Sunday, 7 December 2014

A MAGIC SQUARE TRIBUTE TO DÜRER, 500 YEARS AFTER MELENCOLIA I

Albrecht Dürer's "Melencolia I" contains numerous mathematical and philosophical symbols amongst which we can see a magic square, a polyhedron and a sphere. These 3 symbols are partially interpreted in an earlier article.

1514, the date of the engraving, appears in the lower line of the fourth-order magic square. 500 years later, is it possible to create a fourth-order magic torus tribute to Dürer?

In order to write 2014 we can subtract 1 from all the numbers (1 to 16), and thus obtain the zero that cannot be found in normal fourth-order magic squares. After the subtraction, we can use a flattened viewpoint of the basic magic torus T4.177 (group number T4.05.4.12) that displays not only the Frénicle squares n°99 and 618, but also 14 semi-magic squares. The numbers 0 to 15 are repeated beyond the edges of the square in order to represent the toroidal continuity. As the new figures contain the numbers from 0 to (N²-1) instead of 1 to N², their magic constant is 30 instead of 34.

A magic torus tribute to Dürer - copyright

Detail of the "first state" version of the magic square that appears on Dürer's "Melencolia I"
(National Gallery of Victoria)
Perhaps Dürer hesitated when engraving the left hand side of his square: The number 5 appears to replace a former number 6. It should also be noted that the print detailed in the first version above is, according to The National Gallery of Victoria is "the very rare first state." The reversed number 9 was later corrected in the revised version below. On page 12 of the 1961 Aukland City Art Gallery catalogue "Albrecht DÜRER forty engravings and woodcuts," it is written that "There are two states - the first where the nine on the numerical table is reversed, the second where the nine is corrected." The contemplation of the two versions of Dürer's magic square may well remind the reader of the duality of the “De Umbrarum Regis Novum Portis” engravings in Roman Polanski's film "The Ninth Gate." David Fritz Finkelstein has written that "Dürer's engraving MELENCOLIA I was circulated in two versions not previously distinguished. Besides their conspicuous early Renaissance scientific instruments and tools, they contain numerous apparently unreported concealments whose detection reveals heresies expressed in the work." Frickelstein believes that the second version of the engraving is the one in which the number 9 is reversed, and suggests that the numerals 9, 6, and 0 are curled serpents. To find out more please consult Finkelstein's "Melencolia I.1*"
Detail of the revised version of the magic square that appears on Dürer's "Melencolia I"
(Wikimedia Commons)
Dürer's square, (Frénicle n° 175), is one of 8 semi-pandiagonal squares displayed by the semi-pandiagonal torus T4.077 (torus type n° T4.02.2.03). The seven other semi-pandiagonal squares that are displayed by the same "Dürer torus" are the Frénilce index n° 27, 233, 360, 421, 583, 803, and 850. The semi-pandiagonal "Dürer torus" T4.077 also displays 8 semi-magic squares, and is completely covered by 8 sub-magic squares. More details of the fourth-order magic tori can be found in earlier articles such as:
http://carresmagiques.blogspot.fr/2012/01/4-to-power-of-4-4th-order-magic-tori.html
http://carresmagiques.blogspot.fr/2012/10/table-of-fourth-order-magic-tori.html

A magic torus tribute to Dürer - copyright

Miguel Angel Amela has very kindly authorised me to publish an extraordinary pandiagonal square he created for this anniversary. Again, this is not a normal 4x4 square; as the magic constant of all the columns, rows, and diagonals is not 34, but 1514 - the year that Dürer engraved Melencolia I! Miguel has also managed to write 2014 at the centre of the bottom row, in just the same place that Dürer used to date his square.
Miguel Amela's tribute to Dürer : 2014 - 1514 = 500 years. - copyright

The following figure shows a water retention diagram of Miguel Amela's square that was created by Craig Knecht.

