(10) European Chautisa Yantra: Tona Lisa!
Pankaj Khanna
9424810575
Previous/Next Blog Posts:
(1) Beauty Squarely Introduction & Kuber Yantra
(2) Murphy Radio!? अले वाह!! My first Experience of Magic Square.
(3) Decoding the Quadratum Mirabile! How to solve 3x3 Magic Squares.
(4) Lo Shu Square History of Chinese Magic Square.
(5) The Unhurried Odyssey of a Turtle!! History of Magic Squares in short.
Brief Introduction.
(7) Khajuraho Magic: Introduction.
(8) Chautisa Yantra: Mytho-math Spice.
(9) Dürer Square
(10) European Chautisa and Tona Lisa.
There’s an undeniable aesthetic charm in Dürer’s Magic Square—something that Melencolia I, for all its brilliance, doesn’t quite radiate in the same way. So I decided to surround the square with a touch of Sixteenth-Century European glamour. After all, a masterpiece deserves fitting company… just like the magnificent Chautisa Yantra of Khajuraho!
The world’s most famous model of all time, Mona Lisa (1479–1542), was actually a contemporary of Albrecht Dürer (1471–1528). Imagine, how glorious it would have been to see Mona Lisa smiling cryptically beside equally cryptic Dürer’s Magic Square in the same painting or engraving! A Renaissance crossover episode of epic proportions. Alas… pure wishful thinking!
Reality, however, had other plans. Instead of sharing the frame with the elegant Mona Lisa, Dürer’s Square ended up paired with a perpetually stressed, brooding model in Melencolia I. Truly unfortunate casting!
So what do we do? Simple—we create our own legendary model: Tona Lisa! Cheer leader for magic squares and maths!
If Dürer’s Square can mesmerize viewers with its mathematical sorcery, Tona Lisa will match it spell for spell with her beguiling Jadoo-Tona !
Durer-Tona: a perfect duo for our very own European Chautisa Yantra!
European Chautisa Yantra and Tona Lisa! Desperately waiting to be displayed at Luvre! But we will have to wait till eternity. Museums are run by Art-Historians, Curators and not by Mathematicians!
A ray of hope for those who fail to see beauty in Magic Squares: They may find, at least, Tona Lisa charming!
Quest for the mysteries of Magic Squares is like a Mathematical thriller! A thriller without a heroine is unimaginable! Hence Tona Lisa!
James Bond without a heroine!? Na!!
Coming back to maths and Dürer Magic Square lovers. They insist that this magic square is not following the rules and is aggressively enforcing them on the viewers! Quite true!
They further claim that this isn't just an arrangement of numbers; it's a cosmic ledger! Yes, magic squares are compact cosmic ledgers and one can probably attain numeric Nirvana too after going through them!
So, let's now dive headfirst urgently into the Wondrous World of Dürer's Square to dig out the great numerical marvels liked by the magic square aficionados all over the world.
First act of this mathematical sideshow is dedicated to the square's shocking predictability as shown above!
The Row Rhapsody: Add up any one of the four horizontal rows. It doesn't matter which one you pick! The total is consistently, relentlessly, 34!
The Column Conspiracy: Go vertical! Sum any of the four columns. What's the shocking result? 34. (Chautisa!)
The Diagonal Drama: Draw a line from corner to opposite corner. The two main diagonals? They're also in on the secret, summing up perfectly to Chautisa.l!
The Inner Sanctum Shuffle: Now for the advanced party trick! Take the five smaller, inner 2x2 squares (including the very center and the four corner quadrants) as highlighted with a personal touch, in the above diagram. They, too, are completely incapable of producing any result other than the official party line: Chautisa!!
In summary: Every major linear, columnar, and quadrant path leads to the same mathematically inescapable conclusion: Chautisa the magic sum! Again and again!
However unlike Khajuraho Square where the sum of all nine 2x2 magic squares is the same; in case of European Chautisa Yantra, sum of only five 2x2 magic squares is the same as it is not a Pandiagonal Square.
Yet nothing deters Dürer’s devotees— they keep digging, decoding, and downright obsessing over that irresistible magic sum in his iconic square.
There are some great properties in it which can be seen in several pleasing patterns. For example: Sum of numbers of all four corners is 34. And If you consider all four 3x3 squares of the Durer Square, sums of corners of these four squares are also the same: as illustrated below:
The 86 images above showcase the Chautisa in stunning, varied patterns—each one revealing why this ancient construct is so irresistible, just like our Tona Lisa.
At the heart of any magic square lies its magic sum, obtained by adding four numbers. For the Chautisa, the mathematical question becomes:
How many distinct quadruples (A, B, C, D) of natural numbers satisfy
A + B + C + D = 34,
with 1 ≤ A < B < C < D ≤ 16?
The answer is 86.
Meaning, there are exactly 86 unique ways to form the magic sum of 34 using four distinct numbers from this square. You can derive this either by a bit of clever arithmetic, or through combinatorics using the Inclusion–Exclusion Principle. To view the full list of all 86 combinations, click here.
But simply listing the 86 solutions in a PDF is, frankly, yawn-inducing. Showing them as images—as done above—transforms the same mathematics into something elegant, vivid, memorable and far more beautiful!
The fascinating part?
The Khajuraho Magic Square also generates the same 86-fold magic sum! The patterns differ from Dürer’s Square, yet they are just as enchanting—perhaps even more. Because both the squares are essentially Chautisa squares, mathematical cousins separated by centuries.
And speaking of centuries, it is rather astonishing that the Dürer Square gained recognition more than 500 years after the Khajuraho square had already appeared. Or perhaps… it’s just another case of lost manuscripts, forgotten memories, and unexplored archives in our own country?
We’ve already met a few of these dazzling patterns in the Khajuraho Magic blog post—just enough to tease the mathematical appetite. The full parade of all 86 Khajuraho Chautisa formations will arrive later, in neatly crafted diagrams, when the stars and this blogger's schedule align!
And yes, the Tona Lisa aka European Chautisa saga continues in the next post also. Stay tuned; the drama is far from over!
Pankaj Khanna
9424810575
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मेरे कुछ अन्य ब्लॉग:
हिन्दी में:
तवा संगीत : ग्रामोफोन का संगीत और कुछ किस्सागोई।
रेल संगीत: रेल और रेल पर बने हिंदी गानों के बारे में।
साइकल संगीत: साइकल पर आधारित हिंदी गाने।
कुछ भी: विभिन्न विषयों पर लेख।
तवा भाजी: वन्य भाजियों को बनाने की विधियां!
मालवा का ठिलवा बैंड: पिंचिस का आर्केस्टा!
ईक्षक इंदौरी: इंदौर के पर्यटक स्थल। (लेखन जारी है।)
अंग्रेजी में:
Love Thy Numbers : गणित में रुचि रखने वालों के लिए।
Epeolatry: अंग्रेजी भाषा में रुचि रखने वालों के लिए।
CAT-a-LOG: CAT-IIM कोचिंग।छात्र और पालक सभी पढ़ें।
Corruption in Oil Companies: HPCL के बारे में जहां 1984 से 2007 तक काम किया।
