With this article I begin a series of lessons on organic 3D modeling. This article is specifically about the principles of modeling, i.e. absolutely does not depend on the features of your (any) 3D package. The series of articles will cover the following topics:

• form,
• proportions,
• poles,
• topology
• and much more.

There are a huge number of modeling methods and all of them have their advantages and disadvantages, so there is no such thing as "The best modeling method".

The reason why I took this path forms- she works. I also always wanted to become a sculptor. Before going into detail, I like to sketch out a rough shape. It is because of this that I have achieved so much and that is why I decided to write this article to help beginners in organic 3D modeling and show them the shape before they start doing anything.

The first thing I started with was the shape of the head and I got frustrated because I tried to make it without any reference information (no references- from English reference), only using your imagination. Instead of sketching out a rough shape, my mind was busy with questions like: "How many cuts are needed? Why? Where and When?"

I was worried not only about my head, but also about my eyes, nose and mouth (and I hadn't even gotten to them yet). My brain was confused and I was completely at a loss as to how to create this head... until one day when I managed to sketch out a basic boxing head and behold... see the moment of truth! I was so excited that I decided to do it again! And then again and again, until I got tired of it and was exhausted.

Looking back, it seems so basic and simple to me. All that was needed was to create a box and make a couple of cuts and edits!

However, if it is so simple, then why did I struggle with it for so long? Can we all do this without the problems I experienced? Well, my answer is YES! But only if you mean it the right mindset. For example, I didn’t have one when I first started.

What I realized now is that when we learn 3D modeling, then we are just we don’t teach 3D at all! What we're really doing is looking for the right "mindset." So when you experience difficulties in something, it does not mean that you lack skills or knowledge. This is all because you don't have the right mindset to do what you're trying to do.

Once you have rewired your mind, your rational mind will take over and you will begin to do things naturally. So that's the first thing we have to try to rebuild - the mindset.

Mentality

Drawing a profile (contour): connecting dots

This little example will help you change your mindset.

First, just look at this image. Now we will draw a profile using dots and connect them. If you only had two points (on your forehead and chin) to connect them. How would you do it? Answer: from forehead to chin, because there is simply no other way.

However, if you increase the number of points, they will not only allow you shape the profile more exactly, but they will also allow it to be done in many ways, and this already leads to style formation(artistic).

This is very important to keep in mind when you need to make cuts or know where to finish them.

Key Cut (KR) and Fill Cut (FC).

At first it was very difficult for me to understand where and how many cuts I should make when creating a particular shape. So I was looking for an analogy for this process. This analogy turned out to be Animation.

Animation has a concept Key Personnel(KK). In short, this is characteristic poses character in a certain moment of time. This concept also includes Intermediate Frames(PrK), which fill time intervals between Key Personnel.

This not only speeds up the process, but also makes it easier. The more Intermediate Frames (fill cuts) you have, the smoother and more precise the movement will be.

If you are an animator, then you have the power to control the number of PRKs. This is very similar to cutting polygons in 3D.

Drawing a large number of PKs and managing them all is a very tedious job. The same goes for moving a large number of vertices in 3D - it is very labor-intensive.

The idea behind the CD is joints. When a modeler sketches out a rough shape, he always starts with the KR, which always looks rough. If the editor you're using supports dice, then use it to figure it out. Bend/twist the bones at the joints to see your gross form in the poses.

Once all the CDs are ready, you have two options:

1. Smooth the model.
   Sometimes I create a CR, and then simply let the code responsible for dividing the model into a larger number of polygons (subdivision) complete all the CRs for me. The downside is that it doesn't look realistic. So the next step is to use a soft highlight to correct the shape. Sometimes this can save a lot of time (but it depends on what you're modeling).
2. Add ZR manually.
   In most cases, I prefer manual work, since this way I can control the number of points and their location.

Please note that this concept of Key and Fill cuts is not only useful for creating shapes, but also for detailing your mesh. KR and ZR created using partitioning are one of the ways to optimize the mesh (buttocks, thighs, etc.). Also, sometimes a Fill Cut can become a Key Cut depending on how you look at it. You are the creator, so everything is in your power.

