Shulou(Shulou.com)11/24 Report--

It is not easy to give a proper explanation of geometry. Because the basic concept of this branch of mathematics is either too simple to explain, for example, it is not necessary to say here what is a circle, what is a straight line, what is a plane, etc., or it is more advanced. However, if you have not seen these advanced concepts, you will know nothing about modern geometry. Then, if you understand the two basic concepts, the harvest will be much greater. These two concepts are the relationship between ∶ geometry and symmetry, and the concept of manifolds.

Geometry and symmetry group generally speaking, geometry is the part of mathematics that uses geometric language, such as "point", "straight line", "plane", "space", "curve", "ball", "cube", "distance" and "angle". Are basic and key concepts in geometry. However, there is a more profound view, which is advocated by Klein, that change is the real core of the discipline. So, in addition to the words listed above, add the words "reflection", "rotation", "translation", "stretch", "cut" and "projection". There are also slightly more advanced concepts, such as "conformal mapping" or "continuous deformation".

Transformations are always with groups, so geometry is closely related to group theory. Given a transformation group, there is a corresponding geometry. In particular, if a figure can be transformed into another figure after a transformation in this group, they are said to be equivalent. Different groups derive different concepts of equivalence. Here is a brief description of the most important geometry and the transformation groups associated with it.

Euclidean geometry Euclidean geometry is what most people think of as "ordinary geometry". For example, the theorem that the sum of the interior angles of a triangle is 180 °belongs to Euclidean geometry.

If you want to look at Euclidean geometry from the perspective of transformation, you must first explain how many dimensions the research is in, and of course, you must specify a transformation group. A typical transformation is a rigid transformation. This (rigid) transformation can be considered in two ways. One is that a rigid transformation is a transformation that keeps the distance constant in a plane, in three-dimensional space, or, more generally, in R ^ n. That is to say, given two points x and y, if a transformation T makes the distance between Tx and Ty equal to the distance between x and g, T is said to be a rigid transformation.

It was later found that each of these transformations could be realized by a combination of rotation, reflection and translation. This gives a second and more specific way to think about this group. In other words, Euclidean geometry deals with concepts that are invariant in rotation, reflection and movement, including points, lines, planes, circles, spheres, distances, angles, lengths, areas, and volumes. The rotation in R ^ n constitutes an important group ∶ special orthogonal group, denoted by SO (n). The larger orthogonal group O (n) also includes reflection.

Affine geometry

There are many other linear mappings besides rotation and reflection. What happens if you zoom in on SO (n) or O (n) to include as many of these linear transformations as possible? For a transformation to be an element of a group, it must be reversible, but not all linear transformations are the same, so the group that should be examined is the group GL_n (R) formed by all reversible linear transformations of R ^ n. All these transformations hold the origin still. But if we want, we can also add translation to get a larger group, which includes all the transformations in the form of x → Tx+b. Where b is a fixed vector and T is a reversible linear transformation. The resulting geometry is called affine geometry.

Because stretching and shearing are also included in linear mapping, they can maintain neither distance nor angle, so distance and angle are not the concepts of affine geometry. However, after reversible linear mapping and translation, points, lines and planes are still points, lines and planes, so these concepts belong to affine geometry. Another affine concept is the parallelism of two lines (that is, although linear mappings generally do not keep the angle constant, zero angles are maintained). This means that although there is no such thing as rectangle or square in affine geometry, parallelograms can be discussed. Similarly, although circles cannot be discussed, ellipses can be discussed, because linear maps always turn ellipses into ellipses.

Topology

The idea that geometry associated with a group "studies concepts preserved by all transformations of this group" can be made more accurate by equivalence relations. Let G be a transformation group in R ^ n. You can think of a d-dimensional "graph" as a subset S of R ^ n. In the study of G geometry, S is not distinguished from the set derived from its transformation in G. So at this point we say that the two figures are equivalent. For example, two figures are equivalent in Euclidean geometry if and only if they are congruent in the usual sense, while in two-dimensional affine geometry, all parallelograms and ellipses are equivalent. In a word, we can think that the basic object of G geometry is the equivalent class of graphics, not the graphics themselves.

Topology can be thought of as geometry obtained by applying the loosest concept of equivalence, in which we say that two graphs are equivalent, or homeomorphic in mathematical language, if each of them can be "continuously transformed" into the other. For example, a ball and a cube are equivalent in this sense, as shown in the following figure

Because there are a lot of continuous deformations, it is difficult to say that two graphics are not equivalent in this sense. For example, it seems obvious that a sphere cannot be continuously deformed into a torus because they are essentially different shapes-one with a "hole" and one without. However, it is not easy to turn this intuition into a strict argument. Invariants, algebraic topologies and differential topologies are involved in more detail. We'll discuss it later.

