You might think that mathematical calculations have no meaning for artistic arts. But that's not true. Reflection of the world around us, which to a greater or lesser extent is fine art, is associated with the reproduction geometric shapes and their proportions.
Sometimes these proportions are deliberately violated, creating optical illusions. A master in this field can be called Mauritsa Escher. Dutch artist the middle of the 20th century, in the space of whose drawings everything is possible. One form flows into another, perspective does not converge at one point, objects have neither beginning nor end. This discrepancy with the laws of nature and logic fascinates viewers who are trying to find a solution to the artist’s graphic paintings.
The world of the avant-garde
By presenting art in a new way, using its capabilities in a way that had not been done before, painters of the avant-garde movement (French. avant-garde- walking in front) tried to decompose the objective world into its constituent elements. They represented the feeling through imagery and color associations. From this it turned out that geometric figures in the paintings of painters often expressed both form and content.
An example would be creativity Wassily Vasilyevich Kandinsky. The Russian painter, a representative of the avant-garde of the first half of the 20th century, makes abstraction an end in itself in his paintings. Theoretically, Kandinsky's ideas are justified by the creation next to " real world"The world of abstractions, which outwardly seems to have nothing in common with reality. This is a new system, subject to its own laws.
In painting it important role feeling plays, the artist’s paintings are very musical, although they do not have sounds. The basis of his paintings is an emotional outburst, the figures are subject to this movement, they follow the emotion. Kandinsky wrote that the circle, which so often becomes centrally his paintings of the 20s. can be called "romantic". This romance is deep and contradictory, like fire burning in ice.
Geometry of the creator of the “Black Square”
Suprematism(“superiority”) - an avant-garde direction, the invention of which belongs to Kazimir Malevich. Its formation dates back to the creation of " Black square"(1915). The artist imagined that in his work he had come to the basis of art and stepped beyond it, stepped into that plane where there is no longer form, there is nothing. The geometric form in the paintings of the author of this period is given on its own, without any contexts or specific meanings.
« White on white"1918, where a white square figure is depicted on a white canvas - a retreat into even greater pointlessness. Malevich sought in these paintings to nullify all previous art. A decade later, K. Malevich’s style undergoes changes. The artist moves to a direction later called “Russian neo-Suprematism”. Here colors and shapes express certain thoughts and describe specific events.
« Athletes» 1932 - the picture is completely symmetrical. In the foreground are impersonal figures of athletes, built primarily from vertical lines, which in the background are replaced by contrasting horizontal ones. The colors intersect with each other on both planes.
Very often in the world of artists there are paintings that are significantly different from oil and pastel canvases. They are more reminiscent of drawings, patterns, sketches and are completely incomprehensible to the average viewer. Now we will talk about compositions of geometric shapes, discuss what they are like, what load they carry and why they generally occupy such an honorable place in the art of drawing and painting.
Simple compositions
Every master of the brush who began his journey at an art school will answer you that precise lines and their combinations are the first thing they teach there. Our vision and brain are designed in such a way that if we initially learn to harmoniously combine simple forms, then in the future it will be easier to draw complex pictures. Compositions of geometric shapes allow us to feel the balance of the picture, visually determine its center, calculate the incidence of light, and determine the properties of its components.
It is worth noting that, despite the clarity and directness of such images, they are drawn exclusively by hand, without rulers or other auxiliary items. The parameters of the figures are measured using proportions, which can be located in two dimensions (a flat picture), or can go into perspective, to a single vanishing point of all lines.
Beginning artists draw compositions from geometric shapes in two dimensions. For such paintings, one of the sides is chosen - the plan or the facade. In the first case, all the figures are depicted in a “top view”, that is, the cone and cylinder become a circle, the prism takes on the shape of its base. If figures are depicted in a façade, one of their sides is shown, most often the front. In the picture we see triangles, squares, parallelograms, etc.
3D paintings
In order to develop a sense of perspective, artists learn to depict compositions from three-dimensional geometric shapes that go into perspective. Such an image is considered three-dimensional, and in order to transfer it to paper, you need to clearly imagine everything. Similar drawing techniques are relevant in construction and architectural universities; they are used as exercises. However, students often make real ones out of these “picturesque studies” by drawing incredible insets of figures, dissecting compositions with planes and half-planes, and depicting pictures in cross-section.