A water retention diagram of Miguel Angel Amela's tribute to Dürer - diagram by Craig Knecht - copyright
Harry White and Craig Knecht have created a giant 196 x 196 water retention magic square that spells the dates 1514 and 2014. Craig Knecht has also devised another 14 x 14 square that honours the 500th anniversary of Durer's famous magic square in Melencolia I. The animation of the latter square can be found at the following link.

Lorenzo D. Sisican, Jr.has kindly authorised me to publish two non-normal, but semi-pandiagonal squares that he has created: The first of these squares has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I).  
Lorenzo D. Sisican Jr.'s first tribute to Dürer, 500 years after Melencolia 1 - copyright
The date of the creation of Melencolia I, the year 1514, is obtained by the numbers 15 & 14.
The anniversary year 2014 is obtained by the numbers 20 & 14. Dürer’s date of birth, the 21st May 1471 (21/5/1471), is obtained by the following numbers: (20 + 28 + 167) & 14 & 71 = 215 14 71. Dürer died on April 6th, 1528 (4/6/1528), and his date of death is obtained by the following numbers: (161 + 245) & 15 & 28 = 4 06 1528.


The second of Lorenzo D. Sisican Jr.'s squares is also semi-pandiagonal, and has a magic constant of 500 in its rows, columns and alternate diagonals (500 years after Melencolia I). 
Lorenzo D. Sisican Jr.'s second tribute to Dürer, 500 years after Melencolia 1 - copyright
The date of the creation of Melencolia I, the year 1514,  is obtained by the numbers 151 & 4. The anniversary year 2014 is obtained by the numbers 201 & 4. Melencolia I was created 14 years before Dürer’s death, and the number 14 is also prominent on this square.






Wednesday, 26 March 2014

MAGIC DIAGONAL SEQUENCES OF 4TH-ORDER MAGIC TORI

Magic series of order 4

The number of fourth-order magic series is 86. The complete list of these magic series has been published by Walter Trump who gave me confirmation that all of these magic series were present on fourth-order magic squares, even if the geometrical arrangement of the series was often chaotic. He pointed out that for example the series 7+8+9+10 does not exist in any row, column, or main diagonal of a magic 4x4-square, even when considering all the 24 possible permutations of the four numbers. However the series 7+8+9+10 occurs in the 2x2 centre square or in the four corners of certain 4x4 magic squares. As defined in MathWorld , or in Wikipedia, a magic series is not a specific sequence of numbers, but a set of numbers that add up to the magic constant.

The magic constant (MCn) of an nth order magic square (or torus) can be calculated:
MCn = (n²)² + n² + n    =    n(n² + 1)
                  2                              2
For a fourth-order magic square (or magic torus) the magic constant is therefore:
MC4   =  4(4² + 1)    =   34
                     2

Magic diagonal sequences on fourth-order magic tori

On any magic diagonal of a fourth-order magic torus (or of an order 4 magic square) :
a + b + c + d = 34

Considering the continuous curved surface of a magic torus, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:
a, b, c, d  (sequence type A) - which is also the magic series
a, b, d, c  (sequence type B)
a, c, b, d  (sequence type C)
As they curve round the torus the sequences have no beginning or end. The magic torus concept is explained here: http://carresmagiques.blogspot.fr/2012/03/from-magic-square-to-magic-torus.html
The interactions of the 4th-order magic tori are detailed here: http://carresmagiques.blogspot.fr/2013/12/fourth-order-magic-torus-chart.html

Theoretically the total number of different magic diagonal sequences on fourth-order magic tori would be 86 magic series x 3 sequence types = 258. However, after checking the 255 different fourth-order magic tori I discovered that there are 48 exceptions, leaving only 210 essentially different diagonal magic sequences. My observations have enabled me to determine some of the rules that govern fourth-order
magic diagonal sequences:

Rules for magic diagonal sequences on fourth-order magic tori

Rule 1: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if consecutive numbers with an even successor number occur, then the series cannot be magic along diagonals.

Rule 2: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are odd numbers, then (c + d) - (a + b ) must be a multiple of 3.

Rule 3: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a and b are even numbers, then (c + d) - (a + b) must be a multiple of 7.

