The distance spectra of simple sets and their associations (part 2)

In the first part, we took a hammer in hand (the spectrum matrix of the deviation of squares of distances) and tested it on a pair of nails (a set of three points). And here is my suggestion to the manufacturers of building roulettes while I was busy with these spectra. It is necessary to add a parabolic scale (centimeters squared) on the back of the measuring tape. Since the squares of distances here are simply swarming, and the usual (linear) distances look like a pitiful special case. When building a dacha with such a tape measure, it will be possible to check the squareness of the angles, and other invariants for the distances between points in space and on the plane.

The spectra of vertices of regular polygons and polyhedra

The spectrum of vertices of an equilateral triangle is logical to generalize to the spectra of vertices of regular polygons as a whole.

Let the vertices of a regular polygon fit into a circle of radius R. The center of the own coordinate system (centroid) will be located in the center of the circle. Since all the vertices are on the same plane, the number of eigenvalues ​​of the spectrum will be 2. It is also obvious that the eigenvalues ​​must be equal due to symmetry.

According to formula (2.2) from the first part, the sum of the eigenvalues ​​of the spectrum is equal to the sum of the squares of the distances from the centroid to the vertices, that is, it is equal to the product of the number of vertices and the square of the radius of the circle. As a result, we obtain an expression for the eigenvalues ​​of regular polygons:







Along the way, we note one of the invariants - for a regular n-gon there is a simple expression for the sum of the squares of the distances between a given vertex and the others:

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(To control this invariant when designing an n-coal arbor, a parabolic tape measure would be useful).



To derive a formula for the spectrum of regular polyhedra inscribed in a sphere of radius R , one can use arguments similar to those given above. The only difference is that there are three eigenvalues ​​in space, therefore it is necessary to divide the spectral sum into three:







As is known , there are only 5 types of regular polyhedra with the number of vertices (4, 6, 8, 12, and 20). But the formula (4.7) in itself does not impose any restrictions on the set of vertices. Accordingly, the formula is applicable not only to regular polyhedra, but also to any symmetric vertices of which are located on a sphere. As an example of such a polyhedron, we can give a cube (8 vertices), supplemented by vertices in the center of faces, placed on a sphere (a rhombus - 6 vertices). Total we get a symmetric polyhedron of 14 vertices, for which the formula (4.7) should be applicable (the author did not check).

5. Spectra of the lattices

Lattices are found in our life more often than polyhedra (unfortunately, of course).

Expressions for the lattice spectra of the unit constant

The simplest non-degenerate lattice is a set of points (nodes) located at the same distance from each other on the same line. The distance between the lattice nodes is called the lattice constant . For simplicity, we first consider lattices with a constant of 1.

The spectrum of the one-dimensional lattice will contain only one eigenvalue. We will leave the formal conclusion of the spectrum value to the readers; here we give the final answer. The eigenvalue of a one-dimensional lattice depends cubically on the number of lattice sites:







Degrees in parentheses denote decreasing degrees :







The eigenvalue of a one-dimensional 1-lattice in one form or another is included in the multidimensional spectra of the lattices. Therefore, we give it a special designation g (n) , the name is basic , and we give a table of the first values:

n	2	3	four	five	6	7	eight	9	ten
g (n)	0.5	2	five	ten	17.5	28	42	60	82.5

The spectrum of a square lattice contains two identical eigenvalues. It is expressed through the base:







Here n is the size of the lattice (the number of points) in one direction.

Similarly, to obtain the spectrum of a cubic lattice, it is necessary to multiply the square spectrum by n . The general formula for the spectra of 1-lattices of dimension d is :







If the lattice contains a different number of nodes in different dimensions (which is possible for two or more dimensional lattices), then the eigenvalues ​​of the spectrum in the general case will be unequal. For a rectangular lattice on a plane with dimensions (k: l), the eigenvalues ​​have the form:







For example, for the spectrum of the lattice size (3: 2) we obtain:







Similar formulas hold for a three-dimensional (cubic) lattice of size (k: l: m) :







Observe permutations.

According to formulas (5.4), (5.5), one can determine the mean value of the square of the distances between the nodes of a regular lattice (radius) - the sum of the eigenvalues ​​of the spectrum.