In general, we can say that clarity and linearity are the main properties that any composition of geometric shapes has. At the same time, a drawing can be static or dynamic - it depends on the type of figures depicted and their location. If the picture contains mainly cones, triangular prisms, and balls, then it seems to be “flying” - this is definitely dynamic. Cylinders, squares, tetrahedral prisms are static.
Examples in painting
Geometric shapes have found their place in painting, along with romanticism and other trends. A striking example to this is the artist Juan Gris and his most famous painting“Man in a Cafe”, which, like a mosaic, consists of triangles, squares and circles. Another abstract composition of geometric shapes is the canvas “Pierrot” by artist B. Kubisht. Bright, clear and very unique picture.
DRAWING BY REPRESENTATION: COMPOSITION OF GEOMETRIC BODIES. STEP-BY-STEP GUIDE. REVIEW
Volumetric composition of geometric bodies. How to draw?
A composition of geometric bodies is a group of geometric bodies, the proportions of which are regulated according to a table of modules embedded in each other and thereby forming a single array. Often such a group is also called architectural drawing and architectural composition. Although the formation of a composition, like any other production, begins with a sketch idea - where the general mass and silhouette, foreground and background can be determined, the work must be “built” sequentially. In other words, to have as its beginning a compositional core, and only then, through calculated sections, to “grow” with new volumes. In addition, this allows you to avoid accidental errors - “unknown” sizes, too small indents, awkward insets. Yes, we must immediately make a reservation that topics raised in almost every drawing textbook, such as “Organization of the workplace,” “Varieties of paints, pencils and erasers,” and so on, will not be considered here.
Composition of geometric shapes, drawing
Before moving on to the exam exercise - “Composition of three-dimensional geometric shapes”, you must, obviously, learn how to depict the geometric bodies themselves. And only after that you can move directly to spatial composition from geometric bodies.
How to draw a cube correctly?
Using the example of geometric bodies, it is easiest to master the basics of drawing: perspective, the formation of the volumetric-spatial structure of an object, the patterns of light and shadow. Studying the construction of geometric bodies does not allow you to be distracted by small details, which means you can better understand the basics of drawing. The depiction of volumetric geometric primitives contributes to the competent depiction of more complex geometric shapes. To depict an observed object correctly means to show the hidden structure of the object. But in order to achieve this, the existing tools, even from leading universities, are not enough. So, on the left side, there is a cube tested in a “standard” way, widespread in most art schools, colleges and universities. However, if you check such a cube using the same descriptive geometry, presenting it in plan, then it turns out that this is not a cube at all, but there is some geometric body, with a certain angle, and probably the position of the horizon line and vanishing points only resembles it.
Cubes. Left is wrong, right is right
It’s not enough to put a cube and ask someone to draw it. Most often, such a task leads to proportional and perspective errors, among which the most famous are: reverse perspective, partial replacement of an angular perspective with a frontal one, that is, the replacement of a perspective image with an axonometric one. There is no doubt that these errors are caused by a misunderstanding of the laws of perspective. Knowing perspective helps not only to prevent serious mistakes at the very first stages of form construction, but also stimulates you to analyze your work.
Perspective. Cubes in space
Geometric bodies
It shows combined orthogonal projections of geometric bodies, namely: cube, sphere, tetrahedral prism, cylinder, hexagonal prism, cone and pyramid. The upper left part of the figure shows lateral projections of geometric bodies, and the lower part shows a top view or plan. Such an image is also called a modular scheme, since it regulates the sizes of bodies in the depicted composition. Thus, from the figure it is clear that at the base all geometric bodies have one module (the side of a square), and the height of a cylinder, pyramid, cone, tetrahedral and hexagonal prisms is equal to 1.5 times the size of a cube.
Geometric bodies
Still life of geometric shapes - we go to the composition step by step
However, before moving on to the composition, you should complete a couple of still lifes consisting of geometric bodies. The exercise “Drawing a still life from geometric bodies using orthogonal projections” will be even more beneficial. The exercise is quite difficult, which should be taken with due seriousness. Let's say more: without understanding linear perspective mastering still life using orthogonal projections will be even more difficult.
