N dimensional spaces. N-dimensional arithmetic space

Linear space is a set of objects (of arbitrary nature) for which addition with each other and multiplication of an element by a number are defined. Linear space often called vector

In this case, the following conditions are satisfied:

Elements of the set L called vectors, and the field elements P - scalars.

Linear operations on elements of sets of the same type result in elements of a new set that have the same properties as the original ones. For the direct addition operation it is defined reverse operation subtraction, and for the direct multiplication operation - the inverse division operation. For both direct and inverse operations, one element of the set corresponds to one and only one element of the set B. one-to-one sets. An example of one-to-one sets are vector quantities. The set of all vectors of three-dimensional space forms a vector space. An example of a VP is the so-called n-dimensional arithmetic space . The vectors of this space are ordered systems of n real numbers:  1 ,  2 ,...,  n . The sum of two vectors and the product by a number are determined by the relations:

( 1 ,  2 , …,  n) + ( 1 ,  2 , …,  n) = ( 1 +  1 ,  2 +  2 , …,  n +  n);

( 1 ,  2 , …,  n) = ( 1 ,  2 , …,  n). The basis in this space can be the following system of n vectors e 1 = (1, 0,..., 0), e 2 = (0, 1,..., 0),..., e n = (0, 0,..., 1).

Many R all polynomials  0 +  1 u+ … +  n u n(any degrees n) from one variable with real coefficients  0 ,  1 ,...,  n with the usual algebraic rules for adding polynomials and multiplying polynomials by real numbers forms a polynomial polynomial. Polynomials 1, u, u 2 ,...,u n (for any n) are linearly independent in R, That's why R- infinite-dimensional vector. Any three non-zero vectors that do not lie in the same plane are linearly independent. Polynomials of degree no higher than n form v.p. dimensions n+ 1 ; its basis can be polynomials 1, u, u 2 ,...,u n .

11. Dot product of vectors, its properties.

An operation on two vectors, the result of which is a scalar (number) that does not depend on the coordinate system and characterizes the lengths of the factor vectors and the angle between them. This operation is usually considered to be commutative and linear in each factor. The order in which the factors are written does not matter, that is *=*. If the angle between the vectors And designate through, then them dot product can be expressed by the formula
.If vectors And given by their coordinates:, , then their scalar product can be calculated using the formula. This implies a necessary and sufficient condition for the perpendicularity of two vectors.

To find the angle between vectors you can use the formulas:

,.

6 answers

The dimensions are what you want them to do. For example, depth and time only make sense when you deal with these concepts.

It doesn't have to be about space and time. In fact, the C++ standard calls them extents.

Let's say you have ten different cheeses and you want to estimate the likelihood that someone will prefer them in in a certain order. You can store this in your int; , referring to the extent meaning: favorite cheese, second favorite cheese, third favorite cheese, fourth favorite cheese, fifth favorite cheese and least favorite cheese. The probability that someone prefers cheeses in the order 5-4-6-3-2-1 would be expressed as t.

The point is that the language does not attach domain semantics to extents. It's up to you to do this.

N-dimensional arrays are not just C++. It appears everywhere in mathematics, physics, various other sciences, etc.

Here's an example: Let's say you want to index data by position (x, y, z), time, and "which user created the data." For a data point collected at x1, y1, z1, time1 and generated by user1, you would store it in dataArray = myNewData.

In programming, don't think of multidimensional arrays in terms of traditional geometry unless you're trying to represent the world directly. It's better to think of each successive "dimension" as another array containing arrays. There are several use cases where this might appear. However, if you are using more than three dimensions, I would no longer consider it as arrays or even "arrays of arrays", I prefer trees to be closer to how you program something that requires more than three levels.

