Square Durera solution. Magic squares of Khajuraho, Durera and Golden proportion. "Year of Saturn" and the symbolic meaning of "lion" horoscope

Chapter No. 5.

Magic squares

We call them magic squares or planetary squares. Or seals, cameras, tables. Like many other magical instruments, they are known in different names in various systems, but no matter how none called them, they are dating hundreds or even thousands of years. The earliest of the recorded, is the square 3 on 3, 3rd order, which is now known as Saturn's square, and in China called Lo Shu.

Turtle and Saturn Square

His discovery is attributed to the Great Emperor Yu, "Mudrom King," and refers to about 3000 BC. e. . Once, walking along the river Lo, the influx of the yellow river, the emperor found a turtle. According to the mysterious numerical patterns on her back, the emperor Yu realized that this is a magical creation and took her with him to the palace. The turtle became the object of worship, and the sacred patterns on its shell caused delight of the court scientists.

Fig. 17. Pattern but turtle shell

As it is narrated in this story, the Chinese courtyard admired the unusual clusters of the specks on the turtle shell: in black groups, the number of these specks was odd, and in white - even (Fig. 17). Numerical sequence itself got a name Lo Shu. - Writes of the river Lo. It is found in Chinese mathematical texts, starting from about 2200 BC. e. Much later, in 1275 n. E., Mathematics Jan Hui described in detail the magic squares in the work entitled: "The resumption of ancient mathematical methods to explain the strange qualities of numbers." Jan Hui defended his book with a statement that he relied on the work of earlier mathematicians. He did not explain how most squares received 3, 4 and 5th order, but led a simple formula to build a square Lo Shu. "From scratch" (Fig. 18).

Fig. eighteen. Building Lo Shu.

Write numbers from 1 to 9 in three rows and turn the one written to the right so that 1 turns out to be from above, and 9 - below (a). Change places 1 and 9 (c), and 3 and 7 (s). Then lower 9 so that it will be between 4 and 2 in the upper row (D); Adjust to each other 3, 5 and 7 in the second row and place 1 between 8 and 6 in the lower row (E). Voila!

In order to use the resulting configuration as a magic square, it should be added to the coordinate grid lines by which you can create cells for numbers. And then we will appear the same Lo Shu. Eight external speckling groups on the turtle shell became eight tri grams "I-Jing" (Fig. 19). Lo Shu. Also close to practice feng Shui, nine positions of which are known as "nine wandering stars", trigram bogo Scheme 3 by 3, defining attributes and "healing" of space feng Shui . I found a little of what could clarify all the details of the mentioned story with the sacred turtle, but no doubt - "I-Jing", Saturn's square, On Shu. and feng Shui - Are direct descendants of this ancient animal.

Fig. nineteen. And Jing / Lo Shu

Games with mathematics square Saturn

This is the smallest square and a good object to demonstrate basic information related to functions and terminology. This square is a storeroom of mathematical tricks, the more funny, what you know how easy it is to create it.

Fig. 20. Saturn Saturn

First, being the square of the 3rd order, it has three cells in length, three cells in height and contains numbers from 1 to 9. The largest number in the square corresponds to the number of cells available in it (Fig. 20, 21) .

The sum of numbers in each line (a) is 15, as in each column (B).

The amount of numbers diagonally, both from the top left to the lower right angle, and from the upper right to the lower left (C), is also equal to 15.

Fig. 27.Mathematics Svadrat Saturn

If you fold together all numbers in a square - 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 - sum will be 45.

Divide 45 at the order of the square - 3 - and you will receive 15, the result equal to the sum of any row, column and diagonal.

But that's not all.

Fold a couple of numbers opposite each other. In the middle row it is 3 + 7. In the middle column it is 9 + 1. Angle diagonal pairs - 4 + 6 and 2 + 8. Each pair in sum gives 10.

Now, pay attention to the number 5 in the center of the square, the only number remaining without a pair. Double it, folding with himself: 5 + 5. The amount is equal to 10, which corresponds to pairs located opposite each other. It will also be true for a larger square of the odd order: find the cage in the zero point, double the number in it, and then determine the pairs that give the same amount.

Let's go back to history. Lo Shu. Migrated from China or something like this appeared somewhere else in other places independently?

Fig. 22. Saturn socket

Fig. 23. Fortune telling

Both. Maya was known to this square, as well as North Africans and prehistoric French. Ancient Babylonians entered the eight-pointed star Ishtar in this square to use it to determine directions (Fig. 22). One modern source offers an option to use the square of the Speakers: Draw the grid and the numbers in it with white chalk on black paper. Then place the crystal ball into the center, where the number 5 is usually located (Fig. 23). This version of the square is known as the "Egyptian figure", so, perhaps, the Chinese turtle had the great-grandparents who lived during the times of Pharaoh. Meanwhile, the squares of the 4th order were known in India from about 550 years. e. When the Varachamihir wrote the text about predictions called Brichatsamhita. Some of the 4th order offered by the Varachamihihirahihirachihira contained the encrypted recipes of incense, while others were called the Kacupute, literally "Turtle shell", which again indicates communication with the turtle Lo Shu.

Squares of the 5th and 6th orders were known in Islamic countries by 983 N. e. In the text "Cabs al-Anvar", written by the supervision, about 1384 N. e., Couples seven planets and squares are listed, as shown in Figure 24, in the order, repeated in 1498 of Pachet in "De Viribus" and Cornelius Agrippa in De Occulta Philosophia ("Occult Philosophy") in 1531. This sequence is known as a chaldean order, and it matches the size of each square with the corresponding distance from each planet to the Earth: the further, the less cells, the closer, the cells are larger.

Fig. 24. Chaldean order

From time immemorial, people knew that the sun was closer to Earth than Mars (except for the Great confrontations), Jupiter and Saturn. Watching the movement of the moon, Mercury and Venus ancient astronomers found that sometimes each of these planets took place between the Earth and the Sun, and Mercury and Venus were periodically enveloped him. With the moon, this never happened that prompted our ancestors to come to the logical conclusion that it was the nearest neighbor land. Conversely, Mars, Jupiter and Saturn never turned out between us and our star, on the contrary, describing the circle, they periodically passed behind the Sun, which led to the conviction that these planets were removed from the ground for a longer distance than the sun. Erroneous? Yes, but hardly insanely. Hence the "Chaldean order", which still has a strong effect on the use of magical numbers in the West. The tradition of the long, logic in it is a bit, so make your own conclusions about whether it is worth using the chaldean order and how.

