![]() You can come up with notions for distance and volume in analogy to two and three dimensions, and you can define shapes, such as spheres, using equations. ![]() Points in -dimensional space are given by coordinates. In higher dimensions the same principle applies. If you can't visualise how shapes relate to each other, you use the equations to help you. Points in three dimensions are given by three coordinates, and shapes such as lines, planes or spheres by equations involving the coordinates. If you were one of the many people who struggle visualising things in 3D, you'll have appreciated the help of algebra. To get a grip on higher dimensions, cast your mind back to school when you moved from doing geometry in two dimensions to doing it in three. Higher-dimensional sphere packings are important in communications technology, where they ensure that the messages we send via the internet, a satellite, or a telephone can be understood even if they have been scrambled in transit (find out more here). ![]() The endeavour might seem both useless and impossible to get your head around, but it's neither. They involve packing higher-dimensional spheres into higher-dimensional spaces, namely spaces of dimensions 8 and 24. Pages, Viazovska's proofs are as solid as proofs can Involving no computers and filling just a few This is the face-centred cubic packing, one of the packings with maximal density in dimension three. A panel ofĮxperts eventually decided they were 99% sure that it did, and since then the proof has been verified using formal computer logic. Since no human could possibly check the computerĬalculations in their life time, mathematicians weren't sure if Hales' work really counted as a proof. The proof that you really can't do any better than this only arrived inģ gigabytes of computer code and data for making necessary calculations. The density is about 74%." The precise value of the sphere packing constant in three dimensions is One of them you can see on the market, where oranges are stacked in pyramids (see the figure below, which illustrates this with balls rather than oranges). " we don’t have only one best packing, we have many equally good packings. "In dimension three was known as Kepler’s conjecture and remained open for years," says Viazovska. "This way we will cover slightly over 90% of the area with these equally-sized discs." The precise value of the sphere packing constant in two dimension is If you arrange discs in the same pattern you do get gaps, but the packing turns out to have the highest density possible. In a traditional honeycomb every cell is a hexagon and the hexagons fit neatly together leaving no space between them. "In dimension two the best packing honeycomb," explains Viazovska. " The sphere packing problem is to find this highest proportion, also called the sphere packing constant.įor an easier example, let’s drop down a dimension: instead of packing spheres into 3D space let’s pack discs into 2D space. "It's intuitively clear, though mathematically one has to work a little bit to see this, that there is a maximal possible. But if the box is very large, the effect of the shape is negligible, and the answer depends only on the volume of the box. If the box is small, then the answer depends on the shape of the box. We put as many spheres as we can into the box." The question is, what's the largest number of spheres you can fit in? "To make the problem easier suppose the spheres are of equal size and also hard, so we cannot squeeze them. "Suppose you have a very big box and a supply of spheres," Viazovska explains. (You can also listen to the interview in a podcast.) The problem During a coffee break she gave usĪ condensed description of sphere packing problems and her work. Legendary mathematician Srinivasa Ramanujan, who was elected a fellow Viazovska who in 2016 made a breakthrough in the theoryĮxplained her results at the Royal Society's celebration of the This month we met an expert in the field: the mathematician Maryna Question: how should you pack spheres into a box to make sure the volume Even the simplest possibleįruit shape, the sphere as seen in oranges and apples, causes a problem, because no matter how you pack spheres, ![]() Not only are theyĮasily squashed, they also have shapes that don't lend When it comes to transportation fruit are awkward.
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