What's also important is that this concept also works great for topology/loops (Key and Fill Loops).

Main and Fill is a very interesting concept because it can be applied to almost anything! Next time you look at the topology mesh, try to find the Key Loop, since every head has at least one of these.

Based on what I saw, there are such head topologies:

• C-loop
• X-loop
• E-loop
• And a bunch of others

I'll talk about all this later, but for now let's focus on the form.

Rounding

This is the most common mistake of all beginners. They create the Key Cuts, and then the Filler between them and leave it all without the slightest change. If you don’t round your GR, the result will be square (unnatural, inorganic) and you’ll have to work hard to fix it later. If, every time you create the next Filling Cut, you correctly fit it to the shape, then you will save yourself from constantly reworking the mesh.

Following the lines of the form (body lines, smoothness of lines).

Another common mistake is NOT following the smooth lines of the subject. Remember, this is an organic simulation, so try to think organically. When sketching body parts such as a tail or a body that curves, try to imagine a curving cylinder. And create blocks accordingly.

Fear, haste and doubt

This is a mental level of challenge when you are just starting out in 3D modeling.

Every time you do something for the first time, you experience great difficulty. The point is, don't give up! Everyone goes through this. It is rare to meet a person who has gone through this initial stage and does not talk about how he suffered.

So here’s my advice: take it easy, slow down, there’s no rush here. Try spending a month or two playing with your form. Start with objects that allow you to make a lot of mistakes, such as creatures. And just practice. If it turns out crap, delete it and start over.

At first, everything will work out slowly for you, but as you do similar tasks, your speed will increase all the time. This is exactly why we need practice, to do everything better and faster.

When you create a model for the first time it can be a very fun process. All because of the "look at the whole."

Take, for example, the human figure. Let's say you start with the torso and use extrude to stretch it out. If you don't have legs and arms/head yet, then it all looks very comical. To make "it" look human, you must complete all the remaining body parts.

So there is no need to lose interest due to the terrible result of not having all the pieces in place. You just need to extrude all the body parts and place them in the right places, only then “it” will start to look like a human figure.

Practice

Simulation subject

First let's talk about the modeling subject.

If you are doing character modeling, then you will obviously start at the head and work your way down. A simplified head, torso, and then arms and legs. After a few weeks you will realize that the head is the simplest part of the body, since it is just one block, entirely visible from one point. And all you need to model is to zoom in and out of it (the head).

Other body parts (arms, legs) will be more challenging because they require you to rotate and zoom the model in the viewport. And since you're new to 3D, chances are you're not used to making full use of rotation, spin, pan, and zoom in viewports.

At first, to avoid unnecessary difficulties, use references. And once you get the hang of it, try modeling from memory.

Creating a hand from memory for the first time is hard. So try to use reference pictures/photos first and memory later.

Why do it from memory at all? Just to see if your understanding of the shape of the hand (or whatever object you're creating) has improved.

If you model different creatures, then the situation is the same here. Start with the head, then the body, and then everything below. Don't limit yourself by modeling just one part. Jump from one part to another (for example, I do this), so you (thanks to changing the type of activity) will constantly maintain interest in this process.

Extrude.

Before you start extruding parts like arms and legs, you should know that there are only two ways to do it. This has to do with how to model the angle.

Method A is, of course, faster, but you will still, sooner or later, come to method B. You can also convert A to B using the Polarization method (more on this later). Also note line shape (red).

I have seen many variations of Method A for creating realistic human hand. While method B is suitable for unrealistic characters, for example, cartoons and the like.

If you find it difficult to rotate every time you extrude, then use Method A. But it doesn't really matter which method you choose, since you can convert one topology to another as you go.

This concludes the first part of the article. You can ask questions if something is unclear.

Let me conclude with a few the best.

This is my translation of an excellent series of posts from SomeArtist on subdivisionmodeling.com (which were deleted because the forum ceased to exist).

Subscribe to blog updates(Here ).

