What is the “kissing problem” that has plagued mathematicians for centuries?

It all started in the 16th century with the famous explorer or pirate (depending on your point of view) Sir Walter Raleigh. Which might surprise you after reading the title, since he wasn’t a mathematician and, as far as we know, had no problem with kissing.

What he did was cannonballs, and one question: what would be the most efficient way of stacking them to reduce the space they take up on their ships as much as possible.

It was a math problem, and in math, these bullets are balls and “kissing” is what the points where they touch another ball are called.

Raleigh question That would generate a mathematical puzzle that would occupy brilliant minds for hundreds of years.

He asked who his scientific advisor was on his voyage to America in 1585, the eminent mathematician Thomas Harriot, who gave him the solution:

The best way to store your cannonballs They were arranged in a hierarchy.

In a manuscript of 1591, Harriot made for him a table showing how one might, given the number of cannonballs, calculate the number to be placed at the base of a pyramid with a triangular, square or rectangular base.

But Harriott kept thinking about it, so he thought about the implications for the atomic theory of matter, which was in vogue at the time.

Commenting on that theory in his correspondence with his friend, the famous astronomer Johannes Kepler, he mentioned the packing problem.

Kepler predicted that the best way to reduce the space left by the gaps between the spheres was to make them In each layer the centers of the balls were above Where kisses domains in the bottom layer.

This is what often happens with fruits in the markets.

This, which seems so intuitive, It was very difficult to prove mathematically.

Although it was experimented with by many, including the “Prince of Mathematics” Johann Carl Friedrich Gauss, it was only proven nearly four centuries later, in 1998, by the work of Thomas Hales of the University of Michigan and the power of a computer.

Even this verification did not convince all mathematicians. Even today, there are those who do not consider it worthy of Kepler’s guess.

This wasn’t the only headache that spherical bodies caused.

In fact, there is a broad class of mathematical problems called “field beam problems”.

Their solution has served from exploring the structure of crystals to improving the signals sent from cell phones, space probes, and the Internet.

And like Raleigh with his cannon, Logistics, raw materials and many other industries They largely rely on optimization methods provided by mathematics.

Mathematicians have discovered, for example, that randomly stacked spheres tend to occupy any space with a density of up to 64%. But if you arrange them carefully in certain ways, you can get 74%.

This represents 10% savings not only in transportation costs but also in environmental harm.

But practical applications like this require mathematical proof, and encapsulating the spheres gave rise to particularly difficult unknowns, as did Kepler’s conjecture.

One of them originated from A conversation between Isaac Newton, one of the most outstanding scientists of all time, and David GregoryHe is the first university professor to teach Newton’s advanced theories.

The problem was the number of kisses, but …

Imagine that you have several circles of cardboard of the same size and you want to glue them on a board around one of them.

The number of kisses is equal maximum amount One of the circles you can place by kissing – or touching – is the central circle.

Well, it turns out that mathematicians have shown that can be placed Maximum 6 circles about initialThe number of kisses will be 6.

Now imagine that Instead of cardboard circles, you have rubber ballsall the same size.

Again, the question is: what is the maximum number of balls you can fit around a center?

By adding this third dimension—size—the issue of determining the number of kisses becomes more complicated.

And I took Two and a half centuries Simpler

It began with this famous controversy between Newton and Gregory, which took place in 1694 on the campus of Cambridge University.

Newton was already 51 years old, and Gregory paid a visit that lasted for several days, during which they talked without stopping about science.

The conversation was somewhat one-sided, as Gregory took notes on everything the Grand Master said.

One of the points discussed, and recorded in Gregory’s note, is the number of planets orbiting the sun.

From there, the discussion took off under the question of how many spheres of the same size could be arranged in concentric layers such that they touch a central layer.

Gregory stated – without preamble – that The first layer that surrounds a central ball He was maximum 13 domains.

Newton’s figure The number of kisses was 12.

Gregory and Newton never agreed, and they never knew what the correct answer was.

Today, the fact that the largest number of spheres a power plant can kiss is often called “Newton’s number” reveals who was right.

The debate only stopped in 1953, when German mathematician Kurt Schott and Dutchman BL van der Waerden Show that the number of kisses in Three dimensions I was 12 and he was only 12.

The question was important because a packed group of balls would have an average number of kisses, which helps describe the situation mathematically.

But there are outstanding issues.

Beyond dimensions 1 (intervals), 2 (circles), and 3 (spheres), the problem of kissing is almost unsolved.

There are only two other cases where the number of kisses is known.

In 2016, Ukrainian mathematician Marina Vyazovska proved this number of kisses in dimension 8 is 240 f in dimension 24 He is 196,560.

For other dimensions, mathematicians have slowly narrowed the possibilities down to narrow ranges.

For dimensions greater than 24, or the general theory, The problem is open.

There are several barriers to a complete solution, including computational limitations, but incremental progress on this problem is expected in the coming years.

but, What is the use of grouping spheres of dimension 8, for example?

Algebraic bricklayer Jaume Aguadé answered this question in a 1991 article entitled “One Hundred Years of E8”.

It is used to make phone calls, listen to Mozart on CD, send a fax, watch satellite TV, and connect, via modem, to a computer network.

It is used in all operations that require the efficient transmission of digital information.

“Information theory teaches us that signal transmission codecs are more reliable in higher dimensions and the E8 network, with its amazing consistency and due to the presence of a suitable decoder, It is an essential tool in signal coding and transmission theory.“.

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