reuse the same parallelogram OAPB and note that point B is chosen according to two conditions:
No point lies between O and B along line OB.
The parallelogram formed by OA and OB contains no interior lattice point.
As we now have the parameters ON and OB, we know two sets of point rows, though not the entire set of points yet. Still, we can determine the intersection points of rows from ON, A″... and OB, B″... — and from this, we construct the full network.
Once we have obtained a point series and then a mesh (net), it is not difficult to reconstruct the full lattice.
Let us choose a point D outside of plane OAB, under the condition that no intervening lattice point lies on the segment OD, nor in the parallelepiped formed by A, O, D, and B. We repeat the process as we did for point B in the 2D case, ensuring these spatial conditions are satisfied.
This provides a simple method for directly obtaining the point D. It consists of moving a geometric plane parallel to itself, starting from the position coinciding with plane OAB. As soon as this plane meets a lattice point in its motion, that point is chosen as D, and the distance OD becomes the parameter of the third point series OD, D′, D″...
The solution just described shows that Problem I can be solved in various ways, and it is not hard to see that the number of such solutions is infinite. In fact, the plane ONA has the property of lying as close as possible to plane OAB among the system of all its parallels.
If we consider any lattice point — say, point S — belonging to the mesh carried by this plane, it becomes clear that we can replace the point series OD with a new series GS. This allows us to generate all points of the given lattice as intersections of:
a system of planes parallel to ONA, and
a system of lines parallel to OS, running through all points of the net OB, N′, N″...
Likewise, by choosing OS and OB as initial point series, or OD and OA, we could reconstruct the lattice by using other intersecting planes — such as the mesh of plane GA or the plane OBS — as our reference systems.
This means that we have many different solutions to the same task — and since the number of lattice points is infinite, so is the number of valid reconstructions.
If two rows of points — for instance, OR and OB — in a net are such that not a single point of the lattice lies inside the parallelogram they form, then these point series are said to be conjugate. And when they are chosen as coordinate axes, they are to be called conjugate axes.
The system of point series, which are parallel to two conjugate series like OR and OB, intersect the net into parallelogram-like fields — all equal in size and shape. I will refer to the parallelogram OAPB (see Fig. 1) or OAmB (Fig. 2) as the fundamental parallelogram (or the parallelogrammatic unit cell of the net).
Strips and Mesh Planes
The bounded area between a row of points such as OB, B′, B″... and its neighboring parallel row — such as P, P′, P″ — is called a strip. This strip is characterized by the fact that it contains no lattice points inside it, but only on the two bordering lines.
These two parallel point rows, which enclose a strip, are called adjacent rows. Each row of points has two adjacent rows lying on opposite sides. The plane of such a strip — or of any two parallel point rows — or more generally, any plane defined by three non-collinear lattice points, is called a mesh plane (Netzebene).
Each mesh plane carries a complete lattice net across its surface.
If the parameters of three point series — say OA, OB, and OD (as in Fig. 1) — can serve as the edges of a parallelepipedthat contains no lattice points either in its interior or on its side faces, I will label these three point series as conjugate point rows (konjugierte Punktreihen). When used as coordinate axes, they will be called conjugate axes.
The three mesh planes that contain these rows of points in pairs (i.e., the planes of OAB, AOD, and BOD) will be called the conjugate mesh planes.
A point series will be called conjugate with respect to a mesh plane if it is conjugate to both of the point rows that define the mesh in that plane. The bounded region between one such mesh plane and the next, among a system of parallel planes, will be referred to as a layer (Schicht). No lattice point lies in the interior of a layer. The two mesh planes that bound the layer are called its bounding planes (angrenzende Ebenen). Each mesh plane has two such bounding planes on either side.
The three systems of parallel planes — those that are parallel to the conjugate mesh planes OAB, AOD, and BOD (Fig. 1) — divide space into parallelepipedal cells, each of which has equal volume and shape. I will call the parallelepiped formed by the three conjugate vectors OA, OB, and OD the fundamental parallelepiped, or the core of the lattice (Schaar).
All points of the lattice system can be generated by adjoining such parallelepipeds side by side.
Summary of the Lattice Properties
Now that we have established our terminology, we can summarize the basic characteristics of any lattice as follows:
The lattice points (Gitterpunkte) of a given lattice are arranged on a system of parallel and equidistant planes. Each of these planes carries a net, and the configuration of this net is identical across all planes.
Within each of these nets, the points form systems of parallel, equally spaced, and identically directed point rows.
On every row, the points are equidistant from one another.
The lattice points can be obtained as intersections of three sets of geometric entities:
Three systems of parallel planes, each spaced equally;
Each lattice point is the intersection of one plane from each of the three systems;
Thus, the lattice structure appears as a three-dimensional grid of regularly spaced points.
It is always possible to describe the lattice as composed of parallelepipedal cells, each of which is identical in shape and size and fits together without gaps or overlaps. The corners of these cells coincide with the lattice points.
This division of space into equal parallelepipeds — whose vertices coincide with the lattice points — can be accomplished in an infinite number of different ways, depending on which conjugate axes or point series are chosen.