Still life of geometric bodies
Insets of geometric bodies
Insertion of geometric bodies is such a mutual arrangement of geometric bodies when one body partially enters another - it crashes. Studying the variations of insets will be useful for every draftsman, because it provokes the analysis of one form or another, architectural or living in equal measure. It is always more useful and effective to consider any depicted object from the position of geometric analysis. Sidebars can be roughly divided into simple and complex, but it should be noted that even the so-called “simple sidebars” require great responsibility in the approach to the exercise. That is, in order to make the insertion simple, you should decide in advance where you would like to place the embedded body. The simplest option is such an arrangement when the body is shifted from the previous one in all three coordinates by half the size of the module (that is, half the side of the square). The general search principle for all inserts is the construction of the inserted body from its internal part, that is, the insertion of a body, as well as its formation itself, begins with a section.
Section planes
Composition of geometric shapes, step-by-step exercise
There is a widespread belief that it is easier and faster to form a composition by placing bodies in space through the “chaotic” overlay of their silhouettes on top of each other. Perhaps this is what prompts many teachers to demand the presence of a plan and façade in assignments. This is how, at least, the exercise is already presented in the main Russian architectural universities.
Volumetric-spatial composition of geometric bodies considered in stages
Chiaroscuro
Chiaroscuro is the distribution of illumination observed on an object. It appears in the drawing through tone. Tone is a pictorial means that allows one to convey natural relationships of light and shadows. It is relationships, since even such graphic materials as charcoal pencil and white paper are usually not able to accurately convey the depth of natural shadows and the brightness of natural light.
Basic Concepts
Conclusion
It should be said that geometric accuracy is not inherent in the drawing; Thus, in specialized universities and colleges, using a ruler in classes is strictly prohibited. Trying to correct a drawing using a ruler leads to even more mistakes. Therefore, it is difficult to belittle the importance of practical experience - since only experience can train the eye, consolidate skills and strengthen artistic flair. At the same time, only through the consistent execution of images of geometric bodies, their mutual insets, familiarization with perspective analysis, and aerial perspective, is it possible to develop the necessary skills. In other words, the ability to depict simple geometric bodies, the ability to represent them in space, the ability to connect them with each other and, no less important, with orthogonal projections, opens up broad prospects for mastering more complex geometric shapes, be they everyday objects or the human figure and head, architectural structures and details or cityscapes.
In the first geometry lessons in the 10th grade, the foundations of stereometry are laid, and children become familiar with spatial figures. I gave them an example of impossible spatial figures optical illusions- figures that seem to be an ordinary projection of a three-dimensional object, but upon closer examination, contradictory connections of the elements of the figure are visible, creating the illusion of the impossibility of its existence in three-dimensional space. The guys showed genuine interest, I invite you to plunge into the world of mathematical illusions.
Many will say that mathematics (geometry) is an analytical discipline, fine art is an emotional one, and somehow it just so happened that mathematics and painting are considered something so different, almost opposite and mutually exclusive. Modern figures rarely use images geometric perspective to display the realism of a three-dimensional scene on a canvas or sheet of paper. But there are also artists for whom mathematics with its unprecedented possibilities is the center of attention, and the most common techniques are the depiction of polyhedra, tessellations, impossible figures, Möbius strips, unusual perspectives, fractals.
The Dutch artist Maurice Escher (1898-1972) can be considered the founder of mathematical art; his works are a source of inspiration for many followers. Escher created unique and charming works that use and display wide circle mathematical ideas, and the most interesting ideas of Escher to study are all kinds of divisions of the plane, mosaics, polyhedra and the logic of three-dimensional space.
So, I invite you to the world of optical illusions
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| An absurd resemblance to a cube |
Try climbing the stairs to the top floor with such a cross arrangement of columns. Doesn't work? Why? On the floor of the lower platform, then inside the belvedere, there is a ladder, which two people climb. However, upon reaching the top platform, they will again find themselves outside, under open skies, and again they will have to go inside the belvedere.
Is this waterfall falling or rising? The falling water drives the mill wheel and flows down an upward sloping (?) zigzag chute between the two towers, returning to the point where the waterfall begins again. Both towers appear to be the same height; however, the one on the right appears to be a floor lower than the tower on the left.
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Above and below (High and Low), 1947. Lithograph. |
Is this the house you would like to live in? Two identical floors, but each opens to the observer from different points: the lower part is the scene that he will see standing on the ground, that is, on a platform laid out tiles. Looking up, he will see the same tiled floor, repeated as the ceiling in the center of the composition, but at the same time it serves as the floor for the upper stage. At the top, the tile is repeated again, this time like a real ceiling.