P.S. The barbarian turtle in the title picture was made by American Jesse Sandifer. The simulation was performed entirely in Mudbox, then the whole scene was assembled into 3ds Max and visualized by forces Vray. Photoshop used for texturing and post-processing. For other types of character, as well as a discussion of the work, read

This tutorial is a good start for anyone who wants to learn how to model top-notch characters. Famous in his circle, Jahirul Amin will talk about the importance of correct topology, uniform mesh, the importance of quadrangular polygons and much more.

Before diving into the 3D whirlpool, I suggest having a short educational program and splashing around in the shallow water. Below we will touch on the basics of polygonal modeling, without knowledge of which it is pointless to move on.

Introduction

When geometry becomes a modeler's or animator's aid, the ideal mesh layout comes first. After this, a good topology should come into play, reducing the number of defects in character animation. In other words, a correctly (and on time) created polygon will save not only hours but days of your life.

3-gon vs 4-gon vs N-gon

So what's the difference between 3-, 4-, and N-gon polygons? The answer is obvious: the first has 3 sides, the second has 4, the third has any number of them, more than 4. If you are modeling a character for further animation, we recommend use only quadrilaterals. The process of deforming and dividing quadrangular polygons is much easier, and you will encounter less texture distortion.

It is recommended to hide triangles from your own and other people's eyes. For example, in the armpits or in the groin area of ​​the character. In turn, an unspoken ban is imposed on polygons - they should not exist. They cause distortion and cause a lot of trouble when it comes to rigging and editing vertex groups (aka “weight-painting”).

Finally, a model that consists primarily of quad polygons will be easier to export to other modeling programs such as Mudbox.

The joys of four and three-gon polygons and the horror of the N-gon

The contours of the face, which by definition resemble an N-gon, should be brought as close as possible to a quadrangular format. Little of - the location of the polygons should be as uniform as possible in principle. This is what the geometry of the same name calls for. Following these rules will make it easier to go through the rigging stage and will help when deforming the character during the animation process. In addition, the scale of distortion associated with the use of textures will be reduced, although here we should not forget about the importance of the UV scan itself.

To perform the described task, Maya provides the Sculpt Geometry tool.

The Sculpt Geometry tool in Maya will help you “smooth out” your model’s mesh

Responsible for the smooth transition of each individual edge (aka Edge Flow). It may sound simple, but in practice it is a very insidious thing.

If you set out to create a realistic character, it is recommended to study the basics of anatomy before starting work. By following the structure of the human body and the natural movement of muscles, the animator ultimately obtains a copy that is close to the original. This is especially clearly seen during the deformation process. We recommend starting with the process of wrinkle formation and skin stretching.

For stylized and cartoon characters, Edge Flow is much less important. But still, I highly recommend getting at least a basic understanding of human anatomy.

To make the shape realistic, create a good topology and be sure to take into account the smooth direction of the mesh (edges, polygons).

It is also non-manifold. Means that a three-dimensional object cannot be cut and made flat.

Example: Create a cube, select any edge (edge) and extrude it Edit Mesh > Extrude. In front of you is a somewhat shaped object. (Example below on the left) If the cube were made of paper, then when unfolded you would get a cross-shaped figure with broken proportions. Using such an object in Boolean operations is practically impossible.
To fix the situation, use the Cleanup tool.

Violation of the geometry topology can create dozens of problems. Be vigilant and periodically inspect the figure from different angles.

Each loop (edge ​​edge) must have a target

As a rule, modeling begins with a primitive figure (for example, a cube), the structure of which is subsequently complicated by adding edge loops.

It is important that each new element is created with a specific purpose. There are situations in which “less” equals “better.” Understanding the principles of model optimization comes only with experience, so don’t get discouraged and keep working.

Don't complicate your life: detail should be appropriate

Everything we are trying to do on the screen is a reflection of the world around us in its various forms and manifestations. This is why it is so important to get up from the table from time to time. Important not only for developers, but also for animators, riggers, lighting directors, etc.

Take a closer look at the surface, its structure and shadow. How does it reflect light? How does the deformation process occur? The answer to these and other questions will help you make the right decision when modeling any object.

Topic of conversation: TOPOLOGY.

