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## Mathematical concepts of phantom images.Linear patterns of geometrical figures and drawings.

Linear patterns in geometrical space of circles.

Multi-colored lines form volumetric pictures of phantom visual images.

Geometrical pattern in space of the transformed circle.

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Pattern crossings of lines in geometrical figures of squares.

If to divide a visible plane of space by means of shapes of geometrical figures and outlines of polygons, and also if to connect corners of polygons and points on sides of figures then crossings of connecting lines form geometrical linear patterns or otherwise to tell phantom networks.
A variety of phantom networks and patterns can be indefinitely big and infinite greater, as if to change mathematical parameters of figures and polygons then shapes of patterns changes, As a result it is possible to see that in geometrical crossings of lines there are universal mathematical ratios and global laws according to which patterns vary.
If we delimit and divide geometrical figure’s contour and link points of division with line interceptions geometrical patterns can appear as the result. These patterns’ view varies miscellaneously when division parameters are changed.

At the left it is possible to see linear patterns constructed within the circles. Namely each circle is divided into three parts and each third is divided into 9, 8, 7, 6 parts. Points of divisions are connected by lines which in space of circles form various patterns or otherwise to tell different phantom images of geometrical figures depending on numbers of points which divide third parts of each circle.
Below at the left it is possible to see the circle in which lines of the pattern have different colors. And also t is possible to see the linear pattern which is formed in space of geometrical phantom net of the transformed circle.
Besides it is possible to see geometrical figures of two squares in which various pattern crossings of lines are caused by divisions of sides on eight or seven parts. Namely the first square has equal number of points on each side, and in the second square three sides are divided on eight and one side into seven parts.

The shown images display only some linear patterns, but actually in view of mathematics there is a much greater variety of similar phantom networks.
Surprising pattern crossings of lines exist in spatial fields of hexagons or heptagons.
In essence patterns can be constructed in contours of different geometrical figures and polygons, and also you can calculate spatial figures with special geometrical parameters which are necessary for formation of very diverse phantom images.
These patterns may be created not only in usual geometrical contours but in unusual contours too. The side divisions, which are necessary for some special networks’ creation, may be received as result of special mathematical calculations.

The shown linear patterns or otherwise to tell geometrical phantom networks are computer art graphics, namely images are drawn by means of software programs for visual work with digital pictures on screens of computers.
Unfortunately known computer programs are intended for decisions of general geometrical problems, and consequently cannot be used for digital drawing or designing the shown patterns. Namely known graphic software in interfaces of personal computers allow to draw linear patterns by means of complex mathematical algorithms and operations with graphic tools, but it is necessary to have a special computer program which has simple digital tooling functions.
Such program software for personal or tablet computers in essence can be considered as a computer game which allows to design desirable geometrical constructions and to carry out necessary graphic transformations for drawing every possible linear patterns and phantom networks. Or as a special graphic software for visual design and drawing of digital images, namely for creation of multimedia pictures by artists and designers.
Then any interested user of a personal or tablet computer can use this software program as graphic tools for computer design and digital art, or as a game to draw linear patterns as logic puzzles.
And also users of personal and tablet computers by means of this program software can investigate aspects of geometry and mathematical laws which can be seen in lines of patterns.
To view true variety of patterns it is necessary some program equipped with special set of tools, which allow, with help of some key operations, to construct desired geometrical composition, to use color and to accomplish necessary conversions. And in this case everybody can research multi-sided interesting aspects of this type geometry.

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Linear patterns and phantom networks in seventeen online galleries of this website are drawn with color felt-tip pens and paint markers on paper sheets unlike computer graphic images on this page. Namely patterns in galleries are drawn on a paper by means of writing tools as a basis of real graphic drawings and geometrical pictures in a context of fine arts, but however I do not deny an opportunity of computer visualization and creation of digital pictures by means of multimedia technologies.

