Ceramic Design: Magic Square as Visual Design on Pots                                     

 

An example of 4x4 Magic Squares

 

0

14

13

3

11

5

6

8

7

9

10

4

12

2

1

15

 

 

 

 

 

 

 

 

Sq.1                                                                 

16 integers from 0 to 15 are arranged in 4x4 squares as above. A 4x4 square is magic when numbers have the same sum 30 in the rows, columns, and the diagonals. (For instance, 0+14+13+3=30; 0+11+7+12=30; 0+5+10+15=30; etc.)

 

An example of 4x4 Pan Magic Square

 

0

13

3

14

7

10

4

9

12

1

15

2

11

6

8

5


Sq.2

 

A square is pan magic (or pan-diagonal magic) if it is magic not only in the main diagonals but also in the broken diagonals. (For instance, 0+10+15+5=30; 13+4+2+11=30; 3+9+12+6=30; etc.)

 

This means when the same pan-magic square is repeated next to the original square, any sum of four continuous numbers in the rows, columns, and in the diagonals is the same 30. An example is shown below:

 

0

13

3

14

0

13

3

14

7

10

4

9

7

10

4

9

12

1

15

2

12

1

15

2

11

6

8

5

11

6

8

5

 

 

 

 

 

 

 

 

Sq.3

 

Hexadecimal numbers as Visual Elements

To make these 16 numbers from 0 to 15 visual, I use the following binary decomposition (yes or no) based on 4 basic elements, [1], [2], [4=22], and [8=23].

            0 = no element

            1 = [1]

            2 =          [2]

            3 = [1] + [2]  

            4 =                   [4]

            5 = [1]          + [4]

            6 =          [2] + [4]

            7 = [1] + [2] + [4]

            8 =                            [8]

            9 = [1]                   + [8]

          10 =          [2]          + [8]

          11 = [1] + [2] +       + [8]

          12 =                   [4] + [8]

          13 = [1]          + [4] +[8]

          14 =          [2] + [4] + [8]

          15 = [1] + [2] + [4] + [8]

 

Then, the above example of a pan magic square is written in terms of 4 elements as,

Note that the sum of four numbers in the row, column, and in the diagonal means that every elements appear twice in each row, column, and in each diagonal.

 

Permutation Invariance

This fact also implies that one can create another pan-magic square by inter-changing elements among [1], [2], [4], and [8]. For instance, by inter-changing [1] and [4], one can obtain another pan-magic such as,

Cyclic Invariance

Another consequence of the pan magic square is that the square with the top row moved to the bottom (or vice versa) is also pan magic. Similarly, the square is pan magic when the left end column is moved to the right end (or vice versa).

Rotational Invariance

Though trivial, the pan magic square stays pan magic when the square is rotated clockwise or counter-clockwise by 90 degrees.

Visualization

And assign four different visual elements to the binary elements, [1], [2], [4], and [8].

For instance, four line elements in different orientation are assigned to the binary elements as,

            [1] ®  ¤

            [2] ® \

            [4] ® ¾

            [8] ® ½

See Fig.1.

 

Fig.1

 

Then, we find 16 different graphic elements (hexadecimal) out of combinations of these four elements and zero. They are arranged on the Pot 1 (Hexadecimal), as shown below:

Pot 1: Hexadecimal

 

These hexadecimal elements are arranged to form a pan magic squares. For instance, the above pan magic square (Sq.4) is realized as a square of Fig.2. Note four different colors are used to identify individual elements. Fig.3 illustrates a repeated magic square (Sq.3). Note any 4x4 squares, for example the one identified by a dotted line, and another by a dashed line, are all pan magic.

                      

Fig.2                                                                                                                                       Fig.3

 

Different realization of pan magic squares is used as a design on the Pot 2 and Pot 3  below:

              

Pot 2: Magic Squares                                                                         Pot 3: Magic Squares

 

As noted above, a permutation of elements creates different pan magic square. For instance, consider interchange of element [1] and [4], as shown graphically in Fig.4.

The process of interchanging elements is illustrated in Fig.5. The resulting pan magic square is shown, both graphically (Sq.5) as well as numerically (Sq.6). 

                              

Fig.4                                                                                                                                       Fig.5

 

As another example, consider interchange of element [2], [4], and [8], with [8], [2], and [4], as shown graphically in Fig.6. The resulting pan-magic square is shown in Fig.7.

                      

Fig.6                                                                                                                                       Fig.7

 

 

Variation in Visualization

Different visual elements are used for the binary elements, [1], [2], [4], and [8].

For instance, four triangles (up, down, right, and left) may be assigned. Horizontal bars at four different height, or vertical bars at four different shift, may be also used,, These examples are shown in Fig.8. A pan magic square is shown using triangle elements in Fig.9.

                      

Fig.8                                                                                                                                       Fig.9

 

 

This magic square with triangle elements is used as a design of the Pot.4. The magic squares with vertical bars and horizontal bars are shown on Pot 5 and Pot 6, respectively.

                         

Pot.4: Magic Square w/Triangles                                    Pot.5: Magic Square w/Vertical Bars                                                   Pot.6: Magic Square w/Horizontal Bars

 

So, do you find anything interesting in the design?