Thus, you can safely combine geometry and painting, which is what many modern artists do,creating paintings in Escher's style and in his own style.Mathematical fine art flourishes today, with followers working in a variety of mediums, including sculpture, drawing on flat and three-dimensional surfaces, lithography and computer graphics. Let's see?
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| Where does this door lead? What can be installed in such a display case? |
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| Incredible Tower |
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| Unusual window |
This is the world of mathematical art!
Site pictures
municipal autonomous educational institution
Perevozsky municipal district
Nizhny Novgorod region
« High school No. 2 Perevoz"
ResearchJob
Bymathematics
"Geometry in Painting"
Completed:
student of grade 7 "A"
Shimina Daria
Supervisor:
math teacher
Klementieva M. N.
Transportation 2016
Content
Introduction. ...……………………………………………………………3
Main part. . . .………………………………………………………..............4-.13
1. The concept of geometric technique in painting.…………………… 4
2. Geometric painting. Drawing geometric bodies……….. 5
3.Geometric abstraction……………………………...………….....6
4. Cubism…………………………………………………………………………………. 7
5. Geometric painting by Richard Sarson …...................................8
6. Geometric painting by Simon Birch……………………………9
7.Paintings from straight lines Tadaomi Shibuya…………………………10
8.How to make a picture using the technique of geometric painting………..11-12
9. My paintings using the technique of geometric painting………………..13
Conclusion….………………………………………………………………………………14
INTRODUCTION
Creativity and mathematics are to the same extent the creation of beauty, like the creativity of a painter or poet - a set of ideas, like a set of colors and words, must have internal harmony.
Godfrey Hardy, English mathematician and philosopher.
I have a lot of interests. One of them is drawing. I really like to draw nature, still lifes, people. And I’m good at it! I recently looked up material for a lesson on the Internet. fine arts and came across paintings of geometric shapes. I was very interested in this technique, and I decided to learn as much as possible about it. Using geometric shapes, you can draw portrait sketches, landscapes, ornaments, etc. At school, of all educational subjects I prefer geometry (we started studying it only this academic year).
In my research work, I tried to reflect the technique of geometric painting and show how closely geometry is related to art.
So,target my research work: to study the technique of geometric paintingand use the learned material in practice.
Tasks:
Learn the technique of geometric painting;
draw a picture using the technique of geometric painting with your own hands.
Subject of research: mathematics.
Object of study: geometric drawing technique.
Main part. World of Figures
1. What is geometric painting technique.
Geometric painting technique is one of the early stages in the development of art Ancient Greece(IX-VIII centuries BC). This is manifested in the painting of utensils. The geometric style is characterized by diversity and clarity of patterns; their rigor emphasizes the construction of a constructive object. The ornament is arranged in stripes and is made from broken lines, crosses, and circles. In the later period of the development of Ancient Greece, an image of a human image from geometric shapes appeared.
2.Geometric painting. Drawing geometric solids
In drawing geometric figures, the main thing is to be able to imagine the object in volume and in different angles. It is better to start drawing simple geometric shapes or household objects.
Such exercises help to acquire the necessary skills in drawing and drawings of architectural structures.
The diagram presented below shows all the stages from composition (the correct arrangement of objects in space) to the construction and shading of objects.
Stages of work:
Mark the locations of objects on a sheet of paper with extreme points (composition).
Find the center of the figure, where the construction begins;
Marking with dots;
Outline the lines of the future part, and then finish drawing;
Drawing shadows (light, shadow, penumbra, falling shadow, highlight, reflex);
Create integrity to the drawing.
The latter can be done using clear lines.
The first sketches on the sheet should be pale, and then clearer lines should be added. An eraser should be used extremely rarely - when a mark is made too clearly, but erroneously and will interfere with the further construction of the drawing. It is worth erasing erroneous notes only when the correct ones are found.
If you want to learn how to draw beautifully, then you should try to draw from memory or from life. Before you draw a picture, you need to study and examine the object from all sides and make it with different provisions sketch drawings that will help better reflect the shape being drawn.
3.Geometric abstraction.
Geometric abstraction - a form of abstract art using geometric shapes combined into an abstract composition. Such a picture is created by combining a wide variety of geometric bodies, colored planes, broken and straight lines.
This art has been popular throughoutXXcentury. By using of this art You can share your thoughts and emotional state. It is not always possible to draw such a picture, since it is quite difficult to do. As practice shows, according to psychologists, drawing such pictures can help solve some human problems.