Topology (from ancient Greek τόπος - place and λόγος - word, doctrine) is a branch of mathematics that studies in its most general form the phenomenon of continuity, in particular the properties of space that remain unchanged under continuous deformations, for example, connectivity, orientability. Unlike geometry, topology does not consider the metric properties of objects (for example, the distance between a pair of points). For example, from a topological point of view, a circle and a donut (solid torus) are indistinguishable.

But this is in mathematics. How are things going with the characters? Let me put it in my own words.
Topology is the ability of a mesh to respond correctly to deformations. Be it animation, compression, stretching or other types of deformation. This is achieved by competently constructing a character’s polygonal mesh. There are some rules for this. You can familiarize yourself with some of them.

There is also a concept RE-TOPOLOGY. Changing the topological mesh while preserving the shape of the object as much as possible. The purpose of retopology is to correct the previous (incorrect) topology and/or reduce the number of polygons.

Almost all modern 3D graphics packages have tools for retopology. I personally tried:
1. Maya - both standard tools and plugins.
2. Max - standard tools (horror), plugins and scripts (I liked wrapit, but again not that much)
3. Zbrush - tight and uncomfortable..
4. Topogun - finally found something I liked... if I hadn’t met it
5. 3DCoat.... here I realized that this is so far the most convenient for retopology and UV unwrapping... although it was difficult to figure it out at first... but when I understood the principle of the program - that's it... now retopology is all about it. (don't take this as advertising.)

Well, since this booze has started, I’ll post a couple of my images on the topic of topology.
Head and face

I found an old render of this head.

topology of the face of a humanoid character. You can make both a woman and a child out of him... not to mention a man.
and here's the proof. done quickly, but clearly.
So. a man, an elf, a creature, a woman, and a girl of about 15...
I do not claim that this is the only competent topology, and that this is the ONLY way to do it.
Some studios model characters with their eyes closed. This allows you to get rid of some problems when closing the eye, and avoid deformation of the eyelid when deforming the cheek.

wrist.

I draw your attention to the fact that there are vertexes here that can accommodate 6 hedgehogs... but in these places there are no problems because the deformations are minimal. Naturally, from this brush you can make the hand of a woman, a man, a child... or anyone...
Scull.

male skull. There are many differences between male and female skulls.

The differences are as follows:
Male and female skulls have a number of differences. Namely:
1. The male skull is more massive than the female and has a rather square shape. The woman's skull is slightly pointed towards the top and more rounded.
2. The upper edge of the eye socket is slightly pointed in the female skull, while in the male it has a smoother curve
3. As a result of evolution, the facial muscles have become more developed. Consequently, the place where the muscles attach to the skull is much more noticeable in men. After all, a warrior and a hunter need powerful jaws for combat and struggle.
4. A man's strong lower jaw is square in shape, while a woman's is round in shape.
5. The depth of the skull of men is greater than that of women. This provides relative safety.
6. The brow ridges on a male skull protrude noticeably more. They protect your eyes from direct sunlight.
7. Men's canines are much larger than women's. The warrior and hunter was forced to eat while on the move, and, therefore, actively chew food and do it quite quickly.
Hand and body.
If the body is female or without clearly defined muscles, then you can ignore the magnifying glasses that form the muscles. This applies to the hands. I draw your attention to the white polygons. they come from under the pectoral muscle and go around the deltoid. INTRODUCTION

A future explorer is born

not at 30 years old, studying in graduate school,

and much earlier than the time when

his parents will take him to kindergarten for the first time.

Alexander Ilyich Savenkov

Doctor of Pedagogical Sciences, Professor of Moscow State Pedagogical University

With the development of new technologies, the demand for people with innovative thinking and the ability to pose and solve new problems has sharply increased. Therefore, mathematical preparation of students is becoming more relevant than ever. Here it is appropriate to recall the statement of the great Russian scientist Mikhail Vasilyevich Lomonosov: “Mathematics must be taught only then because it puts the mind in order.”

Every person has a visual concept of space, bodies and geometric shapes. In the school geometry course we will study various bodies and their properties.

