Mathematicians armed with supercomputers have finally determined the value of a complex number that was previously thought impossible to compute.
The number, known as the “Ninth Dedekind Number” or D(9), is actually the tenth digit in the sequence. Each Dedekind number represents the number of possible configurations of a given type of true-false logical operation in different spatial dimensions. (The first number in the sequence is D(0), which represents zero dimensions. This is why D(9), which represents nine dimensions, is the tenth number in the sequence.)
Dedekind’s numbers increase exponentially for each new dimension, making it increasingly difficult to pinpoint. The eighth Dedekind number, which follows the same rules for eight dimensions, was calculated in 1991. But because of the jump in computing power needed to calculate the ninth, some mathematicians considered it impossible to calculate its exact value.
But now, there are two unrelated studies from separate research groups–the Firstly Submitted to the arXiv prepress server on April 5th and second Submitted to the same server on April 6th – I did the impossible. The two studies — each using a supercomputer but running different software — produced exactly the same number.
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Results have not been peer reviewed. But since the studies come to the same conclusion, it is “100%” certain that the number was deciphered correctly, as the lead author said in the second paper, Lennart van HertemThe mathematician at the University of Paderborn in Germany and lead author on the second paper told Live Science.
Van Hirtum and colleagues defend their work during a a lecture at the University of Paderborn on June 27.
What are deadkind numbers?
German mathematician Richard Dedekind first described Dedekind numbers in the 19th century. Numbers are associated with logical problems known as “Montonic Boolean Functions” (MBFs).
Boolean functions are a type of logic that can take as input only one of two values - 0 (false) and 1 (true) – and spit out only those two values. In MBFs, you can swap 0 for 1 in the input, but only if it allows the output to change from 0 to 1, not from 1 to 0. Dedekind numbers are the output of MBFs where the input is the specified spatial dimension.
This concept can be very confusing for non-athletes. Van Hertom explained that it is possible to visualize what is going on by using shapes to represent Dedekind numbers for each dimension. For example, in the second dimension, the Dedekind number is associated with a square, while the third can be represented by a cube, and the fourth and higher by hypercubes.
For each dimension, the vertices or points of a particular shape represent the possible configurations of the MBFs (see image below). To find the Dedekind number, you count the number of times you can color each vertex of each shape with one of two colors (in this case red and white), but with the proviso that one color (in this case white) cannot be layered on top of another (red in this case) .
For zero dimensions, the shape is just one point and D(0) = 2 because the point can be either red or white. For one dimension, the shape is a line with two points and D(1) = 3 because both points can be either the same color or red over white. For the two dimensions it is a square and D(2) = 6 because there are now six possible scenarios where there is no white dot over a red dot. And for the three dimensions, the shape is a cube, and the number of possible configurations jumps to 20, so D(3) = 20.
As the number of dimensions increases, van Hertom said, the virtual shape becomes an increasingly complex hypercube with an exponentially greater number of outcomes.
The following five Dedekind number values are 68, 7581, 7828354, 2414682040998, and 56130437228687557907788.
The newly defined value of D(9) is 286386577668298411128469151667598498812366.
Increasingly complex calculations
Van Hirtum has been working on D(9) for more than three years. To do this, he created a new type of computer program to enable a supercomputer to process data in a specific way. If he used more basic software, he said, it could take him up to 100 years to complete the calculations, even with a sophisticated number-solving machine.
After creating its computer code, Van Hirtum’s team spent more than four months using a supercomputer at the University of Leuven in Belgium to process the data.
However, the calculations didn’t take that long to complete: Van Hertom said the nature of the program meant it was prone to partial errors, which meant the team had to constantly redo.
By comparison, the computer used in 1991 for the D(8) exercise was less powerful than a modern smartphone and completed the task in about 200 hours. A modern laptop computer might have performed these calculations in less than 10 minutes, Van Hertom said.
Van Hirtum believes a similar jump in computer processing power is needed to calculate Dedekind’s tenth number. “If we were doing it now, it would require processing energy equal to the total energy produced by the sun,” he said, making his calculation “practically impossible.”
Van Hertom said that processing power requirements can be reduced by using more sophisticated algorithms.
“But we have kind of been struck by how complex the algorithms are,” he added.
However, other mathematicians are still hopeful that D(10) can eventually be computed, Van Hirtum said.
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