“In the old days, I was very happy because I was only given oranges.” This is a phrase we sometimes hear when older people criticize the large amount of fancy gifts that today’s children receive. One thing that is rarely mentioned is the gift wrap. Suppose you want to give 5 oranges as a gift. How would you arrange the fruit to use as little space and wrapping paper as possible?
After all, there is a lot of mathematics behind this seemingly harmless question. After all, he proves what the fruit trader knew from time immemorial: that the optimal stacking of infinite balls in three-dimensional space can be achieved by arranging the balls in a pyramid. It took him over 400 years. A validated solution to that puzzle, known as the Kepler conjecture, was not published until 2017. However, the situation is quite different when only a finite number of objects are considered.
Surprisingly, mathematicians did not address the latter kind of problem until the late nineteenth century. In 1892, Norwegian geometer Axel Too first studied the optimal arrangement of a finite number of two-dimensional circles. It was not until the next few decades when the Hungarian mathematician Larszló his Fehes his Thoth tackled the subject that significant progress was made in this area.
Optimal arrangement of circles on the plane
To better understand the problem, it is helpful to first consider the simplified two-dimensional case. For example, you can try to arrange multiple coins of the same size in a way that saves space as much as possible. To do this, draw an outline with a string, pull it tight and calculate the area that the string encloses.for n = 2 coins, the best placement is found quickly. Place them in contact with each other.Shortest string that radiates both coins r It will have length (4 + 2π).r.
This length is optimally calculated for each section. Add a straight part of the string (4 x r) and the sum of the round areas surrounding the circle (2π)r). The string encloses a total area of (4 + π).r2. In this case, there is obviously no more space-saving way to place the coins.
On the other hand, if you have three coins, suddenly you have two different arrangements that seem to save space. One is to place the coins either horizontally or along the corners of an equilateral triangle. In the first case, the string is sausage-shaped, so in mathematics it is called a “sausage” pack. The second case is referred to by experts as a “pizza” pack. But which one saves more space, packing sausages or packing pizza?
After all, pizza packs are better.This string is of length (6 + 2π)the law of nature, The area covered is correspondingly (6 + √ 3 + π).r2On the other hand, the cord of the sausage pack is (8 + 2π)r It is long and encloses a region of (8 + π).r2. If you look closely, you can also see this difference directly from the photo. The space between the coins in the sausage arrangement is larger than in a pack of pizza.
Pizza always wins in 2D
In fact, a general formula can be given for required string length and limited space.if someone arranges n Sausage-shaped coin. A string of length 4 is required (n – 1 + 2π)the law of nature, This is 4(n – 1)r2 +pr2 . On the other hand, if the coins are arranged along a triangular lattice that is as close to a regular hexagon as possible, all you need is a string of length 2 (n +p)r (surrounding area 2)n + √ 3(n – 2) + p)r2.
Therefore, a pack of pizzas turned out to be more space efficient than a sausage shape for any number of pieces. n Circle.but is it true everytime Optimal? Determining that is a much more difficult task. After all, there can be completely chaotic arrangements of circles occupying an even smaller area. It turns out that such cases are very difficult to rule out. This is where the Hungarian mathematician Laszlo Fejes Thoth comes into play. He speculated in his 1975 that the following optimal packing is possible. n A circle is an arrangement of triangular lattices forming hexagons that are as regular as possible.
In 2011 mathematician Dominic Ken was able to show that this idea holds true for nearly all values of n. And indeed, the limited case of covering an infinite plane with an infinite number of coins can also be proved. In 1773, the physicist and mathematician Joseph Louis Lagrange discovered that placement along a triangular lattice is optimal as long as only regular packing is considered. It wasn’t until 1940 that Fejes his Thoth finally showed that this solution was more space efficient than a chaotic circle arrangement.
When sausage takes the lead
But what about spheres? Perhaps it should come as no surprise that the 3D case raises even more questions than optimal circular packing in the 2D world. We have at least one clue to get you started. Kepler’s conjecture states that three-dimensional space can be best filled by stacking an infinite number of identical spheres like cannonballs. The first level places them along a triangular grid like coins in the 2D case, and the second level places a sphere in each gap. The third level is again identical to his first level, and so on. (In other words, these spheres look like the pyramidal piles of oranges in the grocery store.)