But that will be in the future, but for now I’m interested in the question: “What is a Möbius strip?” You will ask me why I am interested in this. I will answer. I really love to read. Especially science fiction. One of my favorite science fiction writers is Arthur C. Clarke.

In his story “The Wall of Darkness,” one of the characters travels through an unusual planet, curved in the shape of a Mobius strip. I became interested in what kind of figure this is and what its properties are.

Having studied the relevant literature and Internet sources, I learned that this issue is studied in a separate branch of mathematics - topology. That is why my work is devoted to solving the simplest research problem in this field.

The purpose of the work can be formulated as gaining an understanding of one of the most interesting and unusual branches of mathematics, namely topology and the study of the topological properties of some objects.

To achieve the goal, I solved the following tasks:

understand what this science studies;

study the history of its origin;

consider the topological properties of some objects;

learn about the practical application of topology.

The relevance of the chosen topic lies in the fact that recently this science has increasingly penetrated into such fundamental areas of human knowledge as physics, chemistry, and biology. Therefore, knowledge of its basics becomes significant for a technically educated person living inXXIcentury.

MAIN PART

Topology as a science and the prerequisites for its emergence

Unlike other branches of geometry, where the ratio of lengths, areas, angles and other quantitative characteristics of objects are of great importance, topology is not interested in all this, since other, qualitative questions about geometric structures are studied here.

Let's start understanding the basics of this fascinating science. If we turn to literary sources, we can find the following definition of this concept.

Topology - a branch of mathematics that studies the properties of figures (or spaces) that are preserved under continuous deformations, such as stretching, compression or bending.

Let us explain the concept of “continuous deformation” encountered here. Continuous deformation is a deformation of a figure in which there are no breaks (that is, a violation of the integrity of the figure) or gluing (that is, the identification of its points).

Every branch of mathematics has a core idea. Topology is no exception. The main idea of ​​topology is the idea of ​​continuity, that is, topology studies those properties of geometric objects that are preserved under continuous transformations.

Continuous transformations are characterized by the fact that points located “close to each other” before the transformation remain so after the transformation is completed. During topological transformations, objects are allowed to stretch and bend, but they are not allowed to tear or break.

To visualize the definition of topology, it should be said that from the point of view of this science, objects such as a tea cup and a donut are indistinguishable from each other. That is why there is a catchphrase among scientists that says that a mathematician who studies topology is a person who cannot distinguish a bagel from a tea cup. This statement is true because by squeezing and stretching the piece of rubber from which these objects are made, you can move from one body to the second.

Drawing 1The process of converting a cup into a donut (torus)

Let's take a historical excursion and return toXVIIIcentury when the foundations of this science were laid.

One of the scientists who stood at the origins of this science is a German mathematician and mechanicXVIIIcentury Leonhard Euler. In 1752, he proved Descartes' formula expressing the relationship between the number of vertices, edges and faces of simple polyhedra:

Where, .

Euler's next contribution to the development of topology was the solution of the famous bridge problem. It was about an island on the Pregol River in Königsberg (at the place where the river divides into two branches - Old and New Pregol) and seven bridges connecting the island with the banks (Fig. 2).

It was necessary to find out whether it was possible to go around all seven bridges along a continuous route, visiting each one only once and returning to the starting point. Euler replaced land masses with dots and bridges with lines. Euler called the resulting schemecount (Fig. 3), the points are its vertices, and the lines are its edges.

Drawing 2The Koenigsberg Bridges Problem

L - left bank , R - right bank ,

Drawing 3Graph

The scientist divided the vertices into even and odd, depending on the number of edges coming out of the vertex. Euler proved that all the edges of a graph can be traversed exactly once along a continuous closed route only if the graph contains only even vertices.

Since the graph in the Königsberg bridges problem contains only odd vertices, the required walking route does not exist.

This problem illustrates the practical application of the concept of “unicursal graph”, which appeared in the dictionary of topology inXXcentury. The graph is calledunicursal , if it can be “drawn with one stroke,” i.e. go through it all in a continuous motion, without going through the same edge twice.

Thus, the graph of the Königsberg bridges problem is not unicursal and therefore the problem has no solution.