Spherical geometry so far, we have been gradually relaxing the requirement that two graphics are equivalent, allowing more and more transformations. Now we're going to tighten it again and think about spherical geometry. Now the universe is no longer R ^ n but n-dimensional sphere S ^ n, that is, the surface of a (nasty 1)-dimensional sphere with radius 1, or expressed by algebraic method, that is, the suitable equation in R ^ (nasty 1).

The point of (xylene 1 ~ x 2, … , a collection of x_n+1 Just as the surface of a 3D sphere is two-dimensional, this set is n-dimensional. We will only discuss the case of nasty 2, but it is easy to extend to the larger n.

Now the appropriate transformation group is SO (3), which consists of all such rotations, the axes of which are straight lines passing through the origin (you can also allow reflection to take O (3), they are symmetries of spherical S ^ 2; look at them this way in spherical geometry, not as transformations in R ^ 3 as a whole).

Meaningful concepts in spherical geometry are straight lines, distances, and angles. It may seem strange to be confined to the surface of a sphere and talk about straight lines, but the "spherical line" is not a straight line in the usual sense, but a subset of ∶ obtained by S ^ 2 in the following way, which is a subset of S ^ 2 intersected with S ^ 2 by a plane passing through the origin (center of the sphere) (called a large circle), that is, a circle with a radius of 1, that is, a spherical line.

The important reason to think of a large circle as some kind of straight line is that the shortest path between the two points on S ^ 2 is the big circle. Of course, the path should be limited to S ^ 2.

The distance between two points x and y is defined as the length of the shortest path that connects x and y and lies entirely on S ^ 2. How to define the angle between two spherical lines? A spherical line is defined as the intersection of a plane and S ^ 2, so the angle of intersection of two spherical lines can be defined as the angle of the two planes in the sense of Euclidean geometry. There is also an aesthetic point of view that does not involve anything other than a sphere at all. The idea is to look at a small neighborhood at the intersection of the two spherical lines, when this small part of the sphere is almost flat and the two lines are almost straight. So this angle can be defined as the Euclidean angle of the "limit line" on the "limit plane".

Hyperbolic geometry

The idea of looking at geometry with reference to a set of transformations (that is, transformation groups) is just a useful way to look at the subject. However, when it comes to hyperbolic geometry, the way of transformation is indispensable.

The transformation group that produces hyperbolic geometry is a two-dimensional special projective linear group, recorded as

One of the ways to explain this group is as follows: the ∶ special linear group SL_2 (R) is all matrices with determinant 1.

, that is, the set of such matrices that fit the relational expression ad-bc = 1 (they do form a group, because if the determinant of both matrices is 1, so is their product). In order to make it "projective", make matrix An equivalent to-A, for example, matrix

In order to derive a kind of geometry from this group, it must first be interpreted as a transformation group of a set of two-dimensional points. Once this is done, this set of two-dimensional points is called a model of hyperbolic geometry. The subtlety is that none of the hyperbolic geometry seems to be the most natural model, such as a sphere is a model of spherical geometry. The three most commonly used models of hyperbolic geometry are half-plane model, disk model and hyperboloid model.

The half-plane model is the model that is most directly related to the group PSL_2 (R). The set of two-dimensional plane points required is the upper half plane of the complex plane C, that is, the set of all complex z-dimensional points > 0. After a matrix is given, the corresponding transformation of the matrix is to change the point z to (az+b) / (cz+d). The conditional ad-bc=1 is used to prove that the transformed point is still on the upper half plane, and it is also used to prove that the transformation is reversible.

What hasn't been done here is that ∶ hasn't said anything about distance. It is in this kind of geometry that groups are needed to "generate" geometry. If you want to have a concept of distance from the point of view of the transformation group, it is important that the transformation keep the distance constant. That is, if T is such a transformation, and z and w are in the upper half plane, then T (z) and T (w) are also in the upper half plane, and

It can be proved that, in essence, only one method of defining distance has this property, and this is what the geometry of "generation" by transformation means.

This distance has some strange properties at first glance. For example, the shape of a typical hyperbolic line is a semicircular arc that ends on the real axis. However, to say that it is a semicircle is a semicircle from the point of view of Euclidean geometry on C; from the point of view of hyperbolic geometry, the straight line of Euclidean geometry is equally strange. The real difference between the two distances is that the closer the hyperbolic distance is to the real axis, the larger the Euclidean distance becomes. So to go from point z to point w, the "detour" deviates from the real axis, and the distance is shorter. The best bend is a semicircular arc along the connection point z and point w and at right angles to the real axis.

