Mathematical Secrets of Ancient Tablet Unlocked After Nearly a Century of Study

Dating from 1,000 years before Pythagoras’s theorem, the Babylonian clay tablet is a trigonometric table more accurate than any today, say researchers.



At least 1,000 years before the Greek mathematician Pythagoras looked at a right angled triangle and worked out that the square of the longest side is always equal to the sum of the squares of the other two, an unknown Babylonian genius took a clay tablet and a reed pen and marked out not just the same theorem, but a series of trigonometry tables which scientists claim are more accurate than any available today.

The 3,700-year-old broken clay tablet survives in the collections of Columbia University, and scientists now believe they have cracked its secrets.

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The Formula That Plots (Almost) Everything

Hold onto your logic hats! In this article we're going to explore one of the most amazing formulas in maths: Tupper's self-referential formula.

The protagonist of our story is the following inequality:

`1/2<\floor{mod(\floor{\frac{y}{17}}2^(-17\floor{x}-mod(\floor{y},17)),2))`

The plot works by either coloring a square or not coloring it: a square with coordinates (x, y) is colored if the inequality is true for x and y. If not the square is left blank.

If you plot the plot for many values of and , the outcome is the following:



I'll let that sink in a moment. No, your eyes are not deceiving you, the formula plots a bitmap picture of itself! Hence the name Tupper's self-referential formula (though Tupper never called this function that himself in his 2001 paper).

There is one missing detail, however. I haven’t told you the value of the number N on the y-axis.

Click here to read more information and see where Euler's equation appears.