The term “topology” first appears in a letter to his school teacher Muller, which the German mathematician and physicist, professor at the University of Göttingen Johann Listing wrote in 1836. General topology, originating inXIXcentury, finally formed into an independent mathematical discipline in the second halfXXcentury. This was largely facilitated by the works of Academician P.S. Alexandrova.

Topological properties of objects

Topology in popular science literature is often called rubber geometry. To understand this, you need to imagine that a geometric object is made of rubber and at the same time has the following properties: it can be compressed, stretched, twisted (that is, subjected to all kinds of deformation), but it cannot be torn and glued together.

For example, a small ball can be inflated to the size of a large one, then turned into an ellipse, then deformed into a dumbbell.

Drawing 4The process of deforming objects

In a similar way, you can turn the surface of a ball into the surface of a cube, cone and other figures. There are properties in mathematics that are not violated under any continuous deformations. That's what it istopological properties . One of the branches of topology, general topology, studies these properties.

The properties that are studied in school (Euclidean) geometry are not topological. For example, straightness is not a topological property, since a straight line can be bent and become crooked. Triangularity is also not a topological property, since a triangle can be continuously deformed into a circle.

Lengths of segments, angles, areas - all these concepts change with continuous transformations. An example of a topological property is the presence of a “hole” in a torus (donut). Moreover, it is important that the hole is not part of the torus. No matter how much continuous deformation the torus undergoes, the hole will remain.

One-sided surfaces

Each of us has an idea of ​​what “surface” is. We are simply surrounded by various surfaces: the surface of a sheet of paper, the surface of a lake, the surface of the globe...

As a rule, we imagine a surface with two sides: outer and inner, front and back, etc. Could there be anything unexpected and even mysterious in such an ordinary concept? It turns out that it can.

In 1858, the German mathematician and astronomer August Ferdinand Möbius (1790-1868) discovered a surface that later became known as the “Möbius strip”. According to legend, Mobius was helped to discover his “leaf” by a maid who sewed the ends of an ordinary ribbon incorrectly.

A Möbius strip is the simplest one-sided surface with an edge. It is possible to get from one point of such a surface to another without crossing the edges.

Let's repeat this discovery. Let's create the surface under study and study its properties.

For work we need an A4 sheet of paper, a ruler, a pencil, scissors and glue.

Drawing 5Tools

On a sheet of paper, draw two strips 4 cm wide and cut them out. These will be the blanks from which we will make our tape (sheet).

Drawing 6Creating a blank

From one strip we will glue an ordinary ring, and from the other - a Möbius strip. To do this, turn the second strip half a turn and glue the ends together.

Drawing 7Stages of work

This is what we should get.

Drawing 8Result of work

Let's start researching the properties of the resulting figures. It is impossible to distinguish the front side from the back side of a Möbius strip. They continuously transform into each other. The task of painting different sides of the ring with different colors will not cause any difficulty. Let's see this with a simple example. Take a felt-tip pen, mark a dot and start continuously painting one side. You will see that only its inner surface will be painted over.

Drawing 9Ring coloring

But will this be true for our second paper object? Let's repeat the experiment, choosing as the experimental surface not a ring, but a Möbius strip.

Drawing 10Coloring the Möbius strip

You see that the entire sheet has become colored. But we still only drew the felt-tip pen on one side. From this we can conclude thatthat the strip from which the Möbius strip is made has two sides, and the strip itself has one .

If we move along the edge of the Möbius strip, then after a full turn we will find ourselves on the other edge and come from the opposite side.

Let's continue our research and consider the question of how our two figures (the ring and the Möbius strip) will behave when they are cut. If you cut the ring along the midline, you will get two narrower rings

Drawing 11Cutting the ring

Drawing 12Result of ring cutting

If you cut a Möbius strip along the middle line, it will not split into two rings, as was the case in the ring experiment. We will get one ring, but twice as long (the resulting ring will have a double-sided surface).

Drawing 13Cutting a Möbius strip along the midline

What happens if you cut a Möbius strip along a line lying close to the edge? To get to the beginning of the cut, we will have to go twice as long as cutting this sheet along the midline. You will get two interlocking rings, one large and narrow, and the other small and wide. The most interesting fact is that the large ring will have a one-sided surface, and the small one will have a double-sided surface.