4. Cubism.
Cubism - the avant-garde movement in painting appeared at the beginningXXcentury. Many geometric shapes are used in this direction.
In 1912, a new direction was born in Cubism. Art critics began to call it “synthetic cubism.”
There are three branches of fine art: this direction, which reflect different aesthetic concepts. Each of which can exist as a separate independent movement: Cézanne Cubism (1907-1909), Analytical Cubism (1909-1912) and Synthetic Cubism.
Artists and geometry
5. Geometric painting by Richard Sarson
“I always want to play with shapes”
Adore..."
Richard Sarson is graphic artist. He graduated from King's College London, received a bachelor's degree, and later a master's degree. The work of Richard Sarson fascinates with its singularity. Everyone can see anything in them! To create such a picture you don’t need much. Compasses, paper and ballpoint pens will help turn your fantasies into reality.
The drawing of his technique consists of many circles intersecting with each other. As the author says, he creates such paintings at the call of his heart. All the artist’s works have clear lines, and the creator of the work himself considers what is most important is how his work will look as a whole, and not what it is made of. The artist's favorite figure is a circle. “It's incredible to draw a line and come back to where you started,” says Richard.
According to the artist, people think that a drawing made with ballpoint pens will be too rough. Therefore, in addition to drawings with a ballpoint pen, Richard tries three-dimensional drawings, creating them from threads stretched on pins. The amazing thing about such works is that anyone can wind the ball back and correct the failed part of the work, and when creating a drawing from clear lines, one awkward movement can ruin the whole picture.
Forms are what I live by, admits Richard Sarson. He knows so much about them that he senses the smell and taste, the sharpness and smoothness of lines with which he can convey what we cannot convey in words.
6. Geometric painting by Simon Birch
Everyone copes with a serious illness as best they can.
British artist Simon Birch received a terrible diagnosis in 2007. After that he began to create unusual paintings using geometric shapes in them.
Simon Birch was born in 1969 in Great Britain. Upon completion educational institution received a Master's degree in Fine Art from the Royal Melbourne Institute of Technology.
Simon paints on canvas armed with a brush and mastic. His paintings are made with strange strokes, as if he is trying to brush aside all his thoughts. Unusual artistic technique helped him maintain realism in his strange pictures. The works depict the image and emotions of a person using shape and color.
The collection of colorful geometric paintings is called "Laughing With a Mouth Full of Blood." The name is not very pleasant, but the life of an artist is not easy either. Probably, his paintings serve as a form of therapy and do not allow one to lose heart.
7.Paintings from straight lines
Tadaomi Shibuya
Straightforwardness is a character trait of honest and open people, as well as paintings created by Japanese resident Tadaomi Shibuya.
It is very difficult to find a perfectly straight line in the world. Tadaomi Shibuya's paintings, according to the artist, help create harmony and balance.
They say that a painting made using geometric shapes can help judge a person’s character. But this does not mean that Tadaomi Shibuya is strict and boring.
Some people note in the work of Tadaomi Shibuya not the original forms and creative ideas, but the primitiveness and angularity of the execution of the born idea.
8.How to make a picture using the technique of geometric painting.
After reading the theory and getting to know this technique better, I suddenly wanted to create such a miracle myself. But it turned out to be not so easy! To begin with, I decided to make a picture based on the sample. In order for everything to work out, it was necessary to calculate everything down to the smallest detail. I could just take it and draw it, but I need to make sure that everything is cleaned up. I had to calculate how many and what geometric shapes I would need to create a picture according to all the rules and proportions.
To create such wonderful picture I needed an ordinary sheet of paper (I have A4), geometric shapes (and there are a lot of them in the world... Everything around us is geometry), colored pencils (you can make such a picture from any material) and a little imagination (and I have a lot of it! ).
9.My paintings using the technique of geometric painting.
Conclusion
It seems to me that in the course of my research I have collected and studied quite a lot interesting material:
There is a style in painting that is closely related to geometry.
This style has been used since ancient times and was invented in Ancient Greece.
I recognized many artists who use this technique.
I created my paintings using the technique of geometric painting.
I did what I consider very useful, namely: I created my own painting using this wonderful technique. And most importantly, I fell in love with the subject of geometry even more! The material I collected can be used on various activities in geometry. I think that my classmates will be interested in learning about this fascinating and educational technique, which serves to develop geometric thinking, and also allows you to display inner world and the character of a person. For some it can even become a hobby!