I consider the shown graphic drawings in galleries as phantom fantastic networks or linear textures, or otherwise to tell patterns of geometrical figures in which there are phantoms of imagination and pictures of mental images.
If eyesight looks a little further or nearer of an image plane then lines of patterns seem as abstract distances and intervals which are compressed or stretched as if refract in multivariate space of volumetric perspectives of an illusory reality where figures of linear outlines form hypothetical drawings and pictures of imagination. Also it seems that visible illusions of visual reality form contours of supernatural shapes and phantoms of fantastic phenomenal essences which change and turn to diverse objects of psychophysical perception in geometrical structures of the shown patterns, namely figures of linear outlines have transformations by eyesight and conversion by a brain to art realities of human imagination.
Pattern lines create something like refractions in multidimensional space and it seems us that supernatural forms bleed through an image plane and obtain specific outlines.

The visible validity is a flat projection of the world because perspectives of space are perceived by eyesight, and neural cells in a retina of eyes have a flat surface. It is possible to tell that eyes perceive perspectives of volumetric space, but our brain builds the image of visible validity on a flat surface of retina by means of squares, circles, triangles or other geometrical figures the same as any drawings or plans, or geodetic maps are under construction in human ideas. And consciousness in visual zone of a brain carry out geometrical transformations of visual sensations, and people create volumetric pictures of the world.
Our consciousness executes geometrical transformations of phose visual sensations and constructs environmental world image which consist of squares, circles, triangles and other patterns of geometrical figures.
Therefore the shown linear patterns of two-dimensional geometrical figures turn to volumetric pictures of phantom visual images.

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Our eyesight and consciousness perceives elementary figures of geometry, but surrounding space includes latent geometrical contents which we do not realize as it is not obvious for eyes. Namely there are more complex figures which are inaccessible to human understanding, but however if to understand laws of geometry it is possible to realize psychophysical principles according to which our brain calculates mental images of the global validity. And also it is possible to see multivariate linear patterns which exist in complex geometry of visible world.
Any space structure includes closed geometrical content which is not apparent but determines many laws of human perception.

Global validity is constructed according to mathematical principles of symmetry and proportions. We are surrounded with grandiose geometry of space in which harmony of lines and systematic outlines of diverse world objects have expansion, but we can see and realize geometry of surrounding shapes if eyesight divides visible measurements of the world, and then contours of global validity become obvious.

Many concepts of space have been investigated and formulated, or designated by images in pictures of artists or painters who marked in ideas a strict geometrical essence of spatial perspectives, and who saw exact mathematical proportions in structure of the created world. Strange geometrical objects in art compositions of Kandinski or abstract geometrical figures in pictures of Malevich, and also spatial forms of many other abstract artists are attempts to see geometry of the world, but nobody can measure all world finally.
Geometrical patterns and phantom figures of images in art galleries on pages of this website too are research of spatial aspects of world geometry. Namely graphic drawings or pictures in galleries are art images which designate mathematical principles of spatial perspectives and proportions of a foreseeable reality, and including designate geometrical proportions of a human body.
World culture comprehends the geometrical image of environmental reality miscellaneously. This global image of the world is a subject of investigation for many works or pieces of the fine arts.

It is possible to argue on visual harmony or disharmony, and also about proportions and disproportions in abstract pictures or graphic drawings, or other art works, but in essence any rules of the fine arts and criteria of beauty of geometrical forms or shapes, and in particular principles of beauty in pictorial figures of human bodies are identical to universal laws of world space which have exact parameters. Namely global spatial harmonies can be measured and calculated, and also can be proved in view of scientific geometry.
Including there can be disputable proportions of art shapes in the shown phantom images, but however visual compositions of drawings are constructed according to correct geometrical figures in space of harmonious linear patterns, that is criterion of beauty and integral essence of phantom images in galleries of this website.
The main “Phantom pictures” concept is the fact that geometrical patterns are integrated part of this drawings’ essence, in which environmental world image is reflected.

Content of this page in the future will be increased, as the theme about patterns of geometrical figures in a context of the fine arts is very extensive.
If you have own notions about geometry of space and geometrical patterns of world around, and your ideas can be really represented, then your representations can be shown in a context of online of gallery on pages of this website, or I can give links to websites of your images in the global network of Internet.

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