One example is a tree, where you have a root node, which has nodes, which also have nodes. If you want to pick something, wood is a great tool. Let's say you wanted to sort a bunch of numbers that came in random order. You would do the first number that appeared in the root. If the first number is 5 and the next number is 7, then you should put 7 at the "right" root of node 5. And if you have 3 then 4, you should insert 3 to the "left" of 5 and then to 4 to the "correct" one from 3. If you traverse this tree in order (always going left down the tree, only going back when there are no new nodes, then right), you will get a sorted list: 3, 4, 5, 7.

5 / \ 3 7 \ 4

Here you can see the tree structure. If you were doing this in C, you would use structures that would look like this (I'm using pseudocode):

Struct Node( int val; Node left; Node right; )

There is a lot of material about binary trees (which I explain), but primarily I wanted you to move away from the concept of arrays, "like dimensions in space", and much more just a data structure that can store elements. Sometimes a binary tree or other data structure is too complex, and a 5 or more dimensional array may be more convenient for storing data. I can't think of an example right now, but they have been used before.

As physical 3D beings, we cannot "visualize" what 4, 5, 6 (or higher) physical dimensions represent.

The 4th dimension would increase our perception to the 4th direction, which would be orthogonal to the directions of height, width and depth that we naturally perceive. Yes - the geometry was strange!!

To give us a sense of this idea, in this video Carl Sagan imagines what it would feel like as a perfectly straight 2 meter being (small square) living in the 2nd world to meet a mysterious 3D being.
This three-dimensional creature (suspiciously like an apple) exists primarily in this mysterious third dimension that the little square cannot "see". It only perceives points on the apple that intersect with its 2d flat world, i.e. His projection...

The video looks old fashioned by today's standards, but from a physics/geometry perspective it's still the best explanation I've seen out there.

Discussions about the possibility of a space greater than three dimensions are ongoing. Such opinions are inspired by the concept of abstract multidimensional spaces in mathematics and physics. In physics this concept is used as convenient way descriptions when time and a number of other parameters are added to three spatial coordinates. If the number of such parameters together with space-time characteristics is n, then it is considered that they form n-dimensional space. When enough large quantities properties and interrelated variables, one can come to the concept of multidimensional and even infinite space, but this concept will be rather conditional, since it will be used to characterize completely different properties.

If you take, for example, a stack of sheets of paper (everything that is in the plane of the sheet is a space of two dimensions) and pierce this stack with a fork, then in each sheet (a space of two dimensions) there will be a trace of the fork in the form of four holes. Those who “live” in the space of two dimensions will never be able to connect these four holes into one whole, i.e. imagine that they are a “trace” of the fork and interaction with it. Thus, from the perspective of two-dimensional space and its “inhabitants,” the fork is inaccessible to representation.

Another example. If you imagine horizontal plane, crossing the top of the tree parallel to the ground, then on this plane the sections of the branches will seem separate and completely unrelated to each other. And in our space it is a cross-section of the branches of one tree, together constituting one top, feeding from one root, having one shadow. So maybe three dimensional bodies of our space are there images in our sphere of four-dimensional bodies incomprehensible to us? Or maybe all kinds of anomalous phenomena are “traces” left in our three-dimensional space inhabitants of the four-dimensional?

We are already accustomed to the concept of the “fourth dimension”, or simply “another dimension”, from where all sorts of “evil spirits” sometimes “crawl out” into our modest three-dimensional world, including “aliens” of all stripes, on the one hand, and on the other hand disappear, more often everything irrevocably, people, ships, planes.

The history of the search for “other” dimensions is full of drama, has its own prophets and its own evil geniuses. The ways of science are strange and unpredictable, and what was rejected as a scientific direction at the beginning of the century suddenly aroused intense interest at the end of the century. The history of any scientific direction one should start from its roots. The first hints of the existence of “other” spaces can be found in the works of Giordano Bruno. But only in mid-19th V. Physicists and mathematicians for the first time timidly raised the question of the possibility of the existence of other, higher dimensions. This problem was solved most simply mathematically, and the first to bring it up for discussion was one of the creators of new non-Euclidean geometries, B. Riemann, in his work “On the hypotheses lying in

basis of geometries”, dedicated, in particular, to n-fold extended quantities. Almost at the same time, this problem began to affect physicists, and E. Mach was one of the first to touch upon it: “While still under the influence of atomic theory, I once tried to explain spectral lines gases... The difficulties I encountered in this led me in 1863 to the idea that insensitive things need not necessarily be represented in our sensitive space of three dimensions.”