There are no squares of the 2nd order: a four-cell square does not demonstrate anything surprising when adding numbers 1, 2, 3 and 4. Agrippa gave this circumstance Original explanation: the number 2 was damned due to the actions of the first two people, Adam and Eve, which made The square of the 2nd order is impossible. Other his "proof" was not inferior to the first: he believed that four elements were land, air, fire and water corresponding to here numbers from 1 to 4 are inadequate. Agrippa description And the 1st order square is the only cell containing the number 1, which he identified with God. Perhaps this is a strange justification and caused the inaction of the Inquisition towards the author's authority?

Magic and Magic Squares

Now it's time to distinguish: there are two types of squares that can be conventionally called "magic" and "magical".

Magic Squares are a kind of "entertainment mathematics", something like crosswords for fans of this science. They are called "magic", since they allow you to juggle by numbers through the most incredible combinations. Although their earliest versions had metaphysical attack, for most of the historical or modern magic squares there are no mystical associations. They are simply not intended for these purposes, as well as crosswords can not be a guiding thread for the Chronicles of Akashi.

Squares of the second type, real magic Squares similar to the reerfish of its mathematical component, but, in addition, they have very ancient roots and a long history of the magical and occult use. Here is the magic squares and talk further.

Dürer's Square (Almost Square Jupiter)

Among the many people enchanted by magic / magical squares were the artist Albrecht Dürer (1471-1528) and American politician Benjamin Franklin (1706-1790). Franklin, which served in the late 1730s, long before his political take-off, secretary in the Pennsylvanian Assembly, boredom for the sake of squares. Although both probably enjoyed these puzzles, Franklin (who was Mason) and Durer, of course, were also interested in metaphysical aspects.

The square of Jupiter appears on the Engraving of Dürer "Melancholy" - or almost appears, as Durer allowed himself some of the liberations here (Fig. 25, 26, 27). Why use the square of Jupiter, if melancholy metaphysically corresponds to the planet Saturn? Perhaps healing for the sake of the sake of Jupiter (he, as in the word "cheerful") had to counteract the "Saturnal" gaze?

The painting "Melancholy" is filled with occult associations over which art historians still fight: a complex geometric body, a staircase in seven steps, a compass (showing 51 ° 25? - The value used to create a seven-pointed star or a circle separation to 7) and other details (Fig. 25). It is known that Dureru liked to create visual puzzles to experience and amusing their friends with their help. Probably, "Melancholy" stands in the same row.

Its solution to deploy the Square of Jupiter 180 ° (Fig. 26, 27) may have been due to the specifics of the printing process. Artists who worked in the technique of artists, to obtain a normal writing from an etched on the image plate, had to create their own compositions in a mirror form. This means that any text and numbers were to be originally written on the contrary. Perhaps working on the placement of numbers on the engraving board, Durer wanted to perpetuate the date of creation of the picture? Thus, by turning the traditional square, he received the desired 1514 year, spelled in the lower row. There is another numerical combination, which Durer, of course, knew: each line of the square of Jupiter gives 34 when adding, and in 1514 Albrecht Dürer turned thirty-four years.

Fig. 25 "Melancholy" Dürer

Fig. 26. Square Dürera

Fig. 27. Square Jupiter

We use Dürer's Square to study the possibilities of some squares - magical or magical. The 4th order square has sixteen cells containing numbers from 1 to 16. The principal moment here is the location of each number.

Games with Mathematics Square Jupiter

Figure 28 shows the mathematics of the square of Jupiter.

A, B and C. Rows, columns and diagonals, as in Saturn's square. Each of these combinations in the amount gives 34.

D. The same happens with four angles: 16 + 13 + 4 + 1 \u003d 34, and

E. With four central cells: 10 + 11 + 6 + 7 \u003d 34.

F. And even with pairs of internal numbers located along the exterior edges:

3 + 2 + 15 + 14 (along the upper and lower edge) \u003d 34 5 + 9 + 8 + 12 (along the left and right edge) \u003d 34

Fig. 28. Math Square Jupiter

So, here are already fourteen different ways of addition to 34, possible in this square, but there are others.

G, N, I, J, K and L show another fourteen ways to achieve 34 by adding specific cells in the square of Jupiter, and these methods may even be even more. If a, in and С work in all planetary squares, many of these options are inherent in this square. There are tricks and other squares. I leave them the right to detect them, if, of course, this logic game captures your imagination.

If you are craving for more detailed and in-depth mathematical analysis, then refer to the appropriate literature presented at the end of the book in the bibliography.

Now let's go back to mysticism.

Planetary squares

Here they are shown in ascending order from a smaller to more. It is important to understand that the force of exposure to using these squares does not depend on the thoughtless copying of their appearance; It is concluded in the act of their creation from scratch, in a consistent entry of one number after another. When you draw your own square, use the numbering sequence for meditation. Enume each number in the square in turn - 1, 2, 3, etc. - and not just rocker their row by side. Tip: First, write figures with a pencil, and then, by jumping them with a handle in the appropriate order, - from 1 and further - focus on them all your attention.

Some general comments:

First: If you multipliate the number in the central cell of any square of the odd order on the number of the order itself, then you will receive a total amount of numbers in any row / column of the square. For example, Mars has a square of the 5th order, and the central number is 13, hence 5 x 13 \u003d 65.

Second: If the order of the square is divided into 3, then the total amount of the square is simplified to the number 9. In all other cases, to the number 10 (up to 1).

Third: For all squares of odd order - Saturn, Mars, Venus and Moon - first should be determined by the center. The number 1 is directly under the center of the square, and its largest number is directly above the center. The central square itself will contain the number of "Center": 1-2-3-4-5-6-7-8-9. If you determine the initial, central and endpoint, then the diagram of these squares of the odd order will be revealed by itself.

Squares of even order - Jupiter, Sun and Mercury - begin with numbers 1 in the upper right corner and end in the largest number in the lower left corner. In addition, their sequences are cunning, in any case, in my opinion. Good luck in the detection of their schemes!

Saturn square

Fig. 29. Saturn Saturn

Understanding past experience;

Development of personal discipline;

Proper use of borders and restrictions;

Comprehension of karma.

For more information, refer to the Substitutes section in Chapter No. 4.

Square layout: 3 to 3 mesh 3, 3rd order square. The numbers contained: from 1 to 9.