Rule 4: If the magic series a + b + c + d has complementary pairs a + d = 17 and b + c = 17 (whatever the order of the sequence), and if a - 1 = 8 - b = c - 9 = 16 - d, then the series cannot be magic along diagonals.

Presentation of the list of magic sequences

In the list that follows the index numbers of the magic series are the same as those already published by Walter Trump. I have just added the suffixes A, B, and C to identify the 3 essentially different sequences that are derived from each series. I wish to emphasise that although I have always chosen the lowest number to be the first, none of the sequences have either a beginning or an end, as they each form a continuous loop round their torus.
It was difficult to decide how to present the results. I have chosen to list the different sequence types A, B, and C separately, and I have grouped the sequences by sets of number rhythms in order to facilitate comparisons and reveal patterns. I have also sorted between the complementary (a+d = b+c = n²+1) sequences and the non-complementary (a+d ≠ b+c) sequences.

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Observations

Please note that all of the torus diagonals that sum to the magic constant are taken into account - even those that never coincide with the centre of a traditional magic square - such as for example the third magic diagonal sequence (4, 6, 16, 8) of the Frénicle square n° 275 (torus T4.04.2 - index n°T4.098).

The complementary (a + d = c + b = n² + 1) diagonal series seem quite symmetrical and regular when compared to their orthogonal and sub-magic 2x2 square cousins. I have therefore decided to use larger sets of complementary sequences in this present study.

However there are many other ways of ordering the sequences to accord with the specific number rhythms of the different magic torus types. This is why an ideal list, that will suit all the torus types, remains a difficult objective.

Wednesday, 19 March 2014

MAGIC SEQUENCES OF THE SUB-MAGIC 2x2 SQUARES OF 4TH-ORDER MAGIC TORI

Magic series of order 4

The number of fourth-order magic series is 86. The complete list of these magic series has been published by Walter Trump who gave me confirmation that all of these magic series were present on fourth-order magic squares, even if the geometrical arrangement of the series was often chaotic. He pointed out that for example the series 7+8+9+10 does not exist in any row, column, or main diagonal of a magic 4x4-square, even when considering all the 24 possible permutations of the four numbers. However the series 7+8+9+10 occurs in the 2x2 centre square or in the four corners of certain 4x4 magic squares. As defined in MathWorld, or in Wikipedia, a magic series is not a specific sequence of numbers, but a set of numbers that add up to the magic constant.

The magic constant (MCn) of an nth order magic square (or torus) can be calculated:
MCn = (n²)² + n² + n    =    n(n² + 1)
                  2                              2
For a fourth-order magic square (or magic torus) the magic constant is therefore:
MC4   =  4(4² + 1)    =   34
                     2

Magic sequences of the sub-magic 2x2 squares on fourth-order magic tori

On any sub-magic 2x2 square of a fourth-order magic torus (or of an order 4 magic square) :
a + b + c + d = 34

Considering a clockwise arrangement of the numbers on the sub-magic 2x2 square, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:
 
Clockwise and anti-clockwise sequences are considered equivalent, depending on whether the magic torus is being contemplated from the outside or from within. The magic torus concept is explained here: http://carresmagiques.blogspot.fr/2012/03/from-magic-square-to-magic-torus.html
The interactions of the 4th-order magic tori are detailed here: http://carresmagiques.blogspot.fr/2013/12/fourth-order-magic-torus-chart.html

Theoretically the total number of different sub-magic 2x2 squares on fourth-order magic tori would be 86 magic series x 3 sequence types = 258. However, after checking the 255 different fourth-order magic tori I discovered that there are 134 exceptions, leaving only 124 essentially different sub-magic 2x2 square sequences. These exceptions are of course the result of combination impossibilities that can be tested and eliminated by computer calculation. However, as I do not use computer programming myself, I wondered if it might be possible to find simpler ways of distinguishing the magic sequences from their non-magic counterparts. My observations have enabled me to determine some of the rules that govern fourth-
order sub-magic 2x2 square sequences:

Rules for sub-magic 2x2 square sequences on fourth-order magic tori:

Rule 1: The magic series a + b + c + d (whatever the order of the sequence) must always contain two
even  numbers and two odd numbers.