Einstein's theory of relativity, which appeared at the beginning of the 20th century, then provided enormous scope for the development of physical ideas, even the most extravagant. Einstein was one of the first who encroached on the hitherto unshakable concepts of space and time, showed their dependence in SRT on the frame of reference, speed of movement, and then in GRT - on the strength of the gravitational field.

Later, many scientists began to think about the question why our space has exactly three dimensions or, in other words, what features distinguish geometry and physics in three-dimensional space from geometry and physics in multidimensional spaces?

In 1917, based on general relativity, Einstein created a stationary closed spherical model of the Universe. Characteristic feature This model was the finiteness of space, although, from the point of view of internal geometry, space then appears to be unlimited. There is no contradiction in this. For example, the surface of a balloon, from our point of view, is finite, but from the point of view of a fly crawling along its inner surface, it will be unlimited.

However, when deciding standard equations certain difficulties arose. To receive statistical solutions Einstein was forced to introduce a certain coefficient, the so-called cosmological term I. The equations derived by Einstein are interesting in that they give three solutions and, accordingly, three models of both the Universe and space. Einstein's space-time world is completely static. It can be represented as a cylindrical 4-dimensional world with an unlimited time axis, i.e. according to this model the time

The variable section of the space-time continuum, in contrast to the spatial section, is infinite.

Translated into popular language, Einstein's world is a 3-dimensional physical space, curved and closed on itself due to the presence of matter in it, i.e. A 4-dimensional sphere (hypersphere) that has neither beginning nor end in time. You can bend the three-dimensional world only in a space of 4 or more high order measurements. This clearly implies complete equality of rights fourth dimension in relation to the three existing ones.

During the years of Einstein's life and after, many scientists put forward ideas and presented theories related to the n-dimensionality of space. What Einstein once failed to do is being quite successfully solved by a galaxy of modern theorists, many of whom have already become laureates. Nobel Prizes. This is A. Salash, S. Weinberg, S. Glashow. Within modern theories Great Unification, they managed to bring together within one concept three very different types interactions (gravitational ones have so far remained “overboard”), which can be described using so-called gauge fields. The main property of gauge fields is the existence of abstract symmetries, which give this approach its elegance and opens up broad perspectives. In the Kaluza-Klein theory brought back to life, the symmetries of gauge fields become concrete geometric symmetries associated with additional dimensions of space.

As in the original version, interactions in the theory are introduced by adding additional spatial dimensions to space-time. However, since it is now necessary to give shelter to interactions of three types, it is necessary to introduce not one, but several additional dimensions. A simple calculation of the number of operations included in the Grand Unification theory requires an additional 7 spatial dimensions; if we take into account time, then all space-time has 11 dimensions. Thus, modern version Kaluza-Klein theory postulates an 11-dimensional Universe, 7 spatial co-

whose ordinates are collapsed and therefore are not observed in principle.

Science knows four fundamental interactions in nature:

■ electromagnetic and gravitational on a poppy scale
Romira;

■ weak and strong on a microcosm scale.
However, in recent years V scientific works discuss
the possibility of the existence of another distance is given
ational interaction in the macrocosm - spin, or
torsion bar recording and transmitting information
tion through a torsion field. Physical at
kind of this fifth interaction, apparently, is completely
completely different from the other four interactions,
since the transfer of information here is carried out like
without wasting energy.