Consider the size of the lattice cell

For completeness, it would be nice to find out how the spectrum is influenced by the size of the (constant) cell of the lattice. From considerations of dimension it is logical to assume that the lattice constant should be included in the formula for eigenvalues ​​as a square. So it really is:







If the lattice cell is not square, but rectangular (different lattice constants for each direction), then the eigenvalues ​​will be different. For each measurement - its own lattice constant. For example, for a rectangular lattice with dimensions a and b, the eigenvalues ​​will be:







If both the lattice cell and the lattice itself are asymmetric, then in order to calculate the eigenvalues ​​of such a lattice, it is necessary to follow the formulas for the dimensions and dimensions of the lattice in formulas. For example, the eigenvalues ​​of a two-dimensional lattice of size (k: l) with a rectangular cell of size axb are expressed by the formula:







Here the order of dimensions and dimensions is important, the size a specifies the length of the lattice cell in the direction k , and the size b in the direction l . One can see the connection between a two-dimensional spectrum and a one-dimensional one — the spectrum of one dimension is multiplied by the number of points of another.



Similarly, the formulas for the 3-dimensional lattice will look similar to the size of the abc cell:

Pseudolatts

By a two-dimensional pseudo-lattice we mean a set of two one-dimensional, located perpendicular to each other and intersecting in the center (form a cross). Attaching to this lattice another one-dimensional along the other coordinate (perpendicular to the first two), we obtain a three-dimensional cross (pseudo-lattice).

The eigenvalues ​​of the pseudo-lattice spectrum are determined by the spectrum of a one-dimensional lattice. The only difference is in quantity (dimension). For a two-dimensional lattice, we have two nonzero eigenvalues, for a three-dimensional lattice, three:







Here n is the number of nodes in one coordinate (in one of the directions). (5.6) expresses the spectrum of coordinate axes.

6. Addition of sets

It is convenient to break complex configurations of sets of points into several simple ones. Therefore, it is critical (important) for spectral analysis of sets of distances to consider the question of combining spectra — how the spectra of several sets behave.



First of all, we note that the addition of any set with itself simply doubles the value of the spectrum (since the multiplicity of the set points is doubled).

Let us turn to the lattice spectra. Lattices are convenient because they have relatively simple expressions for the values ​​of the spectrum. This makes it possible to count on obtaining explicit formulas for the spectra of the addition of lattices.

Take for simplicity a one-dimensional lattice. Suppose we have combined the lattice with ourselves and begin to shift one relative to the other. How to behave your own value? The expression for the spectrum of two one-dimensional lattices consisting of n nodes, with a constant a , shifted relative to each other by a distance d has the form:









So, we see that the resulting value of the spectrum of combining two sets can be expressed as the sum of three members - two of them are the spectra of individual sets, and the third is the spectrum between sets (mutual spectrum) .



Let us verify this statement by one more example. We will combine one-dimensional spectra with different lattice constant. In fact, we get a set of points on a straight line, displaying the beat frequency. Some points may coincide (multiplicity 2), and for simplicity, we assume that the two sets are aligned on one of their extreme points (the beginning is the same).





Since we know the spectra of individual lattices, we can isolate the interaction spectrum and find its explicit expression:







That is, the expression for the mutual spectrum (in this context, the spectrum of the square of the distance) between sets consisting of one-dimensional gratings of different constants (but of the same size) looks like:







This expression we just picked up. Is it possible to deduce it in any way from the initial data, the matrix of mutual distances - an open question.



Consider one more union of sets, but not related to Cartesian lattices. We will move apart in space two sets of three equidistant points. As a result, we obtain the vertices of a certain triangular prism with height h . We expect that the third eigenvalue (space) will appear in the spectrum. Its value will be determined by the size of the height (mutual removal of the vertices of two equilateral triangles). The first two values ​​of the spectrum are simply doubled:







In this example, we see that the mutual spectrum can also be selected, as well as the spectra of the original sets. In this configuration, the value of the mutual spectrum forms a new eigenvalue (measurement).

Again respite

We considered the spectra of Cartesian (rectangular) lattices, - we will leave all other types outside the brackets.

The spectra of such gratings have explicit and simple expressions.



The spectra of regular polygons and polyhedra are also simple.



We mainly refer to geometric analogies, but lattices exist not only in physical space. Time counts (bars, rhythms) is an example of a one-dimensional time grid.



It is important to calculate the mutual spectrum of the two sets. This allows you to express the spectra of complex configurations through simple ones.



To numerically verify the spreadsheet capabilities formulas presented here will not be enough - math libraries are needed (NumPy is fine for Python).



Continuation