Total amount of each row, column and diagonal: 15. Total amount of the whole square: 45.

Dividing the total amount by the number of order: 45; 3 \u003d 15.

Square Jupiter

Used to enhance / improve:

Success in court cases;

Expansion of the case;

Good luck, success (and their own feeling of joy?);

Establishing partnerships, unions;

Spiritual growth.

For more information, refer to the "Thursday" section in Chapter No. 4.

Fig. 30. Square Laying Square: Grid 4 to 4, 4th order square.

Total amount of each line, column and diagonal: 34. Total amount of the entire square: 136.

Dividing the total amount by the number of order: 136: 4 \u003d 34.

Square Marsa

Fig. 37. Mars Square

Used to enhance / improve:

Decisions taken;

Physical strength, energy;

Personal courage and willpower;

Control of temperament, passions;

Blessing of vehicles and mechanisms;

Technical abilities;

Commercial cooking.

For more information, refer to the Tuesday section in Chapter No. 4.

Square layout: 5 per 5 mesh, 5th order square. The numbers contained: from 1 to 25.

The total amount of each row, column and diagonal: 65. The total amount of the whole square: 325.

Dividing the total amount by the number of order: 325: 5 \u003d 65.

Square Sun.

Fig. 32. Square Sun.

Self-confidence;

Health, vitality;

Leader's abilities;

Understanding of the goal;

Self-realization;

Success in new projects.

For more information, refer to the materials of the Sunday section in Chapter No. 4.

Square layout: Grid 6 on b, 6th order square. The numbers are: from 1 to 36.

111 Total Square: 666.

666: 6 = 111.

Square Venus

Used to enhance / improve:

Understanding of harmony and beauty;

Ability to friendship and love;

Openness for joy, playfulness and romance;

Love and relationships;

Sensuality;

Home cooking.

For more information, refer to the materials of the Friday section in Chapter No. 4.

Square layout: Grid 7 to 7, square 7th order. The numbers are: from 1 to 49.

Total amount of each row, column and diagonal: 175. Fig. 33. Venus Square Total Square: 1225.

Dividing the total amount by the number of order: 1225: 7 = 175.

Mercury Square

Fig. 34. Mercury Square

Used to enhance / improve:

Clarity of thinking and perception;

Clarity and effectiveness of communication;

Concentration, especially during study;

Intellectual aspirations, knowledge acquisition ability;

Contacts on the spiritual plan;

Security and timeliness of travel.

For more information, refer to the Materials section "Wednesday" in chapter number 4.

Square layout: Grid 8 to 8, square 8th order. The numbers are: from 1 to 64.

Total amount of each row, column and diagonal: 260. Total Square: 2080.

Dividing the total amount by the number of order: 2080: 8 = 260.

Square Moon

Fig. 35. Square Moon

Used to enhance / improve:

Intuition and instinct;

Fertility (defined) and creative abilities;

Emotional mood;

Knowledge of the psyche;

All garden and farmer undertakings;

Travel safety by water.

For more information, refer to the Monday section in Chapter No. 4.

Square layout: Grid 9 to 9, 9th order square.

Total amount of each row, column and diagonal: 369.

Total Square: 3321.

Dividing the total amount by the number of order:3321: 9 = 369.

Use of planetary squares

Select the planet, the traditional topics of which matches your requests. For example, to improve the concentration when preparing for the exam, it is logical to stop your choice on Mercury. The opening of a new case is usually in the competence of the Sun, while the strengthening of the turnover in the already existing business is best to contact Jupiter. All related to restrictions must be addressed to Saturn. If you wish to bless and protect your new vehicle, then here is the best choice - Mars.

One of my girlfriend recently bought a diesel engine, which she redested so that now it can be refilled by the used vegetable oil. Bravo! As a meal to gift the maintenance of the machine and its passengers, you can choose a keyword or a short phrase: "Bless Mercedes" or "Protect my car" or, perhaps, "ABC-987", (fictional) registration number. In this case, the following is suitable: "Rides are good, remain unharmed."

Then we are looking for numbers corresponding to the letters in the selected phrase. Our first task is to use numbers from 1 to 9 as a key for the alphabet (Fig. 36). Many of us are probably already familiar with this key, since it is used in numerology and in simple encryption.

Fig. 36 Alphabetical key 1-9

When working with a scheme 1-9, our phrase will look like in Figure 37.

Fig. 37. "Rides are good, remain unharmed", 1-9

If you use Saturn's square or you need letters from q to Z in the square of Jupiter, then you should use encryption 1-9 shown in Figure 36.

However, we also use the square of Mars, the square of the 5th order, with twenty-five individual cells. Since our phrase does not include the Z, twenty-sixth letter, we can, instead of using one number from code 1-9 for three different letters, assign a separate number for each letter. In order to take advantage of unique numbers, resort to key 1-26 shown in Figure 38.

Fig. 38. Alphabetical key 1-26.

Note: If you simplify each two-digit number, this cipher will correspond to cipher 1-9 in Figure 36.

Now the phrase will look like in Figure 39.

Fig. 39. "Rides are good, be in preservation", 1-26

Hematria

Viewing boningly different words, we sometimes meet amazing numeric parallels. For example, Pop (lion) is expressed in numbers as 3-9-6-5; cheetah (Cheetah) as 3-8-5-5-2-1-8; A. tiger. (Tiger) as 2-9-7_5-9, if working with a set of prime numbers (see Fig. 36). Giving the amounts 23, 32 and 32, they are all simplified to 5.

If you find such coincidences intriguing, hematria can be your favorite occupation. Enjoyed with the above example, but much more complex hematria is based on twenty-two letters of the Hebrew alphabet and five more letters of the same alphabet repeating in several excellent forms when they fulfill the role of expirations. Total twenty seven. Each letter is assigned a numeric value, but in contrast to the alphabets of type A-Z already seen by us, these values \u200b\u200boften reach much more significant values \u200b\u200b- up to three-digit - so when the word is written in a similar way, the amount may be impressive. Other differences: in the hematosis amounts are not simplified to unambiguous numbers; Each letter also has a deep esoteric meaning; In addition, hematria is based on Hebrew, and many of us work in our native language, as the spell must be phonetic.

"Kabbalah calls the Hebrew alphabet" Letters of Angels * 1 ". So Madame Blavatskaya wrote in his book "The Secret Doctrine", because the use of Hemat Terriate's Hematodes is one of the ways to study their divine associations. Hematria is deep, ancient, complex and subtle teaching is the most common determination of what it is engaged.