If you make a Möbius strip that is twisted 3 half turns (540 degrees), and then cut it in half, you will get a Möbius strip twisted in a knot.

You can get interesting things if you fold the paper like an accordion, then make a Möbius strip out of it and cut it in half or one-third. Three interlocking rings will appear before us.

As researchers of the properties of this figure, we were interested in the question: is it always possible to create a Möbius strip? It turned out that if we take a square sheet of paper and cut a strip out of it, we will not be able to get the figure we are interested in.

Then a new question arises: what should be the ratio of the length and width of the strip so that it can always be used to obtain a Möbius strip? It has been mathematically proven that if we take the width of the strip to be 1, then the length should be 1.73.

Practical application of topology

When they talk about topology, the Möbius strip is the first thing that comes to mind for a person familiar with this issue. Therefore, in the field of practical application of this science in various branches of human activity, the use of this particular figure is most often encountered.

The amazing properties of the Möbius strip serve as a source of inspiration for writers and poets. As an example, I would like to give a short excerpt from a poem by Natalia Ivanova:

The Moebius strip is a symbol of mathematics,

What serves as the crown of the highest wisdom...

It is full of unconscious romance:

In it, infinity is curled into a ring.

There is simplicity in it, and with it complexity,

which is inaccessible even to the wise:

Here the plane has transformed before our eyes

Into a surface without beginning or end.

Flatland by Edwin Abbott and its sequel Spherland, written by David Burger in 1976, are rightfully considered a classic book about life in two-dimensional space.

The Flatlander lives on a planet shaped like a two-dimensional surface. If his universe is an infinite plane, then he can travel any distance in any direction. But if the surface on which he lives is closed like a sphere, then it is unlimited and finite.

Whatever direction the Flatlander goes, moving straight and not turning anywhere, he will certainly return to where he started his journey. When a Flatlander travels around the world on a sphere, it is as if he is moving along a strip glued into a ring.

But if an inhabitant of this planet travels along the Mobius strip, then upon returning to the starting point, he will find his heart not on the left, but on the right! A similar situation is described in the fantastic story by H.G. Wells, “The Plattner Story.” A man, having been in the fourth dimension, returned to Earth as his mirror double - with a heart located on the right.

In production, a conveyor belt is made in the form of a Möbius strip. This design feature allows you to increase the service life of the belt, as its surface wears evenly.

Drawing 14Belt Conveyor

Relatively recently, the main device for outputting information from a computer to printing was a dot matrix printer. In its print head, the ink ribbon was also arranged in the form of a Möbius strip.

Drawing 15Matrix printer

Since we are talking about computers, a computer network is used to connect several machines into a single whole. One of the basic terms of network technology is the concept of network topology.Topology – a general diagram of a computer network, showing the physical location of computers and the connections between them.

Drawing 16Examples of computer network topology

The shape of the Möbius strip is quite successfully used in architecture. Let's give a few similar examples.

Drawing 18Logos based on Mobius strip

There is a hypothesis that the DNA spiral itself is a fragment of a Mobius strip and that is why the genetic code is so difficult to decipher and perceive. In addition, such a structure quite logically explains the reason for the onset of biological death - the spiral closes on itself and self-destruction occurs.

Drawing 19DNA helix

Artists and graphic artists also did not ignore the topic that interests us. Indicative in this regard is the work of the Dutch graphic artistXXcentury by Maurice Escher. He is known for his lithographs, in which he masterfully explored the plastic aspects of infinity and symmetry.

He said about his work: “Although I am absolutely ignorant of the exact sciences, it sometimes seems to me that I am closer to mathematicians than to my fellow artists.”

Drawing 20Lithographs by Maurice Escher

CONCLUSION

The topology is the youngest and most

powerful branch of geometry, clearly

demonstrates fruitful influence

contradictions between intuition and logic.

Richard Courant

American mathematician

A Russian folk proverb says: “The end is the crown of the matter.” So my little journey into the fascinating and unusual world of topology has come to an end. It's time to take stock.

