Segi empat sama ajaib

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Dalam matematik rekreasi, sebuah segi empat sama ajaib pada aturan n adalah suatu urutan bilangan n², biasanya integer berlainan, dalam sebuah segi empat sama, seperti mana yang bilangan n dalam semua barisan, semua column, dan kedua jumlah diagonal ke konstan sama.[1] Sebuah segiempat sama ajaib biasa mengandungi integer dari 1 ke n². Istilah "segiempat sama ajaib" juga kadang-kadang digunakan untuk merujukkan pada mana-mana jenis pelbagai segi empat sama kata.

Segiempat sama ajaib bermuncul untuk semua aturan n ≥ 1 kecuali n = 2, walaupun kesnya n = 1 adalah trivial—ia mengandungi suatu sel tunggal yang mengandungi nombor 1. Kes bukan-trivial terkecil, ditunjuk di bawah, adalah aturan 3.

Magicsquareexample.svg

Jumlah konstant dalam setiap row, column dan diagonal digelar konstan ajaib atau jumlah ajaib, M. Konstan ajaib pada segiempat sama ajaib terpulang hanya pada n dan mempunyai nilai

M(n) = \frac{n^3+n}{2}.

Untuk segiempat sama ajaib biasa pada aturan n = 3, 4, 5, …, konstant ajaibnya adalah:

15, 34, 65, 111, 175, 260, … (urutan A006003 pada OEIS).

Sejarah segiempat sama ajaib[sunting | sunting sumber]

Segiempat sama Lo Shu (3×3 segiempat sama)[sunting | sunting sumber]

Sasatera China melatar belakang seawal 650 SM menceritakan lagenda Lo Shu atau "scroll dari sungai Lo".[2] Di China silam, ada suatu banjir yang besar. Rakyatnya cuba untuk memberikan pengorbanan ke dewa sungai pada salah satu sungai banjir, sungai Lo, untuk menyenangkan kemarahannya. Kemudian, di situ bermuncul dari seekor kura-kura dengan suatu angka/corak pada kulitnya; ada titik-titik bulat bilangan yang diatur dalam suatu corak petak sembilan tiga seperti mana jumlah bilangan dalam setiap baris, lajur dan diagonal yang sama; 15. Nombor ini juga sama dengan bilangan hari setiap 24 kitaran tahun matahari China. Corak ini, dalam sesetengah cara, telah digunakan oleh orang-orang yang mengawal sungai itu.

4 9 2
3 5 7
8 1 6

The Lo Shu Square, as the magic square on the turtle shell is called, is the unique normal magic square of order three in which 1 is at the bottom and 2 is in the upper right corner. Every normal magic square of order three is obtained from the Lo Shu by rotation or reflection.

The Square of Lo Shu is also referred to as the Magic Square of Saturn or Cronos. Its numerical value is obtained from the workings of the I Ching when the Trigrams are placed in an order given in the first river map, the Ho Tu or Yellow River. The Ho Tu produces 4 squares of Hexagrams 8 x 8 in its outer values of 1 to 6, 2 to 7, 3 to 8, and 4 to 9, and these outer squares can then be symmetrically added together to give an inner central square of 5 to 10. The central values of the Ho Tu are those of the Lo Shu (so they work together), since in the total value of 15 x 2 (light and dark) is found the number of years in the cycle of equinoctial precession (12,960 x 2 = 25,920). The Ho Tu produces a total of 40 light and 40 dark numbers called the days and nights (the alternations of light and dark), and a total of 8 x 8 x 8 Hexagrams whose opposite symmetrical addition equals 8640, therefore each value of a square is called a season as it equals 2160. 8640 is the number of hours in a 360-day year, and 2160 years equals an aeon (12 aeons = 25,920 yrs).

To validate the values contained in the 2 river maps (Ho Tu and Lo Shu) the I Ching provides numbers of Heaven and Earth that are the 'Original Trigrams' (father and mother) from 1 to 10. Heaven or a Trigram with all unbroken lines (light lines - yang) have odd numbers 1,3,5,7,9, and Earth a Trigram with all broken lines have even numbers 2,4,6,8,10. If each of the Trigram's lines is given a value by multiplying the numbers of Heaven and Earth, then the value of each line in Heaven 1 would be 1 + 2 + 3 = 6, and its partner in the Ho Tu of Earth 6 would be 6 + 12 + 18 = 36, these 2 'Original Trigrams' thereby produce 6 more Trigrams (or children in all their combinations) -- and when the sequences of Trigrams are placed at right angles to each other they produce an 8 x 8 square of Hexagrams (or cubes) that each have 6 lines of values. From this simple point the complex structure of the maths evolves as a hexadecimal progression, and it is the hexagon that is the link to the turtle or tortoise shell. In Chinese texts of the I Ching the moon is symbolic of water (darkness) whose transformations or changes create the light or fire - the dark value 6 creates the light when its number is increased by 1. This same principle can be found in ancient calendars such as the Egyptian, as the 360 day year of 8640 hrs was divided by 72 to produce the 5 extra days or 120 hours on which the gods were born. It takes 72 years for the heavens to move 1 degree through its Precession.

