March 4, 2024

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The Christmas ornaments go back into the box.  How can you do this efficiently in higher dimensional spaces?

The Christmas ornaments go back into the box. How can you do this efficiently in higher dimensional spaces?

If your Christmas ornaments are put away again soon, how will you react? The space in the box is used most efficiently by first layering the Christmas ornaments in a honeycomb grid. Then place the second layer of balls so that each ball falls into a hole in the first layer, and continue repeating this layer after layer.

Mathematicians call the space used the “density” of the beam. If the entire three-dimensional space were filled with equal-sized spheres in infinitely many layers—each layer according to a honeycomb pattern—then the density would be about 74 percent (the exact value is π (pi) divided by the square root of 18). Johannes Kepler in 1611 doubted that this density was optimal. Despite the simplicity of the problem, no proof was found until 1998. It took American Thomas Hills 250 pages full of reasoning and calculations – relying on heavy computer work.

Phitagors theory

Earlier this month, mathematicians Marcelo Campos, Matthew Jensen, Markus Michelin, and Julian Sahasrabudhe developed article Online with a new result about the density of spherical packages in higher dimensional spaces. Fields with dimensions larger than three can no longer be visualized, but can be described mathematically. For the distance between two points on a flat plane, there is the familiar Pythagorean theorem. For example, the distance between a point with coordinates (1, 2) and a point with coordinates (4, 6) is ((4 – 1)2 + (6 – 2)2) = (32 +42) = 25 = 5. In a third-dimensional space or larger, it is done in the same way: the distance between (1, 2, 3, 4) and (4, 6, 15, 88) – two points in a four-dimensional space – is equal to ((4 – 1)2 + (6 – 2)2 + (15 – 3)2 + (88 – 4)2) = (32 +42 +122 +842) = 7,225 = 85.

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Using this higher-dimensional version of Pythagoras, it is possible to describe a sphere with a given center and a given radius, in any dimension: the set of all points that have the same distance (radius) from the center. Mathematicians then wonder how much space these higher-dimensional spheres occupy.

Finding the best packaging for balls of equal size with high dimensions is a very difficult problem. The balls can be stacked in four dimensions at approximately 62 percent density (exactly: π2 Divided by 16). The spheres are located in a four-dimensional grid. It has been shown that no regular stacking is better, but no irregular arrangement has been shown to result in denser packing.

The theory of stacking in higher dimensions is called “completely mysterious.”

Something similar applies to the fifth, sixth, and seventh dimensions: the best known spherical beams have densities of about 47, 37, and 30 percent, respectively. No one expects this to be done more efficiently, but this is unproven.

The exception is the eighth dimension. In 2016, Ukrainian mathematician Marina Viazovska demonstrated that long-suspected stacking is optimal – at a density of 25 percent (exactly: π4 divided by 384) – is actually the best. It earned her the 2022 Fields Medal, the most important award for young mathematicians.

As the dimension increases, the percentage of empty space between the balls also increases. Hardly anything is known about good spherical piles, where the empty space is as limited as possible, and of very high dimensions. Is optimum density achieved through structured stacking? Or specifically because of the stack that shows no regular pattern at all?

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Since each dimension has different obstacles, it is difficult to say anything general about it. It would be great: one ready-made formula that tells you for each dimension the highest density that can be achieved. However, no such formula exists, which is why mathematicians look for lower and upper bounds. The minimum indicates the density that can definitely be achieved – the requirement that it be ideal is no longer required. The easily provable minimum that holds for any dimension n is 1/2n (See Appendix below).

Minimum is better

In 1947, a much better lower bound was found by Briton Claude Rogers. Additional improvements were not significant. Campos, Jensen, Michelin and Sahasrabuddhi have now taken a big step. They proved that for every “sufficiently high” dimension n there exists a population of equal-sized balls with a density “a factor of the order of magnitude of the logarithm of n” better than the density that was the best yet the minimum it used to be.

Their initial version has not been formally peer-reviewed, but experts are excited. Fields Medalist Timothy Gowers

The four mathematicians call the theory of high-dimensional spherical stacks “completely mysterious.” Its spherical stacks are very irregular – there is no symmetrical grid underneath. They always add a new field by means of what is called the Poisson process, a mathematical model that describes the moments when accidental events occur. Renowned mathematician Terence Tao wrote on Mastodon that this “appears to be the first time this method has been successfully applied to the classic ball stacking problem.”

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