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Brief Introduction.
(11) Tona Lisa as a Rock Star!
The European Chautisa is perhaps hiding more secrets than a CIA or KGB archive combined! Is it a bit too much, an exaggeration!? Ok! Ok!! Remove KGB from the above sentence!
Let us begin anew. Secrets must surface, and Tona Lisa must smile and laugh forever; hand in hand with the numbers of Dürer’s square. Here History, Art, Mathematics, and Beauty are not separate realms; they are eternal companions, destined to celebrate together. No further gloom or melancholy here!
(Tona Lisa smiling numerically with numbers 5,6,7,8.)
In the last blog post, we admired a parade of beautiful mathematical patterns hidden inside Dürer’s Square. In this one, the numbers return for an encore—this time flirting shamelessly with Tona Lisa’s beauty, proving once again that even mathematics cannot resist a good smile!
(A) Construction of Durer Square:
First, let us admire the sheer elegance behind the construction of Dürer’s Square. All you need to remember is the simple pattern shown in the four diagrams below. Look closely: the consecutive numbers 5, 6, 7, 8 and 13, 14, 15, 16 curve into smiling and laughing formation respectively. While 1, 2, 3, 4 and 9, 10, 11, 12 stand cresfallen and solemn; perfectly echoing the melancholic mood of Dürer’s engraving. In short it is about: Joy and sorrow, laughter and gravity captured together in perfect balance, just as life itself, sealed inside a magic square!
A single look at the diagram above is enough to build Dürer’s Square in seconds. No fancy academic degrees, no divine inspiration, and definitely no brooding angel required.
(B) Associative or Associated Squares:
An n x n magic square in which every pair of numbers placed symmetrically opposite the center adds up to n²+1; is known as an Associated Magic Square. Also called an Associative, a regular, regmagic, or symmetric magic square.
Dürer’s celebrated square is a perfect example of this elegant property. Here, each such opposite pair sums neatly to 17, a hidden harmony that becomes immediately evident in the diagrams shown below.
All 3 x 3 magic squares are Associated—they’re reliable like that. But 4 x 4 squares? They have commitment issues!
The Khajuraho square, a perfect magic square, is strictly Pandiagonal, leaving no room to be Associated. Meanwhile, the Dürer square is Associated, but definitely neither Pandiagonal nor most perfect. It seems in the geometry of these famous grids; you can't have your cake and eat it too!
(C) Sum of Squares:
This is where Dürer’s Square drops all modesty and turns into a full-blown Arithmetic Rock Star!
Equal sums are impressive, but when the sums of squares line up perfectly, the square isn’t just behaving well; it’s showing off! Good for maths!
We already know Dürer's square is famous for summing to 34. But if you look deeper—by squaring the numbers—you find a fascinating set of "mirror" properties. It seems the square is perfectly balanced not just in simple addition, but in quadratic power as well.
Here is how the symmetry unfolds:
1. The Row Reflections:
The square operates like a sandwich. The top and bottom "bread" layers match, and the inner "filling" layers match.
(A glutton will remember Sandwich and fillings too while discussing maths! Oh! Tawa Bhaji!!😋😋)
The Outer Bounds (Rows 1 & 4):
The sum of squares in the very top row equals the sum of squares in the very bottom row.
16² + 3² + 2² + 13² = 438
4² + 15² + 14²+ 1² = 438
The Inner Core (Rows 2 & 3):
Moving inward, the second row mirrors the third row.
5² + 10² + 11² + 8²= 310
9² + 6² + 7² + 12² = 310
2. The Column Reflections:
The same rule applies vertically. The outer columns balance each other, and the inner columns hold the center.
The Outer Pillars (Columns 1 & 4):
The first column balances perfectly with the last.
16² + 5² + 9²+ 4² = 378
13² + 8² + 12² + 1² = 378
The Inner Pillars (Columns 2 & 3):
Finally, the second column is equal to the third column.
3² + 10² + 6² + 15² = 370
2² + 11² + 7² + 14² = 370
Whether you slice it horizontally or vertically, the outer edges always mirror each other, and the center always holds its own balance.
(D) Sum of Diagonals:
The sum of the numbers on both diagonals (highlighted in green) is=16+10+7+1+4+6+11+ 13 = 68
.
The remaining numbers (shown in red) also add up to =3+2+5+8+9+12+15+ 14 = 68.
.
So far, so good! This is almost business as usual for a well-behaved magic square.
But here comes the real mischief. Square every number on the diagonals and add them up: 16² +10²+ 7²+ 1²+ 4² + 6²+ 11²+13²=256+100+49+1+16+36+121+169= 748
Now do the same for the remaining numbers: 3²+2²+5²+8²+9²+12²+15²+14²= 9+4+25+64+81+144+225+196= 748.
Coincidence? Not quite! But wait, the Durer square isn’t done showing off yet.
Turn the dial up to cubes, and the symmetry still refuses to break: 16³+10³+7³+1³+4³+6³+11³+13³= 4096+1000+343+1+64+216+1331+ 2197= 9248
3³+2³+5³+8³+9³+12³+15³+14³ = 27+8+125+512+729+1728+3375+ 2744= 9248.
Mathematicians and Magic Lovers just love these patterns and mathematical curiosities! it gives them more excuses to host International/ National conferences and a right to write 'Research Papers' on such beauties! And also it gives them a golden opportunity to write Books on them.
And why is this blogger not intending to write Books or research papers on Magic Squares!? Answer in the next blog post! And after that much awaited Ramanujan's Square!
Pankaj Khanna
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