Rule 2: The numbers of the magic series must always be balanced this way:
             1 ≤ a ≤ 7
             2 ≤ b ≤ 8
             9 ≤ c ≤ 15
           10 ≤ d ≤ 16

Rule 3: If there is a single occurrence of consecutive numbers where the successor number is even, then none of the sequences of the series will be magic.
         
Rule 4: If the series contains complementary numbers a + d = 17 and b + c = 17, and if there is a double occurrence of consecutive numbers where the successor number is even, only the sequences that restrict these successive numbers to the diagonals of the sub-magic 2x2 square will be magic. Some of the odd - even - odd - even series of complementary pairs have this characteristic and the only magic sequences of these series are therefore type C.
 
Rule 5: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if a - 1 = 8 - b = c - 9 = 16 - d
             implying that (c + d) - (a + b) = n²
             then only the sequence type A will be magic.
             (as this only occurs with odd-even-odd-even, and even-odd-even-odd series).

Rule 6: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if the series is odd - odd - even - even,
             and if (c + d) - (a + b) is not a multiple of 3,  
             then only the sequence C (odd - even - odd - even) will be magic.

Rule 7: If the series contains complementary numbers a + d = 17 and b + c = 17,
             and if the series is even - even - odd - odd,
             and if (c + d) - (a + b) is not a multiple of 7,  
             then only the sequence C (even - odd - even - odd) will be magic.

Presentation of the list of magic sequences

In the list that follows the index numbers of the magic series are the same as those already published by Walter Trump. I have just added the suffixes A, B, and C to identify the 3 essentially different sequences that are derived from each series. I wish to emphasise that although I have always chosen the lowest number to be the first, none of the sequences have either a beginning or an end, as they each form a continuous circuit within their sub-magic 2x2 square.
It was difficult to decide how to present the results. I have chosen to list the different sequence types A, B, and C separately, and I have grouped the sequences by sets of number rhythms in order to facilitate comparisons and reveal patterns. I have also sorted between the complementary (a+d = b+c = n²+1) sequences and the non-complementary (a+d ≠ b+c) sequences.


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Observations

The complementary series (a + d = c + b = n² + 1) have, where possible, been divided into pairs of sequences which share similar characteristics. Although the sub-magic 2x2 square magic series 1 + 2 + 15 + 16, and 3 + 4 + 13 + 14 seem similar, I have preferred to stick to the same sets already used in my study of the orthogonal sequences. These include 4 exceptions with unique characteristics that cannot be easily paired:
1 + 2 + 15 + 16
3 + 4 + 13 + 14
5 + 6 + 11 + 12
7 + 8 +  9  + 10

There are many other ways of ordering the sequences to accord with the specific number rhythms of the different magic torus types. This is why an ideal list, that will suit all the torus types, remains a difficult objective.



























Tuesday, 18 March 2014

MAGIC ORTHOGONAL SEQUENCES OF 4TH-ORDER MAGIC TORI

Magic series of order 4

The number of fourth-order magic series is 86. The complete list of these magic series has been published by Walter Trump who gave me confirmation that all of these magic series were present on fourth-order magic squares, even if the geometrical arrangement of the series was often chaotic. He pointed out that for example the series 7+8+9+10 does not exist in any row, column, or main diagonal of a magic 4x4-square, even when considering all the 24 possible permutations of the four numbers. However the series 7+8+9+10 occurs in the 2x2 centre square or in the four corners of certain 4x4 magic squares. As defined in MathWorld, or in Wikipedia, a magic series is not a specific sequence of numbers, but a set of numbers that add up to the magic constant.