According to the hematodium, the lion, which we mentioned earlier, has the following numerical value: 30 + 10 + 70 + 50 \u003d 160. Where as a cheetah ( cheetah) It looks much more solid: 60 ( cH) + 8 (long e) + 300 (t) + 1 (a) \u003d 369.

For a deeper interpretation, we turn to the symbolism of the hematodium (Fig. 40). From her point of view, the name of our old acquaintance of a lion (Pop) It is composed of letters having the following metaphorical values: "Wheel incentive", that is, "Prut", "Hand", "Eye" and "Fish". Title letter lameoften translated as "Wheel incentive", in a broader sense can mean something like a "motivator". Iodine Or "Hand", perhaps personifies the desire to embody its ideas in physical reality, that is, literally, fighting fate. AYN. or "Eye" implies a vision and rejection, a look and understanding. Finally, NUN or "Fish" should be talking about non-microgenic for people of the environment and the need for adaptation for survival in a hostile world. As we see, through this deep interpretation, the concept of "lion" acquires much more complex symbolic meaning.

Fig. 40. Hematrichesky Code

Cruel "King of Jungle"? Of course, the characteristics discussed above give a lot of food for reflection and can serve as a guide for human Acts in the conditions of "Jungle" of modern life.

The present practices from the hematodia will be used not only by the complex of metaphorical values \u200b\u200bof letters, laying the name of the object of the study, but also to study the numerical component, in this case the word "lion". What other words give the amount of 160? Is it possible to expand our understanding of the test word with them?

Babylonians also used a hematic system. At King Sargon II (approx. 722-705 BC) was a wall of 16,283 cubet (1 qubit \u003d 0.48 m), built on the basis of the numeric value of its name. This grandiose example can inspire us to independent use of hematodes, perhaps not for the construction of massive walls, but for much more modest measurements and calculations based on the hematic value of their own name or qualities that we would like to get.

Fig. 41. Greek hematodic code

So, if I did a talisman that contributes to the manifestation of previously mentioned lionic qualities, I would decorated it with a number 160.

Greek letters also have numeric values, and there is own tradition of their study (Fig. 41). For example, the Gnostic Deity Abraxas has a numeric value of 365 (1 + 2 + 100 + 1 + 60 + 1 + 200), equal to the number of days a year.

In traditional practice, scientists are used by hematium to sacred texts, looking for words with equal numeric values.

In front of the recorder, found this numerical resonance once, the countless number of directions for the study can be opened. Result? The secret harmony, hidden in the web of the relationship between meaning and completely invisible to the random reader of the same text.

Breathe patience: It can easily be work for decades.

No matter, you use the code from Figure 36 or from Figure 38, in any case, let's start. We are going to "enter" the phrase "Rides well, remain unharmed" into the square of Mars (Fig. 31), using the numbers that we just picked up. From this point on, you must have a tracing, ruler and eraser. To begin with, tie the "test drive" of its phrase on the trash imposed on top of the magic square. Thus, you can determine where your numbers are located, and avoid errors in the square painted by the square, over the creation of which you worked for a long time. To begin with, refer to Figure 42. How can you make sure your phrase graphic has, rather, a pleasant design, and does not look a chaotic ball. Edit and reproduce your drawing - that's why you need tracing and eraser. Draw the line from hand or using a ruler if you want them to be completely straight. When they stretch on the draft, remove the cushion and, focusing on your aspirations, spend the lines on the square.

Fig. 42. The beginning of the phrase on Mars

Fig. 43. The end of the phrase on Mars

Figure 42. The beginning of the phrase in Mars: these are the first four "letters" (R-U-N W), denoted by numbers 18 (with an asterisk), 21, 14 and 23. Nothing compulsory, but remember, short phrases work better than verbal jets. Figure 43 shows how the result should look like.

Figure 43. End of the phrase in Mars: the graphic shape of the desired phrase, the beginning and the end of which are marked with asterisks.

As a rule, squares are created and used, corresponding to the planet at that time in an astrologically favorable position relative to other celestial bodies. You can simply work on a day associated with a concrete planet. If you want, outline the circle and determine the parties of the world before starting work.

Experiment with various colors and types of markers, color pencils, exotic paper, or anything, which can spur your fantasy. When your pattern is ready, it can be cut or embroidered, draw it on an aquatic surface or above the ceremonial subject. Features are limitless. Fix the use of squares. These records will become your reference book or recipe book for future use. If something works especially well, it makes sense to reproduce this procedure again and again, and in the event of the ineffectiveness of something you can make the necessary adjustments.

What else can you do with magic squares

Determine in each square of its "magic line". This is done as follows: Finding the number 1, spend a line from it to number 2, then to the number 3 and so on, according to the increasing, to the final number. As a result, excellent geometric patterns, which can serve and quite practical purposes will appear before your eyes. Garden layout? Business logo? Tattoo? Entertaining on the selected map for vacation?

Follow the line connecting the date and time of your birth. This action should contribute to the activation of positive talents and the potential in stock. Use the square of the planet that manages your sign, or the planet, the position of which you find the most favorable for your witchcraft, or do it with each square, and then compare the results. In this way, it is possible to detect the order of lines, which will later become the basis of a personal magical symbol.

Lunny labyrinth

In search of any more magical wonders of the squares, I turned to the wonderful book of Clifford A. Pikikover "Zen magic squares, circles and stars". Using the magic square of the 9th order (not the square of the moon), the peak detected an interesting geometric pattern, which turned out when unwinding all odd numbers. I copied this idea, imposing a tracing on top of another 9th order, of the present square of the moon (Fig. 35). As I used another square, I got a completely different pattern. Shocked by unexpected recognition, I realized that I see the "seed diagram" in front of him - the basis for the semi-coat labyrinth (see Fig. 44, 45).

The variants of this labyrinth are discovered worldwide, on space from Crete to the south-west of America. I first learned about the "seed diagram" at the labyrinth seminar, where we created a big labyrinth on the sandy shore of the mountain lake. It consists of vertical + (plus sign) in the center, four corner L-shaped forms and four angular points. These components are secretly present in the usual square of the moon and become visible only when unwinding all odd numbers.