However, the situation is quite different when only a finite number of spheres are considered. Now, back to the orange wrapped in wrapping paper example. If you only have one or two oranges, you’ll quickly know how to best place them. With three, the task becomes more complicated. You can line them up (sausage packs), or form triangles (pizza packs) as before. The situation is similar with the three coins, but we’re only dealing with spheres. In this case, compare the volume of the arrangement to find out which pack saves the most space.
First, it helps to decompose the sphere’s shell again into individual geometric shapes and sum their volumes. For sausage packs, this is very simple.The shape can be divided into cylinder and sphere, whose total volume is 16⁄3Pi r3 ≈ 16.76r3. Pizza packs are a little more complicated. We get three semi-cylinders, a triangular prism and a sphere, whose total volume is: 13⁄3Pir3 + 2√3r3 ≈ 17.08r3. So sausage packs are much more space efficient in this case. And it turns out that the sausage arrangement can actually be optimally stuffed, after all.
Add another sphere, n = 4, three different arrangements can be distinguished. Again, balls and oranges can be arranged one after the other (sausage) or distributed on a plane (pizza). However, you can also stack them using all three spatial dimensions. This arrangement is called a “cluster” pack. It turns out that even with 4 balls, the sausage pack that requires the least volume is the best.
However, the situation becomes more complicated when the number of spheres increases. Mathematicians speculate that sausage packs are best when: n = 55 balls. However, in 1992 mathematicians Jorg Wills and Pierre Mario Gandini decided that a cluster pack would save him space for 56 spheres. However, it is unclear what exactly this cluster looks like. Mathematicians were able to find a better arrangement than a sausage pack of balls, but could not show that it was optimal. There may be other arrangements that occupy even less volume.
The abrupt transition from ordered one-dimensional chains to three-dimensional clusters is known among experts as the “sausage catastrophe.” Wills and Gandini proved that arrangements containing 59, 60, 61, or 62 spheres, and all collections containing at least 65 spheres, also cluster optimally. For all other amounts, i.e. n If is less than 56 or is 57, 58, 63, or 64, the sausage pack is considered optimal. That is, sausage packs are best for up to 55 balls, cluster packs are best for 56 balls, and sausages are still the most space-saving arrangement for 57 or 58 balls. will be After 59, 60, or 61 spheres, it’s back to the cluster again.
The answer doesn’t seem particularly intuitive. And no one could prove it beyond a doubt.
A Visit to the 42nd Dimension
A mathematician would not be a mathematician if he stopped in three dimensions. So what is the best packing? n What does a four-dimensional ball in four-dimensional space look like? d A distinction is made between sausages (one-dimensional chains) and clusters (whole balls). d-dimensional space) and pizza packaging. The latter represents a sort of transition from her two other cases. This includes all situations where more than one or less than one sphere is scattered. d size.
After all, the sausage catastrophe seems to exist in the 4th dimension as well, but it happens much later than in the 3rd dimension. Gandini and his colleague Andreana Zucco proved in his 1992 that: d = 4, with at least one, cluster packs take up less space than sausage packs. n = 375,769 balls.
how about pizza? Wills and mathematicians Ulrich Betke and Peter Gritzmann showed in 1982 that pizza is not the optimal package for him in three and four dimensions. The balls either fill the entire space (cluster) or form rows (sausage). Only these two extreme cases can lead to optimal packing arrangements.
In 1975, Fejes Thoth expressed the now-famous “sausage conjecture” of higher dimensions. According to him, his pack of sausages is perfect for any finite number of spheres with dimensions greater than or equal to five. Although this conjecture has not yet been conclusively proven, Betke and his colleague Martin Henk were able to show in his 1998 that the sausage conjecture holds for his 42 or more spatial dimensions.
So if you’re handing out 42 dimensional oranges for Christmas, it’s best to line them up. As in the original question, sausage-style wraps are perfect if you’re only gifting 5 of these three-dimensional fruits.
Imagine how complicated that task would be if you wanted to pack dinosaur figurines or dolls instead of oranges! Gift wrapping is clearly an area of mathematical mystery.
This article was originally published on science spectrum Reprinted with permission.