Modern works J. Wheeler, A. Perose, K. Pribram, P. Davis allow the presence of this fifth fundamental interaction in nature - spintorsion interaction. The fields associated with it (torsion fields) have the ability to transmit information almost energy-free to any part of the Universe, and also ensure the “holographic nature” of information connections in the Universe.

According to the stated paradigm, almost all phenomena associated with sensory perception phenomena and bioenergetic (more precisely bioinformational) influence of healers. Therefore, there is every reason to believe that torsion fields responsible for psychic phenomena.

Nowadays, this area of ​​​​activity has ceased to be exotic. Now many organizations, enterprises, and research institutes are involved in it. The production of synthetic anti-torsion screens from films has been organized for sale to the public, which can be used as protection against geopathogenic radiation, radiation from computers, computers, television receivers and other radio-electronic devices. New structural materials with unique properties are being created. For example, scientists in Russia and Ukraine have created steel that is twice as strong and

six times more flexible than normal. The most various types sensors that respond to torsion fields.

The prospects for using torsion fields are enormous. It is enough to mention the new generations of computers with micro-level elements with truly incredible computing abilities. The discovery of the fifth fundamental interaction will change our understanding of nature. If our century has passed under the sign of electromagnetism, then the next one will be the century of torsion energy.

Construction

Converting 1-tetrahedron to 2-tetrahedron

Converting 2-tetrahedron to 3-tetrahedron

As is known, through any N points one can draw an (N–1)-plane, and there are sets of N+1 points through which the (N–1)-plane cannot be drawn. Thus, N+1 is the minimum number of points in N-space that does not lie in one (N-1)-plane, and can serve as the vertices of an N-polyhedron.

The simplest N-polyhedron with the number of vertices N+1 is called an N-tetrahedron after the name of the three-dimensional member of this family. In the literature, the name “simplex” is also accepted. In lower-dimensional spaces, this definition corresponds to 4 figures:

• 0-tetrahedron (point) – 1 vertex;
• 1–tetrahedron (segment) – 2 vertices;
• 2–tetrahedron (triangle) – 3 vertices;
• 3-tetrahedron (actually a tetrahedron) – 4 vertices.

All these figures have three common properties:

1. In accordance with the definition, the number of vertices of each figure is one greater than the dimension of space;

2. Exists general rule transforming figures of lower dimension into figures of higher dimension. It lies in the fact that from geometric center a perpendicular to the next dimension is constructed, a new vertex is built on this perpendicular and connected by edges to all the vertices of the original tetrahedron;

3. As follows from the procedure described in paragraph 2, any vertex of the tetrahedron is connected by edges to all other vertices.

Described sphere

An N-sphere can be described around any N-tetrahedron.

For a 1-tetrahedron this statement is obvious. The described 1-sphere will be a segment coinciding with the 1-tetrahedron itself, and its radius will be R = a/2. Let's add one more point to the 1-tetrahedron and try to describe a 2-sphere around them.

Let us construct a 2-sphere s 0 with radius a/2 such that the segment AB is its diameter. If point C is located outside the circle s 0, then by increasing the radius of the circle and shifting it towards point C, you can ensure that all three points are on the circle. If point C lies inside the circle s 0, then you can adjust the circle to this point by increasing its radius and shifting it in the direction opposite to point C. As can be seen from the figure, this can be done in any case when point C does not lie on the same straight line with points AB. The asymmetrical location of point C relative to AB is not a hindrance.

Considering general case, suppose that there is an (N–1)-sphere S N-1 of radius r, circumscribed around some (N–1)-dimensional figure. Let's place the center of the sphere at the origin. The equation of the sphere will look like

Let us construct an N-sphere with center at point (0, 0, 0, ... 0, h S) and radius R, and

The equation of this sphere

Substituting x N = 0 into equation (1), we obtain equation (2). Thus, for any h S, the sphere S N-1 is a subset of the sphere S N, namely, its section by the plane x N = 0.