Leaving enough space on the sides, draw a large seed chart on the sandy beach or draw a little on a sheet of paper, and then proceed to creating a labyrinth. By connecting the top of the main vertical line from the top of the right-to-top L (as shown in Fig. 45), continue to create arches, the lines of the line from left to right, as shown in Figure 46 and 47. In general, if you started from the line, then finish the point and vice versa . Note that the U-shaped "angles" of the reversal, forming the labyrinth loops, are also the outer corners of the moon square.

Fig. 44. The square of the moon showing the seed pattern of the labyrinth in the form of lines of odd numbers

Fig. 45. Labyrinth with drawn first arch

Fig. 46. Labyrinth with the second and third arches

Fig. 47. The labyrinth to which the fourth and fifth arches have been added

Fig. 48. The labyrinth, to which the arches are added 6, 7 and 8. Now we have a full 7-circular maze

Like the moon itself, growing and decreasing on the right left, moves on the sky from left to right, and you, being in a maze, should move both the sategorically and anti-melanche. Try to color the labyrinth made on paper with pencils of all the colors of the rainbow, replacing one color on the other in the place of rounding the corners.

Several unusual additional comments:

First: In the square of the moon of 81 cells and, respectively, 81 numbers, and the mass of the moon itself is 1/81 from the mass of the Earth.

Following: The land moves in space at a speed of 28 miles per hour; Moon - at a speed of 2268 miles per hour. This means that the moon moves 81 times faster than the Earth.

Last thing: As carved on the statues of Maya in Palenk "81 Moon is 2392 days." Divide 2392 to 81 and you will receive 29.53 - a number equal to the number of days in the lunar cycle, according to the calculations of modern scientists.

Based on the theoretical analysis of the Pandiaagonal squares of 4 × 4, their features of the "structure" are shown: the invariants of the structure of the Pandiagonal squares of 4 × 4 are pairs of numbers equal in the amount of one of the two numbers of Fibonacci - 13 or 21. It is revealed that any option of many six digits of this and him Similar pantagonal squares of 4 × 4, forming a continuous symmetric configuration, is equal to an integer in an integer number - 51. A geometric figure of "Cube in Cuba" has been constructed, which has the properties of the "gold symmetry" of Pandiagonal squares of 4 × 4. The properties of the "Gold Symmetry" have all the numbers of the diagonals of the cube (two numbers form in one case - in the amount of the number 13, in the other - 21), and all planes having 4 angle (numbers) both internal and external squares of the geometric shape are formed in The sum of Fibonacci is 34.

Introduction

Based on the theoretical analysis of Khajuraho's squares, Dürer and similar squares of 4 × 4, the features of their "structure" were identified: the invariants of the structure of Pandiagonal squares 4 × 4 are pairs of numbers equal in the amount of one of the two Fibonacci numbers - 13 or 21.

The magic square is the square table N × n, filled with N 2 in various numbers in such a way that the amount of numbers in each row, each column and on both diagonals is the same. The earliest unique magic square of 4 × 4 was found in the inscriptions of the XI century in the Indian city of Khajuraho. Square 4 × 4, depicted on the Engraving of Albrecht Dürer "Melancholia", is considered the earliest in European art (1514). The sum of the number of Dürer's square on any horizontal, vertical and diagonal is 34. This amount is also found in all angular squares of 2 × 2, in a central square, in a square of angular cells, in squares built by "horse's move" (2 + 12 + 15 +5 and 3 + 8 + 14 + 9), in the vertices of rectangles parallel to the diagonals (2 + 8 + 15 + 9 and 3 + 12 + 14 + 5), in rectangles formed by medium cell pairs on opposite sides (3 + 2 + 15 + 14 and 5 + 8 + 9 + 12). Most additional symmetries are associated with the fact that the sum of any two centrally symmetrically located numbers is 17.

There are 48 pantagonal squares of 4 × 4 with an accuracy of turns and reflections. If we take into account even symmetry with regard to the toric parallel transports, then only 3 significantly different squares remain (Figure 2).

Main part

My "Structure" of Pandagonal Squares 4 × 4 is analyzed and the invariant parts of their structure (Figure 3) are revealed. The invariants of the structure of the Pandiaagonal squares of 4 × 4 are pairs of numbers equal in the amount of one of the two numbers of fibonacci - 13 or 21. Various variants of symmetric combination of these numeric pairs form a plurality of pandiagonal squares of 4 × 4.

Dürer's square (and to him like pandiaagonal squares 4 × 4) have a golden proportion symmetry. For example, in Figure 4, the variants of symmetries are shown in red and blue squares from the sum of the red components of squares in possible positions (4 or 2, during rotation in different directions) is 51. Thus, the sum of all numbers of the square is 136, Of which 85 - blue, 51 - red. 136/85 \u003d 1.6; 85/51 \u003d 1.667.

On the basis of Dürer's square, we have built a geometric figure of "Cube in Cuba", which has the properties of symmetry of Pandiagonal squares of 4 × 4 (Figure 5). Such a "transformation" became possible when the vertical columns of the Dürer square numbers are arranged at a certain angle, thus forming a cube in Cuba. In this case, the properties of the "gold symmetry" have all the numbers of the cube diagonals (two numbers form in one case - in the amount of the number 13, in the other - 21), and all planes having 4 angle (numbers) both internal and external squares of the built figure Form the number of Fibonacci in the sum - 34.

Conclusion

1. Based on the theoretical analysis of the Pandiagonal squares of 4 × 4, their features of the "structure" are shown: the invariants of the structure of the Pandiagonal squares of 4 × 4 are pairs of numbers equal in the amount of one of the two numbers of fibonacci - 13 or 21.
2. It was revealed that any variant of the set of six digits of the square of Dürer and the similar Pandiagonal squares of 4 × 4, forming a continuous symmetrical configuration, is equal to an amount of an integer - 51.
3. The geometric figure of the "Cube in Cuba", which has the properties of the Golden Symmetry of Pandiagonal Squares of 4 × 4. The properties of the "Gold Symmetry" have all the numbers of the diagonals of the cube (two numbers form in one case - in the amount of the number 13, in the other - 21), and all planes having 4 angle (numbers) both internal and external squares of the geometric shape are formed in The sum of Fibonacci is 34.

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Magic Square Albrecht Dürer. Magic square 4? 4, depicted on the Engraving of Albrecht Dürer "Melancholia I", is considered the earliest in European art. Two average numbers in the lower row indicate the date of creation date (1514). The amount of numbers on any horizontal, vertical and diagonal is 34. This amount is also found in all angular squares 2? 2, in a central square (10 + 11 + 6 + 7), in a square of angular cells (16 + 13 + 4 + 1 ), in squares constructed by the "horse's run" (2 + 8 + 9 + 15 and 3 + 5 + 12 + 14), in rectangles formed by vapor medium cells on opposite sides (3 + 2 + 15 + 14 and 5 + 8 + 9 + 12).

