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The fourth page results logic puzzles made of figures which are received as a result of partitioning the diheptagonal geometrical network. The information on geometrical figures and geometry of the fine arts can be considered in landscape design and garden architecture as landscapes and architectural spaces are perceived harmoniously if follow proportions of correct geometrical figures. |
Any geometrical figures can be considered as objects of logic puzzles as any
geometrical figure is a code of mathematical laws which can be a source of logic
reflections.
The diheptagonal network of lines is remarkable as is the code of geometrical
laws conformable to a human nature and have harmonious influence on human
perception.
Therefore the logic puzzles, constructed from a heptagon and a diheptagon, are
not only logic but also harmonious, in which it is possible to find reflection
not only about laws of geometry but also reflection about laws of harmony.
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The chart shows a logic puzzle which task consists that
it is necessary to cut the diheptagonal network of lines and to isolate
such figures of which the figure shown in the chart can be made. Namely it is necessary to isolate five figures from the diheptagonal network and to make of them the shown figure. The following chart shows a similar puzzle in which it is necessary to combine the shown figure by means of 5 isolated figures. The isolated figures are specified in the first puzzle, and in the second puzzle the isolated figures are latent. The third chart shows a puzzle in which the task consists that it is necessary to isolate seven figures from the diheptagonal network and to combine the shown figure. In each puzzle all figures should be isolated from one diheptagonal network of lines. The shown puzzles are examples which can be samples for many other similar puzzles. Pay attention, if will solve the shown puzzles, that some lines of the isolated figures in puzzles have inexact concurrences that is connected to geometrical properties of the diheptagon, but these inexact concurrences of geometrical figures conclude fundamental property of harmony which key principle consists in inexact concurrences. These inexact concurrences can be named "golden" as they are magnitudes which are necessary for calculating at construction of absolute proportions of world around. From the mathematical point of view for creation of puzzles it is better to use the dihexagonal (twelve-sided) network of lines in which there are no discrepancies and all lines coincide correctly, but nevertheless the diheptagonal (fourteen-sided) network much more interesting and is more salutary for human perception. If to compare geometrical puzzles which are constructed on the basis of a hexagon or a heptagon then puzzles on the basis of a hexagon seem correct, and puzzles on the basis of a heptagon seem beautiful. For example, if to construct abstract compositions by means of figures which are isolated from the dihexagonal and diheptagonal networks of lines then the preference will be given an abstract composition which is constructed from figures of the diheptagonal network. I do not consider other polygons, such as a dipentagon or dioctagon, as research of various polygons is a special theme which can be a source not only entertaining puzzles, but also interesting abstract painting. I bring to your attention the set of geometrical polygons which can be used for the decision of the shown puzzles and for creation of sketches of abstract pictures. Images of polygons are submitted in the PDF file which can be taken, then it is possible to print images and to use for the decision of various geometrical tasks.
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At the end of this page I offer to look three
abstract compositions which are constructed by means of figures isolated from
the diheptagonal network:
(the first and third abstract compositions have a casual arrangement of figures, and in
the second abstract composition the isolated figures are focused concerning lines of the diheptagonal network)
And also I bring to your attention two abstract compositions in which proportional human figures are ciphered:
Besides I give an abstract self-portrait which is constructed according to geometrical
principles and according to position of astrological planets which was in the
morning February, 17, 1967.
The portrait is astrological as it is constructed by means of lines which
connect planets within the framework of an astrological circle:
The principle of an astrological portrait consists that within the framework of
an astrological circle the diheptagonal network of lines is combined with the
symbol of the sun and harmonizes constellations (ratio) of planets in the
horoscope of the person. Then the figure of the person is calculated according
to lines of the diheptagonal network and also according to lines which form
constellations of planets, that is shown in the picture on the right.
Then I have isolated proportional geometrical figures from lines of the
diheptagonal network, which transformed and have compared to symbols of planets,
and also have compared to points where lines of the human figure and the
diheptagonal network are crossed.
It is simply possible to tell that the astrological portrait is constructed
according to proportional structure of the diheptagon, but coordinated with
position of planets in the horoscope of the person.
Hands, legs and the head of a human figure can be compared to any points and
lines which form a portrait, but top of the head, or any top of hands or legs,
should be compared to position of the sun as the symbol of the sun is connected
to structure of the diheptagon.
Colors of geometrical figures not casual also depend on astrological
characteristics of planets which form a multi-colored combination connected with
human perception of planetary radiations.
The information on color ratio of planets I shall try to publish in the
future on pages of the present gallery of phantom images, and up to that time
necessary data on color can be found in other sources of the information.
If it is necessary, the astrological or geometrical portrait of any person can
be drawn.
The shown abstract compositions are computer images, but also can be drawn by
the valid art means and methods.
Besides in portraits it is possible to cipher many other astrological and
geometrical symbolical ratio which specify individual psychological
characteristics of the person.
In each portrait it is possible to connect some figures which are variants of an
individual human body.
Or in portraits figures of several people can be connected.
It is similarly possible to organize an arrangement of rocks of the Japanese garden according to lines of the diheptagonal network and according to position of astrological planets in the horoscope.
I hope that the geometrical concepts submitted on this page of gallery of
phantom images, will allow to see structures and forms of world around in a new
fashion, and will allow to understand abstract art symbols of phantom images.
Though actually phantom images more significant also have many-sided sense about
which I shall try to tell in the future.
The following page results geometrical measurements of concrete human bodies which differ from the ideal human body entered in lines of the perfect diheptagonal network.