Let's assume that point C has coordinates (X 1, X 2, X 3, ..., X N). Let us transform equation (2) to the form

and substitute the coordinates of point C into it:

The expression on the left side is the square of the distance R C from the origin to point C, which allows us to reduce the last equation to the form

from which we can express the parameter h S:

It is obvious that h S exists for any R C , X N and r, except X N = 0. This means that if point C does not lie in the plane of the sphere S N–1, you can always find a parameter h S such that on the sphere S N with center ( 0, 0, 0, ..., h S) both the sphere S N–1 and the point C will lie. Thus, an N–sphere can be described around any N+1 points if N of these points lie on one (N –1)–sphere, and last point does not lie in the same (N–1) plane with them.

Applying the latter by induction, we can state that the N-sphere can be described around any N+1 points if they do not lie in the same (N-1)-plane.

Number of faces of N-tetrahedron

A tetrahedron has N+1 vertices, each of which is connected by edges to all other vertices.

Since all the vertices of a tetrahedron are connected to each other, any subset of its vertices has the same property. This means that any subset of L+1 vertices of a tetrahedron defines its L-dimensional face, and this face is itself an L-tetrahedron. Then for a tetrahedron the number of L-dimensional faces is equal to the number of ways to select L+1 vertices from full set N+1 vertices.

Let us denote by the symbol K(L,N) the number of L-dimensional faces in an N-polyhedron, then for an N-tetrahedron

where is the number of combinations from n to m.

In particular, the number of faces of the highest dimension is equal to the number of vertices and is equal to N+1:

Formulas for a regular N-tetrahedron

Number of L-dimensional faces
Height
Volume
Radius of circumscribed sphere
Radius of the inscribed sphere
Dihedral angle

Some useful ratios

Wikimedia Foundation. 2010.

• Translation

Hello, Habr. Remember the awesome article “You and your work” (+219, 2222 bookmarks, 350k reads)?

So Hamming (yes, yes, self-monitoring and self-correcting Hamming codes) has a whole book written based on his lectures. We're translating it here, because the man speaks his mind.

This is a book not just about IT, it is a book about an incredible way of thinking cool people. "It's not just a charge positive thinking; it describes the conditions that increase the chances of doing great work.”

We have already translated 6 (out of 30) chapters.

Chapter 9. N-dimensional space

When I became a professor after 30 years of active research at Bell Telephone Laboratories, primarily in the department mathematical research, I remembered that professors must comprehend and summarize past experience. I put my feet on the table and began to think about my past. IN early years I was mainly involved in calculations, that is, I was involved in many big projects, requiring calculations. Thinking about how several large engineering systems in which I was partly involved were developed, I began, now at some distance from them, to see that they had a lot common elements. Over time, I began to understand that design problems are in n-dimensional space, where n is the number of independent parameters. Yes, we create 3-dimensional objects, but their design is in multi-dimensional space, 1 dimension for each design parameter.

Many dimensional spaces will be needed to make further proofs intuitively understandable without strict detail. Therefore, we will now consider n-dimensional space.

You think you live in three-dimensional space, but in many cases you live in two-dimensional space. For example, in the random course of life, if you meet someone, you have a reasonable chance of meeting that person again. But in the world of 3 dimensions this chance does not exist! Consider fish in the ocean that potentially live in three dimensions. They move on the surface or on the bottom, limiting things to two dimensions, or they form schools, or gather in one place at the same time, such as an estuary, a beach, the Sargasso Sea, etc. They cannot expect to meet a friend if they wander in open ocean in three dimensions. Or, for example, if you want planes to collide, you have to gather them near an airport, place them in 2D flight levels, or send them in a group; truly random flights would have fewer accidents than they do now!