Kepentingan kebudayaan pada segiempat sama ajaib[sunting | sunting sumber]

Magic squares have fascinated humanity throughout the ages, and have been around for over 4,000 years. They are found in a number of cultures, including Egypt and India, engraved on stone or metal and worn as talismans, the belief being that magic squares had astrological and divinatory qualities, their usage ensuring longevity and prevention of diseases.

The Kubera-Kolam is a floor painting used in India which is in the form of a magic square of order three. It is essentially the same as the Lo Shu Square, but with 19 added to each number, giving a magic constant of 72.

23 28 21
22 24 26
27 20 25

Arabia[sunting | sunting sumber]

Segiempat sama ajaib dikenali pada ahli matematik Arab, mungkin seawal abad ke-7, apabila orang Arab berhubung dengan budaya India atau Asia Selatan, dan mempelajari matematik dan astronomi India, termasuk aspek-aspek matematik berkombinasi. Ia juga telah dicadangkan bahawa gagasan tiba melalui China. Segiempat sama ajaib pertama pada aturan 5 dan 6 bermuncul di sebuah ensiklopedia dari Baghdad sekitar 983 M, Rasa'il Ihkwan al-Safa (Ensiklopedia Brethern of Purity); segiempat sama ajaib yang lebih mudah dikenali pada beberapa ahli matematik awal Arab.[2]

Ahli matematik Arab Ahmad al-Buni, yang bekerja pada segiempat sama ajaib sewaktu 1200 M, menganggap ciri-ciri mistikal pada mereka, walaupun tiada rinci pada ciri-ciri sepatutnya ini dikenali. Ada juga rujukan pada kegunaan segiempat sama ajaib pada pengiraan astrologi, suatu amalan yang kelihatan berasal dari orang Arab.[2]

India[sunting | sunting sumber]

Segiempat sama ajaib 3x3 telah digunakan sebagai sebahagian dari upacara di India sejak zaman vedic, dan berlanjut untuk digunakan untuk sampai ia tidak layak digunakan lagi. Suatu segiempat sama ajaib yang terkenal di India dapat dilihat di Khajuraho di kuil Jain Parshvanath. Ia bermula dari abad ke-10 [3].

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

Ini dirujukkan sebagai Chautisa Yantra, sejak setiap sub-segi empat sama berjumlah ke 34.

Eropah[sunting | sunting sumber]

Pada 1300, membangun pada karya Arab Al-Buni, sarjana Greek Byzantine Manuel Moschopoulos menulis suatu perjanjian matematik pada tajuk segiempat sama ajaib, mengetepikan mistikisme pewaris terdahulunya.[4] Moschopoulos is thought to be the first Westerner to have written on the subject. In the 1450s the Italian Luca Pacioli studied magic squares and collected a large number of examples.[2]

Pada sekitar 1510 Heinrich Cornelius Agrippa menulis De Occulta Philosophia, menulis pada Hermetik dan karya ajaib Marsilio Ficino dan Pico della Mirandola, dan dalam ia dia expounded pada magical virtues aturan 3 ke 9, setiapnya berkaitan dengan salam satu planet astrologi. Buku ini mempunyai pengaruh yang sangat besar sepanjang Eropah sehingga counter-reformation, dan segiempat sama ajaib Agrippa, kadang-kadang digelar Kamea, berlanjut digunakan dalam ajaib majlis moden dalam cara yang hampir sama dengan yang diprajelasnya terdahulunya.[2][5]

Zuhal=15
4 9 2
3 5 7
8 1 6
Musytari=34
4 14 15 1
9 7 6 12
5 11 10 8
16 2 3 13
Marikh=65
11 24 7 20 3
4 12 25 8 16
17 5 13 21 9
10 18 1 14 22
23 6 19 2 15
Sol=111
6 32 3 34 35 1
7 11 27 28 8 30
19 14 16 15 23 24
18 20 22 21 17 13
25 29 10 9 26 12
36 5 33 4 2 31
Venus=175
22 47 16 41 10 35 4
5 23 48 17 42 11 29
30 6 24 49 18 36 12
13 31 7 25 43 19 37
38 14 32 1 26 44 20
21 39 8 33 2 27 45
46 15 40 9 34 3 28
Utarid=260
8 58 59 5 4 62 63 1
49 15 14 52 53 11 10 56
41 23 22 44 45 19 18 48
32 34 35 29 28 38 39 25
40 26 27 37 36 30 31 33
17 47 46 20 21 43 42 24
9 55 54 12 13 51 50 16
64 2 3 61 60 6 7 57
Luna=369
37 78 29 70 21 62 13 54 5
6 38 79 30 71 22 63 14 46
47 7 39 80 31 72 23 55 15
16 48 8 40 81 32 64 24 56
57 17 49 9 41 73 33 65 25
26 58 18 50 1 42 74 34 66
67 27 59 10 51 2 43 75 35
36 68 19 60 11 52 3 44 76
77 28 69 20 61 12 53 4 45
Fail:Hagiel sigil derivation.svg
Asalnya sigil dari Hagiel, planetary intelligence Venus, diletakkan pada segiempat sama ajaib Venus. Setiap huruf Hebrew memberikan suatu nilai angka, memberikan verteks-verteks sigil.