One of the most famous properties of two-dimensional hyperbolic geometry is that it is a kind of geometry that makes the Euclidean parallel axiom untenable. That is to say, you can find a hyperbolic line L and its outer point x, so that the crossing point x can draw two lines that do not intersect L. After proper explanation, all other axioms of Euclidean geometry hold true in hyperbolic geometry. It can be seen that it is impossible to deduce parallel postulates from those axioms. This discovery solves a problem that has plagued mathematicians for more than two thousand years.

Another property complements the property of the sum of the interior angles of Euclidean triangles and spherical triangles. There is a natural concept of hyperbolic area. The area of hyperbolic triangles with vertex angles α, β and γ is π-a-β-γ. So in the hyperbolic plane, a + β + σ is always less than pi, and when the triangle is very small, it's almost equal to pi. The properties of the sum of interior angles reflect the fact that the ∶ sphere has positive curvature, the Euclidean plane is "flat", and the hyperbolic plane has negative curvature. In this way, the sums of interior angles of hyperbolic triangles, Euclidean triangles and spherical triangles are less than, equal to and greater than π, respectively, and their differences are proportional to curvature, and the curvature of these spaces is also negative, zero and positive. It says "make up" the corresponding nature, and that's what it means.

The disk model was conceived by Poincare at a famous moment when he boarded a bus. its set of points is the open unit disk of the C plane, that is, the set D of the complex number whose module is less than 1. Now, the typical transformation shape is as follows. Take a complex number an in D and the real number θ, and this transformation changes the z point into a point.

It is not entirely obvious that these transformations become a group, and it is even less obvious that this group is isomorphic to PSL_2 (R). However, it can be proved that the function of changing z to-(iz+1) / (Zioni) maps the unit disk to the upper half plane, and vice versa. This proves that the two kinds of geometry are the same, and you can use this function to turn the result of one geometry into the result of another.

Like the half-plane model, when close to the edge of the disk, the hyperbolic distance is larger than the Euclidean distance. From the hyperbolic geometry point of view, the diameter of the disk is infinite, it actually has no edge.

A mosaic pavement of a hyperbolic disc shows that some congruent graphics can be used to tessellation the disc. To say that these figures are congruent means that any one of them can be transformed into any other by one transformation in the group. So, although these figures do not all look the same, they are the same size and shape from the perspective of hyperbolic geometry. The straight line in the disk model is either a (Euclidean) arc that intersects at right angles to the circumference of the unit circle, or a straight line segment that passes through the center of the disk.

The hyperbolic model can explain why this geometry is called hyperbolic geometry. This time, the point set is

This is a single-leaf rotational hyperboloid, which is generated by the rotation of the hyperbola x ^ 2 = 1 + z ^ 2 on the plane z ^ 0 around the z axis. The general transformation in PSL_2 (R) is some kind of "rotation" on the hyperbolic surface of this single leaf rotation, which can be synthesized from the real rotation around the z axis and the "hyperbolic rotation" on the xz plane. The so-called hyperbolic rotation is the matrix.

The transformation. Just as a normal rotation maintains a unit circumference, a hyperbolic rotation maintains a hyperbolic x ^ 2 = 1 + z ^ 2, while its inner points change with each other. Similarly, it is not obvious that this transformation will give the same group as above, but this is indeed the case, so that the hyperbolic model is equivalent to the above two models.

Lorentz geometry

This is a geometry for special relativity, modeled on four-dimensional space-time, also known as Minkowski space. The main difference between it and 4-dimensional Euclidean geometry is that it does not take into account the usual distance between two points (trech zreagem g Z. Z) and (tweak pr é cor pr é cor phaptyre y è re z'), but the following quantities.

If it is not the extremely important negative sign in front of it, it is the square of Euclidean distance. This reflects the fact that time and space are very different (although they are intertwined).

Lorentz transformation is a linear mapping from R ^ 4 to R ^ 4 and keeps the "generalized distance" unchanged. Let g be a linear mapping from (trecrology xreagem z) to (- trecoveryrep z), and G is the corresponding matrix of g (the elements on the main diagonal are-1, 1, and the other elements are matrices of 0). We can abstractly define Lorentz transformation as

For a point (trem xrem y rem z), if

Let's say that this point is empty; and if

It is said that it is time-like; and if

Let's just say it's on the light cone. All of these are the real concepts of Lorentz geometry because they are preserved by Lorentz transformations.

Lorentz geometry is also of fundamental importance to general relativity, which can be said to be the study of Lorentz manifolds. All these are closely related to Riemannian manifolds.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

Welcome to subscribe "Shulou Technology Information " to get latest news, interesting things and hot topics in the IT industry, and controls the hottest and latest Internet news, technology news and IT industry trends.