Sphere how many dimensions

Solving the sphere-packing problem in dimensions amounts to figuring out how many codewords we can pack into some defined region in dimensional space. The more efficiently we can pack codewords, the more distinct messages we can send using our dimensional message scheme. One might justifiably ask how important solving the sphere-packing problem is for finding error-correcting codes. Is pretty good good enough? High-dimensional spheres get spiky. Hence it is all the more important to find the best sphere packings so we can get as much as possible out of or into?

Viazovska's breakthrough on sphere packing in eight dimensions—and the subsequent solution to the dimensional problem—did not have a practical effect on creating error-correcting codes using those dimensions. Researchers already knew the correct answer to within minute fractions of a percent, so anyone packing spheres in those dimensions was already using the right model. But her work is a proof of concept that a new approach to sphere packing has the potential to make a difference in other dimensions and have a big impact on how we communicate with each other.

The views expressed are those of the author s and are not necessarily those of Scientific American. Follow Evelyn Lamb on Twitter. Already a subscriber? Sign in. Thanks for reading Scientific American. Create your free account or Sign in to continue. See Subscription Options. Go Paperless with Digital. Image: Evelyn Lamb. Get smart. There are many real-world objects that we see around us which are spherical in shape. Since a sphere is a three-dimensional shape it also has a volume and a surface area.

Our planet Earth is not in a perfect shape of a sphere, but it is called a spheroid. The reason it is called a spheroid is it is almost similar to that of a sphere. In geometry , a sphere is a three-dimensional solid figure, which is round in shape. From a mathematical perspective, a sphere is a set of points connected with one common point at equal distances.

Some examples of a sphere include a football, a soap bubble. The important elements of a sphere are as follows. As we discussed in the previous section, a sphere has a radius, diameter, circumference, surface area, and volume. Considering a sphere to have a radius of 'r', the following table lists the important formulas of a sphere.

A sphere is a three-dimensional object that has all the points on its outer surface to be equidistant from the center. The following properties of a sphere help to identify a sphere easily. They are as follows:. A circle and a sphere are two different shapes.

The important differences between a circle and a sphere are as follows:. The area covered by the outer surface of the sphere is known as the surface area of a sphere. The surface area of a sphere is the total area of the faces surrounding it. The surface area of a sphere is given in square units.

Hence, the formula to find the surface area of a sphere is:. Check out the surface area of the sphere section for more details. The volume of a sphere is the measure of space that can be occupied by a sphere. We can determine the volume of a sphere if a string runs along the diameter of a circular disc and when rotated along that string. The unit of volume of a sphere is given as the unit 3.

There are two kinds of spheres - solid sphere and hollow sphere. The volume for both these is different. Likewise a surface can only live in two dimensions if it is flat. Surfaces exist in many dimensions, but they look like a plane when you take a small enough piece. If we are creating a coherent useful idea of dimension, it turns out that we want a curve to be one-dimensional and a surface to be two dimensional - but what do we mean by a curve or a surface if we have many-dimensional space?

The idea of a manifold captures this - if our curve is locally like a line, and the local pieces can be coherently glued together to make the whole, it is a one-dimensional manifold. Likewise a surface is a two dimensional manifold if it looks locally like a plane wherever you look and you can glue the local pieces together in a coherent way.

There are some technicalities about precise definitions, to make these intuitive ideas rigorous. But that is why the dimensions come out as they do. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Ask Question. Asked 7 years, 5 months ago. Active 7 years, 5 months ago.

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