How ancient war trickery Is alive in math today + The journey to define dimension

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How ancient war trickery Is alive in math today

INTRO: Imagine you’re a general in ancient times and you want to keep your troop counts secret from your enemies. But you also need to know this information yourself. So you turn to a math trick that allows you to achieve both aims.

In a morning drill you ask your soldiers to line up in rows of five. You note that you end up with three soldiers in the last row. Then you have them re-form in rows of eight, which leaves seven in the last row, and then rows of nine, which leaves two. At no point have you counted all your soldiers, but now you have enough information to determine the total number without having to state an explicit count that an enemy could intercept.

Legend suggests ancient Chinese generals actually employed this technique, though it’s unclear if they really did. What we do know is that the mathematical technique now known as the Chinese remainder theorem was devised sometime between the third and fifth centuries CE by the Chinese mathematician Sun Tzu (not to be confused with Sun Tzu who wrote The Art of War almost 1,000 years earlier).

The theorem allows you to find an unknown number if you know its remainders when it’s divided by certain numbers that are “pairwise coprime,” meaning they do not have any prime factors in common. Sun Tzu never proved this formally, but later the Indian mathematician and astronomer Aryabhata developed a process for solving any given instance of the theorem.

“The Chinese remainder theorem gives you an actual recipe for making a number,”said Daniel Litt of the University of Georgia. To better understand how the theorem works... (MORE)
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NEXT: "The Journey to Define Dimension"
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The journey to define dimension

EXCERPTS: The notion of dimension at first seems intuitive. [...] But as we’ll see, finding an explicit definition for the concept of dimension and pushing its boundaries has proved exceptionally difficult for mathematicians. It’s taken hundreds of years of thought experiments and imaginative comparisons to arrive at our current rigorous understanding of the concept.

The ancients knew that we live in three dimensions. Aristotle wrote, “Of magnitude that which (extends) one way is a line, that which (extends) two ways is a plane, and that which (extends) three ways a body. And there is no magnitude besides these, because the dimensions are all that there are.”

Yet mathematicians, among others, have enjoyed the mental exercise of imagining more dimensions. What would a fourth dimension — somehow perpendicular to our three — look like?

[...] This all adds up to an intuitive understanding that an abstract space is n-dimensional if there are n degrees of freedom within it (as those birds had), or if it requires n coordinates to describe the location of a point. Yet, as we shall see, mathematicians discovered that dimension is more complex than these simplistic descriptions imply.

The formal study of higher dimensions emerged in the 19th century and became quite sophisticated within decades [...]

[...] in 1890, Giuseppe Peano discovered that it is possible to wrap a one-dimensional curve so tightly — and continuously — that it fills every point in a two-dimensional square. This was the first space-filling curve. But Peano’s example was also not a good basis for a coordinate system because the curve intersected itself infinitely many times; returning to the Manhattan analogy, it was like giving some buildings multiple addresses.

These and other surprising examples made it clear that mathematicians needed to prove that dimension is a real notion and that, for instance, n- and m-dimension Euclidean spaces are different in some fundamental way when n ≠ m. This objective became known as the “invariance of dimension” problem.

Finally, in 1912, almost half a century after Cantor’s discovery, and after many failed attempts to prove the invariance of dimension, L.E.J. Brouwer succeeded by employing some methods of his own creation. In essence, he proved that it is impossible to put a higher-dimensional object inside one of smaller dimension, or to place one of smaller dimension into one of larger dimension and fill the entire space, without breaking the object into many pieces, as Cantor did, or allowing it to intersect itself, as Peano did. Moreover, around this time Brouwer and others gave a variety of rigorous definitions, which, for example, could assign dimension inductively based on the fact that the boundaries of balls in n-dimensional space are (n − 1)-dimensional.

Although Brouwer’s work put the notion of dimension on strong mathematical footing, it did not help with our intuition regarding higher-dimensional spaces: Our familiarity with three-dimensional space too easily leads us astray. As Thomas Banchoff wrote, “All of us are slaves to the prejudices of our own dimension.”

[...] The surprising realities of high-dimensional space cause problems in statistics and data analysis, known collectively as the “curse of dimensionality.” The number of sample points required for many statistical techniques goes up exponentially with the dimension. Also, as dimensions increase, points will cluster together less often. Thus, it’s often important to find ways to reduce the dimension of high-dimensional data.

The story of dimension didn’t end with Brouwer. Just a few years afterward, Felix Hausdorff developed a definition of dimension that — generations later — proved essential for modern math...

[...] One surprising consequence of Hausdorff’s definition is that objects could have non-integer dimensions. Decades later, this turned out to be just what Benoit B. Mandelbrot needed when he asked, “How long is the coast of Britain?” A coastline can be so jagged that it cannot be measured precisely with any ruler — the shorter the ruler, the larger and more precise the measurement. Mandelbrot argued that the Hausdorff dimension provides a way to quantify this jaggedness, and in 1975 he coined the term “fractal” to describe such infinitely complex shapes.

[...] Lastly, some readers may be thinking, “Isn’t time the fourth dimension?” Indeed, as the inventor said in H.G. Wells’ 1895 novel The Time Machine, “There is no difference between Time and any of the three dimensions of Space except that our consciousness moves along it.” Time as the fourth dimension exploded in the public imagination in 1919, when a solar eclipse allowed scientists to confirm Albert Einstein’s general theory of relativity and the curvature of Hermann Minkowski’s flat four-dimensional space-time. As Minkowski foretold in a 1908 lecture, “Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve independent reality.” (MORE - missing details)

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