N-dimensional space is a mathematical construct that we must explore to understand what happens to us as we wander around there while solving design problems. In two dimensions we have the Pythagorean theorem, for right triangle square of the hypotenuse equal to the sum squares of other sides. In three dimensions, we are interested in the length of the diagonal of the parallelepiped, Fig. 9.1. To find it, we first draw the diagonal of one face, apply the Pythagorean theorem, then take it as one of the sides with the other side of the third dimension, which is perpendicular to it, and again from the Pythagorean theorem we find that the square of the diagonal is the sum of the squares of three perpendicular sides. It obviously follows from this proof and the necessary symmetry of the formula that if you go up to more high dimensions your diagonal square will also be equal to the sum of the squares of pairwise mutually perpendicular sides,

Where x i is the length of the sides of a rectangular block in n-dimensional space.

Rice. 9.I

Continuing the geometric approach, the planes in space will simply be linear combinations of x i , and the sphere around a point will be all points that are at one fixed distance from a given one.

We need the volume of an n-dimensional sphere to understand the idea of ​​the size of a piece of limited space. But first we need the Stirling approximation for n!, which I will derive so that you understand most of the details and are confident in the correctness of what follows, rather than taking our word for it.

With a product of type n! difficult to handle, so we take log n!, which becomes

Where, of course, ln is the logarithm to base e. Sums remind us that they are related to integrals, so we'll start with an integral like this

We use integration by parts (since we know that ln x comes from the integration of an algebraic function and can therefore be eliminated in the next step). Let U=ln x, dV=dx, then

On the other hand, if we apply the trapezoidal formula to the integral ln x we ​​get, see Fig. 9.II,

Since ln 1 = 0, adding (1/2) * ln n to both sides of the equality we ultimately get

Let's get rid of logarithms by raising e to the power of both sides.

Where C is a certain constant (close to e), independent of n, since we approximated the integral by a trapezoidal formula, the error grows more and more slowly as n increases

Rice. 9.II

More and more, C has a limit. This is the first form of the Stirling formula. We will not waste time calculating the limit of the constant C as it approaches infinity, which turns out to be √(2*π)=2.5066... ​​(e=2.71828...). Thus we finally obtain the Stirling formula for the factorial

The following table shows the error of the Stirling approximation for n!

Please note that as the numbers increase, the coefficient approaches one, but the difference becomes larger and larger!

If you consider 2 functions

Then the limit of the ratio f(n)/g(n) with n tending to infinity is equal to 1, but as in the table the difference

Gets bigger and bigger as n increases.

We need to expand the concept of factorial to the set of all positive real numbers, for this we introduce the gamma function in the form of an integral

Which exists for all n>0. For n>1 we integrate by parts again, this time using dV=e^(-x)dx and U = x^(n-1). For two limits, the integrable part is 0 and we have the following given formula

Thus, the gamma function takes values ​​(n-1)! for all positive integers n and naturally extends the concept of factorial to all positive numbers, since the integral exists for all n > 0.

We will need

Let's denote x=t^2, then dx=2t*dt, and we get (using symmetry in the last step)

Now we use the standard approach to calculate this integral. We get the product of two integrals, one with respect to the variable x and one with respect to the variable y.

X^2 + y^2 imply polar coordinates, so let's transform it to the form

Angle integration is simple. Exponential integration is now also simple, and we end up with.

Thus,

Now let's return to the volume of the n-dimensional sphere (or hypersphere, if you like). It is clear that the volume of an n-dimensional cube with side x is equal to x^n. After a little thought, you will understand that the formula for the volume of an n-dimensional sphere should look like

Where C n is the corresponding constant. For the case n = 2, the constant is equal to π, for the case n=1 it is equal to 2 (if you think about it). In the three-dimensional case we have C 3 = 4*π/3.

We'll start with the same trick we used for the gamma function of 1/2, except this time we'll take the product of n integrals, each with its own variable x i . The volume of a sphere can be represented as the sum of the volumes of the surfaces, each term of this sum corresponds to the surface area multiplied by the thickness dr. For a sphere, the value of the surface area can be obtained by differentiating the volume of the sphere with respect to the radius r,

And therefore the volume terms are equal

Equating r^2=t, we have

Where do we get it from?

