What is the ‘kissing problem’ that has plagued mathematicians for centuries – 06/18/2023 – Science

It all started in the 16th century with the famous explorer or pirate (depending on your point of view) Walter Raleigh. But he wasn’t a mathematician nor, as far as we know, did he have a problem with kissing.

What he had were cannonballs and a question: what was the most effective way to stack them to minimize the space they took up on his vessels as much as possible?

It was a math problem — and in math these bullets are spheres and “kiss” (or kiss) can be a way of calling the points where one sphere touches the other.

Raleigh’s question would spawn a mathematical mystery that would populate brilliant minds for hundreds of years.

He posed the question to his scientific adviser on a trip to America in 1585, the distinguished mathematician Thomas Harriot, who gave him a solution: The best way to store your cannonballs was to arrange them in a pyramid shape.

In a 1591 manuscript, Harriot made him a table showing how, given the number of cannonballs, one could calculate how many to place at the base of a pyramid with a triangular, square, or oblong (elongated) base.

But Harriot continued to think about it, and considered the implications for the atomic theory of matter, which was then in vogue.

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

Kepler surmised that the ideal way to minimize the space left by the gaps between spheres was to have the centers of the spheres in each layer above where the spheres at the bottom “kissed”.

This is what is often done with fruit in markets, for example.

This form, which seems so intuitively obvious, has turned out to be extremely difficult to prove mathematically.

Although many have tried, including Johann Carl Friedrich Gauss, “the prince of mathematics”, it was only proven almost four centuries later, in 1998, with the work of Thomas Hales, from the University of Michigan, in the USA, and the power of a computer.

And even this verification did not convince all mathematicians; even today there are those who do not consider it worthy of Kepler’s conjecture — which indicates that if we stack equal spheres, the maximum density is reached with a pyramidal stacking of centered faces.

The unknowns of the spheres

That wasn’t the only headache caused by spherical objects.

In fact, a broad category of mathematical problems is called “sphere-packing problems”.

Solving them served from exploring the structure of crystals to optimizing the signals sent by cell phones, space probes and the internet.

And just like Raleigh with its cannonballs, the logistics, raw materials and many other industries rely heavily on optimization methods provided by mathematics.

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

That 10% represents savings not only in transportation costs, but also in damage to the environment.

But practical applications like this require mathematical proof, and sphere packing has brought up particularly difficult unknowns, as has Kepler’s conjecture.

One arose from a conversation between Isaac Newton, one of the greatest scientists of all time, and David Gregory, the first university professor to teach Newton’s cutting-edge theories.

It was a number of “kisses” problem, but…

What are?

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

The number of “kisses” is equal to the maximum number of circles you can place by “kissing” — or touching — the center one.

That simple.

It turns out that mathematicians have shown that a maximum of 6 circles can be placed around the initial one, so the number of “kisses” is 6.

Now imagine that instead of cardboard circles, you have rubber balls, all the same size.

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

By adding this third dimension, volume, the question of specifying the number of “kisses” becomes more complicated.

And it took two and a half centuries to uncomplicate it.

Newton and Gregory

The issue began with that famous argument between Newton and Gregory that took place in 1694 on the campus of the University of Cambridge in the United Kingdom.

Newton was already 51 years old, and Gregory paid a visit for several days, during which they talked nonstop about science.

The conversation was pretty one-sided, with Gregory taking down everything the great teacher said.

One of the points discussed and recorded in Gregory’s memo was how many planets revolve around the Sun.

From there, the discussion went off on a tangent, to the question of how many spheres of the same size can be arranged in concentric layers so that they touch a central one.

Gregory stated — without much preamble — that the first layer around a central ball had at most 13 spheres.

For Newton, the number of “kisses” would be 12.

Gregory and Newton never reached an agreement and never knew what the right answer was.

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

The debate only stopped in 1953, when the German mathematician Kurt Schütte and the Dutch BL van der Waerden showed that the number of “kisses” in three dimensions was 12 — and only 12.

The question was important because a group of packed spheres will have an average number of “kisses”, which helps mathematically describe the situation.

But there are unresolved issues.

Thousands of kisses

Beyond dimensions 1 (intervals), 2 (circles) and 3 (spheres), the “kiss” problem is almost unresolved.

There are only two other cases where this number of “kisses” is known.

In 2016, Ukrainian mathematician Maryna Viazovska established that the number of kisses in dimension 8 is 240, and in dimension 24 it is 196,560.

For the other dimensions, mathematicians slowly whittled the possibilities down to narrow bands.

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

There are several obstacles to a complete solution, including computational limitations, but the expectation is that there will be an important advance in this problem in the coming years.

What’s the point, though, of packing 8-dimensional spheres, for example?

The algebraic topologist Jaume Aguadé answered this question in a 1991 article entitled “One Hundred Years of E8”.

“It’s used to make phone calls, listen to Mozart on a CD, send a fax, watch satellite television, connect via a modem to a computer network.”

“It serves for all processes in which the efficient transmission of digital information is required.”

“Information theory teaches us that codes for transmitting signals are more reliable in higher dimensions, and the E8 lattice, with its surprising symmetry and given the existence of an appropriate decoder, is a fundamental tool in encoding and transmission theory of signs.”

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