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.

wrist.

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.

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.

Mathematical Structures and Modeling 2000, no. 6, p. 107-114

UDC 530.12:531.18

TIME AND TOPOLOGY OF THE HUMAN BODY

The Philosopher Kant declared the time is given us a piori, i.e. is defined for person from birth. Has it relation with the topology and geometry of the human body? In Minkowsky space-time the four-dimensional topology of the human body is trivial and diffeomorphic to R = x B, where BcRl Such topology allows to perceive the sensations by consecutively any point of the body. If body has other four-dimensional topology which is not diffeomorphic to R, then there exists full collapse of memory in an effort to observe the sensation consecutively. Hence, other topology of the body means the absence of time in that form to which we have become accustomed.

This article was written with the aim of a comprehensive study of the consequences of the theory of absolute space-time. It is known that the material body is described in the theory of relativity by a set of world lines, but physics is not interested in the human body. Let us, however, try to find out how the pseudo-Euclidean geometry of space-time correlates with the four-dimensional topology of a body that a living organism can have in the absolute Minkowski World of Events.

1. The illusion of time

Human life occurs in time. We organize the events that happen to us by dating them. We know thoroughly that the past in our lives is something that is irretrievably gone, and the future that awaits us is unknown because it has not yet arrived. But we know that death awaits us ahead.

At birth a person receives a body. From the point of view of mathematics, life is a four-dimensional region R, which has a topological structure diffeomorphic D1 xB, where D1 is a one-dimensional disk, a period of time that a person is destined to live, and B is his body in three-dimensional space, the topology of which is simplified in Fig. 1. The modern theory of space and time suggests that the World of Events is a so-called four-dimensional pseudo-Euclidean space V4, called space-time. An event is a point in space-time V4. The life path of an elementary material object is a curve, a world line, in the V4 Event World. Therefore, a person’s life as the totality of all events occurring in his life is a smooth embedding h: D1 x B -> V4. World line

© 2000 A.K. Guts

Email: [email protected] Omsk State University

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.

Let's be clear

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