Slide 13. From the presentation "Square in life". The size of the archive with a presentation of 388 KB.

Geometry class 8

Summary of other presentations

"Determining the correct polygons" - solving problems. Any correct polygon is convex. Oral work. Built figure. Parquets from the right polygons. Formula for calculating the angle of the correct N-carbon. What is equal to the sum of the external angles of the correct N-Corner. What is each of the corners of the correct polygon. Creative task. Polygons of different species. Convex polygon. Tasks lesson. The plane without lumen can be covered with correct triangles.

"Types of rectangles" - diagonal. Perpendicular. Rectangle. Square is a parallelogram. Find all unknown square angle. Exercises. Exercises on planimetry on finished drawings. Exercises on planimetry. Reverse statement. Sign. Parallelogram. Roma side. Sign of rhombus. Special property of rectangle. Parallelogram AVD. Little side of the rectangle. Height. Property rhombus. Prove. Find the perimeter of the square.

"Building tangent to the circle" - Circle and direct have one common point. Total points. Chord. Tangent to circumference. Reiteration. Circle and straight. Diameter. Theorem on segments of tangents. Mutual arrangement of direct and circumference. Decision. Circle.

"Calculation of the polygon area" is a polygon composed of several polygons. Test. Polygon area. Work in notebooks. AVD-parallelogram. Properties of squares. Oral solution tasks. As you understand. What are the basic properties of the square you know. Square area is equal to the square of it. Middle sides of the rhombus. In the rectangle diagonal are equal. Objectives lesson. Units of measuring areas. Work on ready-made drawings.

"Tasks for signs of the similarity of triangles" - determining the height of the subject on the puddle. Individual map. Solving tasks on ready-made drawings. Measuring the height of large objects. Similarity of triangles. Shadow from a stick. Determination of the height of the object in the mirror. Solving practical tasks. Falez method. Name like triangles. Determination of the height of the pyramid. Independent work. Determination of the height of the subject. Gymnastics for eyes. The motto of the lesson.

"Vector concept" - vector length. Vectors. Direction of vectors. Equality of vectors. Collinear vectors. Mark in the drawing. Equal trapezium. Historical reference. Two nonzero vector. A task. Two nonzero collinear vector. Zero vector. Geometric vector concept. What is a vector. Setting the vector from this point. Parallelogram.

Durer Albrecht (1471-1528), German painter, draftsman, engraver, art theorist.

he studied at his father.
Father, jewelry master, wanted to attract her son to work in a jewelry workshop, but Albrecht did not express any desire. He loved and stretched to painting.

Nuremberg artist Wolgemut Durer mastered not only painting, but also engraving on wood.
The works of the artist Martin Schongauer, with whom he never met, Albrecht went a lot and went everywhere, studied, studied ...

But the time has come when Albrecht had to marry. And then he chose himself from the old and respected Nuremberg family Agnes Frey, his father's daughter. Marriage with Agnessee was childless, and the spouses were different in character, which made the family not very happy.

Nevertheless, he opened his case and in his workshop created a significant part of his engravings.
In Venice, rumors had rumors about his love for both sexes ... Perhaps Dürer practiced same-sex love with a heart friend, an expert of antique literature Pirkheimer.

Long, curled hot way, hair, dance lessons, fear of sifilis in Venice and buying medicines against this disease in the Netherlands, elegant clothes, petty vanity in everything that concerns its beauty and appearance, melancholy, narcissism and exhibitionism, complex of Christ, a childless marriage, subordination to his wife, tender friendship with a libertine Pirkheimer, whom he himself in the October letter of 1506 was jokingly offritable -

all this is combined in a durer with a tender care about the mother and brothers, with many years of hard work, frequent complaints about poverty, illness, unhappiness, allegedly pursued it.

Be god faithful!
Gain health
And eternal life in heaven
Like the Major Virgin Mary.
Tells you Albrecht Durer -
Swim in sin
Until the last day of the post,
And shut up the devil to the mouth
Overpower unclean.
May Lord will help you Jesus Christ
Together in good!
Think more often about death
About the burial of your bodies.
It comes to the soul
Distracts from evil
And the sinful world
From the oppression of the flesh
And the science of the devil ...

When 1498, Coberger published"Apocalypse",

Durer created 15 xylography, which brought him European fame. The recognition with the Venetian school had a strong influence on the picturesque manner of the artist.
In Venice, the artist fulfilled by the German merchants "Holiday wreaths from roses"and then there were other suggestions, paintings that left the indelible impression of the multi-faceted colors and plots.

Emperor himself Maximilian I.

before the art of Albrecht Dürer.
Dürer adhered to the views of the "iconoborets, however, in the later works of A. Dyurra, some researchers find sympathy to Protestantism.

At the end of his life, Durer worked a lot like a painter, during this period they created the deepest works in which acquaintance with the Netherlands art.

One of the most important paintings of recent years - diptych "Four Apostles"The artist was present to the city council in 1526.

In the Netherlands, Durer became a victim of an unknown disease (perhaps malaria), from whose attacks suffered to the end of life.

Albrech made up the so-called magic square, depicted on one of his most perfect engravings -"Melancholia". Dürera's merit It is that he managed to enter into the distinguished square of the number from 1 to 16, that sum 34 was obtained not only when adding numbers vertically, horizontally and diagonally, but in all four quarters, in a central four-corner and even when adding four angular cells. Also, Durer managed to enter into a table of the year of creating engraving ""(1514).

There are three famous engravings on the tree in the works of Albrecht Dürer, with the image of the south and northern hemispheres of the starry sky and the Eastern Hemisphere of the Earth, which became the first in history printed by typographic way.

In 1494, the book of Sebastian Brant is published under the symbolic name"Ship Fools" (Das Narrenschiff Oder Das Schiff Von Narragonia).
During the obligatory for the workshop, the wanderings on Rhine, Durer fulfilled several machine-made engravings in the spirit of late Gothic, illustrations to the "ship of fools" S. Branta,

on which the fleet crosses the sea. Circle is full of fools. Here laugh at the fools of sailors and the ships of the empire.