It's easy to see that

And we can calculate the following table.

Thus, we see that the coefficient C n increases to n=5 and then decreases to 0. For spheres of unit radius, this means that the volume of the sphere tends to zero as the dimension increases. If the radius is equal to r, then for the volume, denoting n=2k for convenience (since real numbers change smoothly as n increases and spheres of odd dimensions are more difficult to calculate),

Rice. 9.III

No matter how large the radius r is, increasing the number of dimensions n gives rise to a sphere of arbitrarily small volume.

Now consider the relative amount of volume located close to the surface of an n-dimensional sphere. Let r be the radius of the sphere and the internal radius of the surface r(1-ε), then the relative volume of the surface is

For large n, no matter how thin (relative to the radius) the surface is, there is almost nothing inside it. As we say, the volume is almost entirely on the surface. Even in 3-dimensional space, a unit sphere has 7/8 of the volume inside a surface 1/2 radius thick. In n-dimensional space 1-(1/2)^n inside half the radius from the surface.

This is important in design; It turns out after the calculations and data transformations above that the optimal design will almost certainly be on the surface, and not deep inside as you might think. Computational methods are usually not suitable for searching for the optimum in multidimensional spaces. This is not strange at all; generally speaking best design- this is to bring one or more parameters to its extreme - obviously, you will end up on the surface of the visible design area!

Next we will look at the diagonal of an n-dimensional cube, in other words the vector from the origin to the point with coordinates (1,1,...,1). The cosine of the angle between this line and any axis is given by definition as the ratio of the coordinate value of the length of the projection onto a given axis, which is obviously equal to 1, to the length of the vector, which is equal to √n. Hence

It follows that for large n the diagonal is almost perpendicular to each coordinate axis!

If we consider points with coordinates (±1, ±1,..., ±1) then there will be 2n such diagonals, which are all almost perpendicular to each coordinate axis. For n=10, for example, their number is 1024 such almost perpendicular lines.

I need the angle between two vectors, and while you may remember that this is the dot product of vectors, I suggest printing it out again to better understand what's going on. [Remark; I have found that it is very useful in important situations to review all the underlying derivations involved to get a feel for what is going on.] Take two points x and y, with corresponding coordinates x i and y i . Rice. 9.III. Applying the cosine theorem in the plane of 3 points x, y and the origin, we have

Where X and Y are the lengths of the segments to points x and y. But C can be obtained using the differences in coordinates along each axis

Equating the two expressions we see

Let us now apply this formula to two segments drawn from the origin to random point from a set of coordinates

(±1, ±1,..., ±1)

The dot product of two such factors, taken at random, is again equal to ±1 and this must be summed n times, with the length of each segment being √n, therefore (note n in the denominator)

And by weak law large numbers this tends to 0 as n increases almost certainly. But there are 2^n random vectors and for given vector all the rest of the 2^n random vectors are almost certainly almost perpendicular to this one! n-dimensionality is truly vast!

In linear algebra and other disciplines, you learn to find a set of perpendicular axes and then represent everything else in that coordinate system, but you see that in n-dimensional space after you have found n mutually perpendicular coordinate axes, there are 2^n other directions almost perpendicular themes that you found! Theory and practice linear algebra completely different!

Finally, to further prove that your intuition about n-dimensional space is not very good, I will produce another paradox that I will need in later chapters. Let's start with a 4x4 square divided by 4 unit square, in each of which we draw unit circle, Rice. 9.IV. Next, we will draw a circle with its center at the center of the square, touching the rest with inside. Its radius should be from Fig. 9.IV,

In 3-dimensional space we have a 4x4x4 cube and 8 spheres of unit radius. The inner sphere touching the others at a point lying on the segments connecting the centers has a radius

Think about why its radius is greater than 2 dimensions.