Kegunaan yang terumum untuk Kamea-Kamea ini adalah untuk memberikan suatu corak yang mana untuk membinakan sigil pada mambang, malaikat atau jin; huruf-huruf nama entitinya ditukarkan ke bilangan, dan barisannya dikesan balik melalui corak yang bilangan berlanjutan ini membuat pada kamea. Dalam suatu konteks ajaib, istilah segiempat sama ajaib juga digunakan pada pelbagai segiempat sama kata atau segiempat sama nombor yang didapati pada grimoires, termasuk sesetengah yang tidak mengikut mana-mana corak yang terbuka, dan walaupun dengan yang berlainan nombor pada barisan dan column. They are generally intended for use as talismans. For instance the following squares are: The Sator square, one of the most famous magic squares found in a number of grimoires including the Key of Solomon; a square "to overcome envy", from The Book of Power;[6] and two squares from The Book of the Sacred Magic of Abramelin the Mage, the first to cause the illusion of a superb palace to appear, and the second to be worn on the head of a child during an angelic invocation:

S A T O R
A R E P O
T E N E T
O P E R A
R O T A S
6 66 848 938
8 11 544 839
1 11 383 839
2 73 774 447
H E S E B
E Q A L
S
E G
B
A D A M
D A R A
A R A D
M A D A
H O M O

Albrecht Dürer's magic square[sunting | sunting sumber]

Detail of Melencolia I

The order-4 magic square in Albrecht Dürer's engraving Melencolia I is believed to be the first seen in European art. It is very similar to Yang Hui's square, which was created in China about 250 years before Dürer's time. The sum 34 can be found in the rows, columns, diagonals, each of the quadrants, the center four squares, the corner squares, the four outer numbers clockwise from the corners (3+8+14+9) and likewise the four counter-clockwise (the locations of four queens in the two solutions of the 4 queens puzzle [1]), the two sets of four symmetrical numbers (2+8+9+15 and 3+5+12+14) and the sum of the middle two entries of the two outer columns and rows (e.g. 5+9+8+12), as well as several kite-shaped quartets, e.g. 3+5+11+15; the two numbers in the middle of the bottom row give the date of the engraving: 1514.

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

The Sagrada Família magic square[sunting | sunting sumber]

A magic square on the Sagrada Família church façade.

The Passion façade of the Sagrada Família church in Barcelona, designed by sculptor Josep Subirachs, features a 4×4 magic square:

The magic constant of the square is 33, the age of Jesus at the time of the Passion. Structurally, it is very similar to the Melancholia magic square, but it has had the numbers in four of the cells reduced by 1.

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

While having the same pattern of summation, this is not a normal magic square as above, as two numbers (10 and 14) are duplicated and two (12 and 16) are absent, failing the 1→n² rule.


Lihat juga[sunting | sunting sumber]

Nota[sunting | sunting sumber]

  1. "Magic Square" by Onkar Singh, The Wolfram Demonstrations Project.
  2. 2.0 2.1 2.2 2.3 2.4 Swaney, Mark. History of Magic Squares.
  3. Magic Squares and Cubes By William Symes Andrews, 1908, Open court publish company
  4. Manuel Moschopoulos - Mathematics and the Liberal Arts
  5. Drury, Nevill (1992). Dictionary of Mysticism and the Esoteric Traditions. Bridport, Dorset: Prism Press. ISBN 1-85327-075-X. 
  6. "The Book of Power: Cabbalistic Secrets of Master Aptolcater, Mage of Adrianople", transl. 1724. In Shah, Idries (1957). The Secret Lore of Magic. London: Frederick Muller Ltd. 

Rujukan[sunting | sunting sumber]

Wikisource-logo.svg
Wikisource mempunyai sumber asli 1911 Encyclopædia Britannica teks berkaitan kepada: Magic Square.

Bacaan lanjut[sunting | sunting sumber]

  • Charney, Noah The Art Thief Atria (2007), a novel with a key plot point involving a magic square.
  • McCranie, Judson (November 1988). "Magic Squares of All Orders". Mathematics Teacher: 674–78. 
  • King, J. R. (1963). "Magic Square Numbers".