During the course of my work, I became acquainted with a new area of ​​mathematics for me - topology. I examined some of the simplest concepts used by this science and accessible to understanding without serious mathematical training.

In practice, he recreated the most famous topological surface - the Möbius strip and studied its general properties. I also became acquainted with the practical application of topological surfaces in various spheres of human activity.

Thus, all the tasks I set at the beginning of this work were successfully solved. I hope that my acquaintance with this area of ​​mathematics in the future will not be so superficial, which provides grounds for continuing work on the chosen topic as my mathematical knowledge accumulates.

BIBLIOGRAPHY

Mathematical encyclopedic dictionary / Yu.V. Prokhorov [and others]. – M.: Publishing house “Soviet Encyclopedia”, 1988. – 340 p.

Boltyansky, V.G. Visual topology / V.G. Boltyansky, V.A. Efremovich – M.: Nauka, 1975. – 160 p.

Starova, O.A. Topology / O.A. Starova // Mathematics. Everything for the teacher. – 2013. – No. 9. – p.28-34.

Stewart, J. Topology / J. Stewart // Quantum. – 1992. – No. 7. – p. 28-30.

Project for gifted children: Scarlet Sails [Electronic resource] – Access mode:http:// nportal. ru/ ap/ blog/ scientifically- technical- tvorchestvo/ list- myobiusa– access date: 01/18/2017

Prasolov, V.V. Visual topology / V.V. Prasolov. – M.: MTsNMO, 1995. – 110 p.

Abbott, E. Flatland / E. Abbott. – M.: Mir, 1976. – 130 p.

Topology- a rather beautiful, sonorous word, very popular in some non-mathematical circles, interested me back in the 9th grade. Of course, I didn’t have an exact idea, however, I suspected that everything was tied to geometry.

The words and text were selected in such a way that everything was “intuitively clear.” The result is a complete lack of mathematical literacy.

What is topology ? I’ll say right away that there are at least two terms “Topology” - one of them simply denotes a certain mathematical structure, the second one carries with it a whole science. This science consists of studying the properties of an object that will not change when it is deformed.

Illustrative example 1. Bagel cup.

We see that the mug, through continuous deformations, turns into a donut (in common parlance, a “two-dimensional torus”). It was noted that topology studies what remains unchanged under such deformations. In this case, the number of “holes” in the object remains unchanged - there is only one. For now we’ll leave it as it is, we’ll figure it out a little later)

Illustrative example 2. Topological man.

By continuous deformations, a person (see picture) can unravel his fingers - a fact. It's not immediately obvious, but you can guess. But if our topological man had the foresight to put a watch on one hand, then our task will become impossible.

Let's be clear

So, I hope a couple of examples brought some clarity to what is happening.
Let's try to formalize all this in a childish way.
We will assume that we are working with plasticine figures, and plasticine can stretch, compress, while gluing different points and tearing are prohibited. Homeomorphic are figures that are transformed into each other by continuous deformations described a little earlier.

A very useful case is a sphere with handles. A sphere can have 0 handles - then it’s just a sphere, maybe one - then it’s a donut (in common parlance, a “two-dimensional torus”), etc.
So why does a sphere with handles stand out among other figures? Everything is very simple - any figure is homeomorphic to a sphere with a certain number of handles. That is, in essence, we have nothing else O_o Any three-dimensional object is structured like a sphere with a certain number of handles. Be it a cup, spoon, fork (spoon=fork!), computer mouse, person.

This is a fairly meaningful theorem that has been proven. Not by us and not now. More precisely, it has been proven for a much more general situation. Let me explain: we limited ourselves to considering figures molded from plasticine and without cavities. This entails the following troubles:
1) we can’t get a non-orientable surface (Klein bottle, Möbius strip, projective plane),
2) we limit ourselves to two-dimensional surfaces (n/a: sphere - two-dimensional surface),
3) we cannot obtain surfaces, figures extending to infinity (of course we can imagine this, but no amount of plasticine will be enough).

The Mobius strip

Klein bottle