Moving to n dimensions, we have a 4x4x...x4 cube and 2^n spheres, one in each corner, each touching the other n adjacent ones. The inner sphere touching all the others from the inside will have a radius

Check it out carefully! Are you sure? If not, why not? Where is the error in reasoning?
Having verified that this is true, we apply it to the case of n=10 measurements. For the inner sphere we have the radius

Rice. 9.IV

And in 10-dimensional space, the inner sphere went beyond the boundaries of the cube. Yes, the sphere is convex, yes, it touches the other 1024 from the inside, and at the same time it extends beyond the boundaries of the cube!

This is too much for your sensitive intuition about n-dimensional space, but remember that n-dimensional space is where the design of complex objects usually occurs. You should try to get a better feel for n-dimensional space by thinking about the things just described until you begin to see how they could be true, or rather why they should be true. Otherwise you will have problems when you decide difficult task design. Perhaps you should recalculate the radii different sizes, and also return to the angles between the diagonals and coordinate axes and see how it turns out.

Now it must be strictly noted that I did all this in classical Euclidean space, using the Pythagorean distance, where the sum of the squared differences of the corresponding coordinates is equal to the square of the distance between the points. Mathematicians call this distance L 2 .

L1 space does not use the sum of the squares of coordinate differences, but rather the sum of distances, as if you were traveling through a city with a rectangular grid of streets. It's the sum of the differences between two points that tells you how far you'll have to go. In computing, this is often called the "Hamming distance" for reasons that will become clear in later chapters. In this space, a circle in two dimensions looks like a square standing on top, Fig. 9.V. In three-dimensional space it is like a cube standing on top, etc. Now you can better see how the paradoxical inner sphere from the example above can extend beyond the cube.

There is a third, often used, metric (they are all metrics = functions of distance) called L∞, or Chebyshev distance. Here, the maximum difference in coordinates is taken as the distance, regardless of other differences, Fig. 9.VI. In this space, a circle is a square, a 3D sphere is a cube, and you see that in this case the inner sphere from the paradox has zero radius in all directions.

These were examples of metrics, measures of distance. The conditions for determining the metric D(x,y) between two points x and y are as follows:

1. D(x,y) ≥ 0 (non-negative)
2. D(x,y) = 0 if and only if x=y (identity)
3. D(x,y) = D(y,x) (symmetry)
4. D(x,y) + D(y,z) ≥ D(x,z) (triangle inequality).

Rice. 9.V

Rice. 9.VI

I leave it to you to check that the three metrics L ∞ , L 2 and L 1 (Chebyshev, Pythagorean and Hamming) all satisfy these conditions.

The truth is that in complex design for different coordinates we can use any of these metrics mixed together, so that the design space is not a complete picture, but a mixture of bits and pieces. The L 2 metric is obviously related to least squares, and the remaining two L ∞ and L 1 are more similar to comparisons. When comparing in real life you usually use either the maximum difference L ∞ in any one characteristic as sufficient condition to distinguish two items, or sometimes, as in bit strings, it is the number of mismatches that is significant, and the sum of squares is not suitable, which means that the L 1 metric is used. This is in to a greater extent true for identifying patterns in AI.

Unfortunately, although all of the above is true, it is rarely revealed to you. Nobody ever told me about this! I'll need a lot of the results in later chapters, but generally speaking, after this demonstration you should be better prepared than you were before for complex design and for thorough analysis space in which design is carried out, as I tried to do. Clutter is basically where design comes in, and you have to find a workable solution.

Since L 1 and L ∞ are not generally known, let me make a few comments about the three metrics. L2 is a natural distance function for use in physical and geometric cases, including data extraction from physical measurements. That's why you find L 2 everywhere in physics. But when the subject concerns intellectual judgments, the other 2 metrics are more appropriate, although this is slow to perceive, which is why we often see frequent use chi-square estimators, which are obviously a measure of L2, where other more appropriate estimators should be used.

To be continued...

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