It is believed that in addition to A. Dyurra, several charter ponder-cutters worked on the project ... Picture "Ship Fools"- Posted and famous artistJerome Bosch.

Figure Durera "Ship Fools"

Upstairs on the right fools on the cart, the ship is floating near the sea surrounded by boats, and all fools on the ship and boats.
Many illustrations for "ship fools", as commentators celebrate, are little related to the content of the book itself.
As it turns out, the Bookt's book itself was elected only as a reason, an excuse, to promulgate a large number of engravings (hundred sixteen) on the topic of the "ship of fools".

There are u Albrecht Dürer and such a picture like "Holiday of all saints" (Altar Landauer) 1511. Museum of the history of the Arts, Vienna. This picture also brought a huge fame to the artist.

We better remember the geniuses of whims,
That the secret of their creation is stored:
Immortal smile Mona Lisa
And nude maha waiting look ...
Was Durer Jeweler and the poet
Secrets of words and numbers learned.
He traveled a lot, while
Great engravings created.
Scientist was. After all, it is for the first time
Picked wave of their ideas,
Created science - anthropometry,
What studies the skull of people.

Versatile, like any genius!
In everything I was looking for harmony law.
Projects of fortresses and fortifications,
And the first star satin, also he!
Great Artist Work, -
Some strange characters are full, -
As if you want to tell us something
Open Wishes the secret to us.
The whole world of art is 5 centuries,
In an effort to hear the true motive:
What is the winged creation
Sits from the Duma sad labored?

The history of art silent,
Who is he.
And everyone sees his own there.
And in it and the human little,
And the angelic we do not know.
He was considered "MUSOE" once,
But where such "muse" you will find.
After all, our winged is clearly bearded.
Well, it looks like Durera.
And hair and long clothes
Worn before often men.
What can we say about the former modes?
We are with you these modes are not experts.

In fortune telling and disputes there is little sense -
Centuries between us laid ...
I am a master call him because
With it, the rules, ticks and saw.
Wouldn't they see in the root of the wonder?
What then, do not understand, the dispute?
For "virgin" and for "muses" of this
More cosmetic set!
Saw with the plane - Dürer handles!
What are we arguing about many centuries?
He master on engravings is the best
And for engravings needed a board.

On the head - like the crown, -
From the Herbs of Wonderful, Wizard Wreath ...
Understand the viewer:
And here is the ball? Sleeping near the legs?
The winged "man" sits in sorrow.
What is sad? What does he remember?
Nothing around himself notices -
The wizard's eyes are aspirated!
Around a set of things is quite strange,
Attracts our inquisitive look:
Stepladder, near the multi-faceted stone,
Hourglass, Scales, Square ...

I'll tell you about the square, -
It is associated with flesh, spirit and fire.
The meaning of the secret numbers will open you too,
Captured on the square volume.
View over square bell will find out.
On the day of the vessel we heard his ringing ...
Angry Angel writes something.
We see: he is very displeased.

Lighted by the Siagan Sun gave ...
But I am addicted to me simplicity:
After all, people with impatience expected
In that "terrible" the coming of Christ.
In the sky unfamiliar comet.
Everything brighter became, everything is visible!
It seemed to people: "The terrible sign is, -
It remains to live the earth not many days! "

Prophecy everyone remembers John,
What will be bright light before the end.
And panic covers countries
And God turned to God.
The artist was sad: "Made a little
And before God in the wines! "
His soul sad and suffered
That still sinful feelings are full!

In those years, the idea first sounded,
That people are the universe of the crown,
And, in God's kingdom, taking his beginning,
Here, on Earth, the man himself! ...
Can be the wizard sorry for the first time
About those creatures, which gave the light
Where in the faces of Jesus and Mary
There is almost completely divine.

Suddenly realized that he would not see eternity,
He who lives, earthly earthly.
What does not raise humanity in the souls,
Divinity revounds from himself.
It seemed to him that he didn't do everything,
Moorful waves did not carry there ...
The average of the "trample" of the soul was sitting,
Magnarounding up from the ground.

The soul was looking for grace not in God,
And in the satiety, in weakness and warm ...
Angry Angel led results.
What a master made on earth.
And in the starry sky melted comet.
Magic Square trouble broadcast ...
Now you will understand why all this.
In the soul gave birth to Durer sadness.

On the illustration of the famous mysterious engravy of Albrecht Dürer "Melancholia-1"

From the Internet:
"In 1514, Durer followed a bright comet that appeared on the sky. A lot in the image of engraving is associated with this comet and symbolism of the planet Saturn, which is a patronage of melancholic. It was believed that the God of Saturn is older than other gods, and only he is inherent in the highest intellect, and only melancholics are available to the joy of discoveries ...

In the center of the composition we see a woman with wings and in a wreath, personifying logic - this is a Muz Dürer. Motionally sitting on the porch, it is immersed in melancholic thoughtfulness and sadness: a woman at least has wings, but cannot penetrate the curtain of the mystery of the universe. All that is around happens - passes without her participation. It inhibits her and brings melancholic mood. A sullen boy with wings, side sitting on the millstone, something thoughtfully writes on the wax plank. This is a push, symbolizing the harbinger of the Angelic Spirit.
Arravings are scattered around the heroes of engraving and construction tools. And at the feet, curling into the ball, sleeping a dog, symbolizing the melancholic temperament. "