The magic constant (MCn) of an nth order magic square (or torus) can be calculated:
MCn = (n²)² + n² + n    =    n(n² + 1)
                2                           2
For a fourth-order magic square (or magic torus) the magic constant is therefore:
MC4   =  4(4² + 1)    =   34
                     2

Magic orthogonal sequences of fourth-order magic tori

On any orthogonal line of a magic torus (column and row of an order 4 magic square) :
a + b + c + d = 34

This equation can be expressed differently to express the balance of the numbers and facilitate the analysis of the series:
a + b + c + d = 16+1
   2         2
Therefore:
a + b = 16 + 1 - c + d         and       c + d = 16 + 1 - a + b
   2                       2                             2                       2

Considering the continuous curved surface of a magic torus, each of the 86 4th-order magic series can be expressed in 3 essentially different sequences:
a, b, c, d  (sequence type A) - which is also the magic series
a, b, d, c  (sequence type B)
a, c, b, d  (sequence type C)
As they curve round the torus the sequences have no beginning or end. The magic torus concept is explained here: http://carresmagiques.blogspot.fr/2012/03/from-magic-square-to-magic-torus.html
The interactions of the 4th-order magic tori are detailed here: http://carresmagiques.blogspot.fr/2013/12/fourth-order-magic-torus-chart.html

Rules for magic orthogonal sequences on fourth-order magic tori

Rule 1: The magic series a + b + c + d (whatever the order of the sequence) must always contain two even numbers and two odd numbers.

Rule 2: The numbers of the magic series must always be balanced this way:
             1 ≤ a ≤ 7
             2 ≤ b ≤ 8
             9 ≤ c ≤ 15
           10 ≤ d ≤ 16

Rule 3: If b - a = 1 and if b is an even number, then the series will only be magic if :
           d - c = 1, and 17-c = b.
           If d - c = 1 and if d is an even number, then the series will only be magic if :
           b - a = 1, and 17-c = b.
Exception 3a : If a third set of consecutive numbers is introduced ( c - b = 1), the series will not be magic. This excludes the magic series 7 + 8 + 9 + 10 in orthogonal sequences.
Exception 3b : If all the numbers are situated between n + 1 and 3n - 1 then only the types A and C sequences (where complementary numbers are adjacent) will be magic. This excludes the orthogonal sequence (5, 6, 12, 11).

Presentation of the list of magic sequences

In the list that follows the index numbers of the magic series are the same as those already published by Walter Trump. I have just added the suffixes A, B, and C to identify the 3 essentially different sequences that are derived from each series. I wish to emphasise that although I have always chosen the lowest number to be the first, none of the sequences have either a beginning or an end, as they each form a continuous loop round their torus.
It was difficult to decide how to present the results. I have chosen to list the different sequence types A, B, and C separately, and I have grouped the sequences by sets of number rhythms in order to facilitate comparisons and reveal patterns. I have also sorted between the complementary (a+d = b+c = n²+1) sequences and the non-complementary (a+d ≠ b+c) sequences.


Magic torus diagram copyright
Magic torus diagram copyright
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Observations

The complementary series (a + d = c + b = n² + 1) have, where possible, been divided into pairs of sequences which share similar characteristics. There are 4 exceptions with unique characteristics that cannot be paired. Interestingly these 4 series are inversely related :
1 + 2 + 15 + 16
3 + 4 + 13 + 14
5 + 6 + 11 + 12
7 + 8 +  9  + 10

There are many other ways of ordering the sequences to accord with the specific number rhythms of the different magic torus types. This is why an ideal list, that will suit all the torus types, remains a difficult objective.














Thursday, 26 December 2013

FOURTH-ORDER MAGIC TORUS CHART

This latest study shows how the 255 4th-order magic tori are linked, and how they interact. The 4th-order magic torus chart indicates the position of each magic torus in the system. All of Bernard Frénicle de Bessy's 880 4x4 magic squares, and all of Henry Dudeney's 12 square types are accounted for.


For presentation reasons a landscape format was chosen for this paper. However, this is not ideal for vertical blog pages, and I wish to apologise for the inconvenience caused to readers! Please note that if you click on any page of the paper a full size view will be displayed.


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Although this paper was first edited on Christmas Eve and then published on Boxing Day, this is a complete coincidence!