/ End quote /

Those who are looking for in the winning creature, on the Durer engraving, the female began persistently do not want to notice the beard and mustache on the face of "Music" and a joiner tool, scattered near his legs. They see only long hair and a dress that looks from our point of view, female.
But take a look at the Self-portrait of Dürer 1498. Here you have a long, curled hair masters, here's his dress that resembles us "female".
So, on the Engraving "Melancholy" Durer portrayed himself. And this is not my discovery at all. For the first time, this was said in the last century Soviet and Russian art historian, historian of culture. Honored Art Worker of the RSFSR, Paola Volkov.
And the wings are God's gift to all creative people. This is known to everyone.
Now the tool that the master worked, making his amazing engravings on the tree and the skinny homeless dog, whom Albrecht sheltered is quite appropriate - this is his everyday work environment.
Do not forget that Durer was an excellent mathematician:
The ruler near his legs, the circulation in hand, the ball for calculating the parameters of the lines of lines is also often necessary for it.
I think that everything is clear with the winged being.
But, if this is still Albrecht Durer, what is he so sad on his own engraving?
Unlike Polala, I believe that the master depicted not entirely, but his soul. And it is she in a state of melancholy.
In this state of the soul, the master fell not random, but as a result of a proportion of a number of circumstances:
On April 11, 1490, Albrecht Durer left his native Nuremberg for several years and went on a trip to Europe. In early 1492, he arrived in Colmar / now France /, where he hoped to meet with the famous engraver Martin Schongauer. By the time of the arrival of Albrecht, Schongauer was already dead, but the young man was welcomed by the brothers Martin. They even allowed Albrecht to work in his workshop.
When the time came to say goodbye, one of the brothers extended Albrecht a small square copper plate with figures applied to her:
-The memory!
"What is it?" Durer was surprised.
- Formula of Harmony, - Drew the answer.-Martin ordered to give someone who wants
Excele all the famous world of painters.
Durer, like many great, passionately dreamed of it.
If Albrecht asked what his biggest hobby had after his painting, he probably would call mathematics.
Dierer had an irrevocable desire to measure and imprinted in the form of mathematical formulas everything he saw. He invented its own devices for measuring the curves of lines and solved the most complicated measuring tasks using an ordinary line and a circulation. He even wrote several methodological manuals for working with measuring instruments.
From the Internet:
"Durer earned wide fame as a mathematician, first of all, a geometer (at that time, German scientists were almost not dealt with geometric tasks), who studied the theory of perspectives, constructing geometric shapes and font development. The results obtained were highly valued in the writings of the following centuries, and in the second half of the XIX century their scientific analysis was made. According to Johann Lambert, the later works on the theory of perspectives did not reach Durer's heights. In the history of mathematics, Durer is put on a par with known scientists of his time and is considered one of the founders of the theory of curves and descriptive geometry "
/ End quote /

The sign is interested. In the chaos scattered on her numbers, he immediately saw harmony.

16- 3- 2- 13
5- 10- 11- 8
9- 6- 7- 12
4 - 15- 14- 1

Harmony - in philosophy, the coordination of heterogeneous elements, in the aesthetics, the coherence of the whole, born from the union of differentness in the quality of entities.
The master even remembered the lines of some ancient poet dedicated to harmony:

"And the summer heat, and winter coolness,
And the return of birds in your own land ...
All on time and as much as necessary, -
Life on earth can not be different!
In harmony, all sacred will be born,
The greatness of spiritual full,
Gaze, attracting the wonderful beauty,
She draws us to the Lord! ... "

The master began with the simplest arithmetic action - addition of numbers and was amazed by the result achieved:
The addition of 4-digits in each row horizontally, vertically, diagonally, in the corners and even in sectors were given the same amount - 34!

Now Dürer did not have more doubt that it deals with numerology.

From the Internet:

The main provisions of the current version of Western Numerology were developed in the VI century BC. e. Ancient Greek philosopher and mathematician Pythagoras, who united the mathematical systems of the Arabs, Druids, Phoenicians and Egyptians with the sciences of human nature [the special importance of Numerology received in Kabbalah, where its variety is known called hematria. Kabbalists expanded Pythagorov concept using numbers in magic squares for various purposes. With the discovery in the XIX century, scientists of the nature of light, electricity and magnetism, the vintage occult values \u200b\u200battributed to the numbers began to attribute both energy vibrations. Modern numerology [gives preference to a simplified numerical and alphabetic code based on the theories of Pythagora.
One of these magic squares was now the case of Albrecht Durer.
But how can this square help him in his writings.?
And then it as if painted:
34 is 3 + 4 \u003d 7, that is, the formula for converting the solid ground / 4 / consciousness / 3 / in the hardness of heaven / 7 /.
It was and remains the main task of any creativity, the main activity of any person endowed with God talent.
For the task of any master is to spiritive the power of his talent around the world. Do not give him to become soulless!
But there was the era of Renaissance, with her humanism and the reformation of the church. Humanism looked at the task of the painter otherwise: make a person in art by the main person. Give the whole Divine feature of the human. Not a person must return to himself the lost similarity of God, but God must decline to human existence, to become a clear person. Practically, everything should be aimed at acquired by man on earth of happiness.
No matter how sad, but it was the humanism of the revival epoch that became a fertile soil for liberalism, who called on people to abandon all restrictions, including moral. During its existence, the liberalism of the spiritual revival of Europe turned first into the liberalism of the revolutionary restructuring of the world, and after the modern liberalism of permissiveness.
Dureru The ideas of humanism and the reformation were much clearer and closer than foggy messages from eternity.
Albrecht could not but interest the numbers in the base of the square:

1514
Something had to happen in 22 years? Maybe the end of the world, about which in the Middle Ages, as in our time, did not stop talking? Durer and believed and did not believe in the easiest coming of the Lord.
1514 came. Waiting for trouble did not leave the masters all this time.
In 1514, a bright comet appeared in the sky of Europe. Mae died hot beloved Mother Albrecht. Durer was in a state of melancholy for a long time. The clock was sitting among your working tools scattered on the floor, looked at one point and waited for the end of the world.
The wings of a gentle creature are a bright indication that we have a wizard wizard, but the soul is not fixed to God, but tied to everyday terrestrial concerns. In a word -gumanism.
The end of the world did not come.
The painter finally managed to get together with the Spirit and depicted the state of his teaching soul on Engraving "Melancholia 1"
Why "1"?
It is necessary to turn to numerology again:

From the Internet:
From the unit begins great things. For the figures, one no barriers, because she is accompanied by success. Studies are forced to open new horizons in any areas that are later becoming a benchmark for everyone.
End of quotes /

So it is worth assumed that Durer decided to start some kind of new stage of his creativity, make a jump to those transcendental vertices that he dreamed of.
Nothing significant in 1514 did not happen:
Albrecht went to the creative business trip to Portugal. There he created for his emperor Maximilian of the first engraving brought there from India of the wonderful beast - Rhino! Maximilian, at the sight of a rhino, came to indescribable delight and appointed a lifetime rent from the pocket of Nureberg orders.
A bright comet from the European Neighbor soon disappeared, and Durer continued to portray the Virgin and the Lord similar on the ordinary Burgers of Germany, going along the path of reformation and liberalism.
However, in the Pantheon of the Great, he still entered, although he failed to transcend those in whose works more divine than the earth.
Is it comparable at least one Madonna Durer, in which so little heaven and so much earthly, with Madonna Rafael?
And it is necessary to pray before the face of the Lord, but not in front of the magic square.
…...........................................................