Sphere

A sphere comes from the Ancient Greek word (sphaîra) meaning 'ball'. It is a special shape in three-dimensional geometry, similar to how a circle looks in two dimensions. Imagine all the points that are the same distance from one central point in space — that distance is called the radius, and the collection of all those points forms a sphere.

Spheres are very important in many areas of mathematics and appear often in nature and everyday life. For example, bubbles like soap bubbles naturally form spheres when they are calm. Our Earth is often thought of as a sphere when we study geography, and astronomers use the idea of a celestial sphere to understand the night sky.

Many human-made objects are based on spheres too. Things like pressure vessels, curved mirrors, and lenses often have spherical shapes. Because spheres can roll easily in any direction, most balls used in sports and toys are made this way, as well as ball bearings that help machines move smoothly.

Basic terminology

A radius is any line from the center of a sphere to a point on its surface. If you extend this line through the center to the opposite side, you get a diameter, which is the longest line you can draw between two points on the sphere. The diameter is always twice as long as the radius.

A unit sphere has a radius of exactly 1. Often, the center of a sphere is placed at the starting point of a coordinate system for easy calculations.

On a sphere, a great circle has the same center and radius as the sphere itself and divides it into two equal parts called hemispheres. Even though Earth isn’t a perfect sphere, we can use terms from geography to describe spheres. For example, an imaginary line through the center can act like Earth’s axis, with points called poles. The great circle in the middle, far from the poles, is like an equator. Circles around the sphere parallel to this middle circle are like lines of latitude.

Equations

A sphere can be described using special math rules. If you know the center point of the sphere (called x₀, y₀, z₀) and its distance from the center (called the radius r), you can find every point on the sphere. This is done with a simple rule:

( x − x₀ )² + ( y − y₀ )² + ( z − z₀ )² = r².

This rule helps us understand how spheres work in math and science.

The symbols used here are the same as those used in spherical coordinates.

Properties

Enclosed volume

In three dimensions, the space inside a sphere (called a ball) can be measured. If the sphere has a radius of r and a diameter of d, the volume V is given by:

V = 4/3 π r³ = π/6 d³

This means that the volume inside a sphere is a special number (about 0.5236) times the width of the sphere cubed.

Surface area

The outside area of a sphere can also be measured. If the sphere has a radius of r, the surface area A is:

A = 4 π r²

Other geometric properties

A sphere can be made by spinning a circle around one of its lines. Spheres are special because every point on them is the same distance from the center. They also have some unique shapes and properties.

Properties of the sphere

Here are some special things about spheres:

All points on a sphere are the same distance from its center.
Slices of a sphere look like circles.
Spheres have special shapes that stay the same no matter how you look at them.
Every point on a sphere behaves the same way.
Spheres don’t have special center points like some other shapes.
The shortest paths on a sphere are big circles.
Spheres are the shapes with the least outside area for their size, and the most space inside for their outside area.
Spheres have a special kind of curve that stays the same everywhere.
Spheres have the same kind of curve everywhere.
Spheres have a special bending that stays the same everywhere.
Turning a sphere around will always leave it looking the same.

Treatment by area of mathematics

Spherical geometry

Main article: Spherical geometry

In spherical geometry, points are used just like in regular geometry. But instead of straight lines, we use special paths called great circles. These are the biggest circles you can draw on a sphere, and they go through the center of the sphere. The shortest way to get from one point to another on a sphere is to follow the smaller part of a great circle between those points.

Some ideas from regular geometry still work on a sphere, but not all of them do. For example, on a sphere, the angles of a triangle always add up to more than 180 degrees. Also, any two triangles that have the same shape on a sphere will also have the same size.

Differential geometry

A sphere is a smooth shape where every part curves the same way. This curving is linked to the size of the sphere. Because of this, we can’t flatten a sphere onto a flat surface without changing sizes or angles, which is why maps of the Earth look the way they do — some parts get stretched or squished.

Topology

It’s possible to turn a sphere inside out in three-dimensional space without tearing it or creasing it, though it might cross over itself during the process.

Curves on a sphere

Main article: Circle of a sphere

Main article: Rhumb line

Main article: Clélie

Main article: Spherical conic

Circles on a sphere are similar to circles on a flat surface. They are made up of all points that are the same distance from a fixed point on the sphere. When a sphere meets a flat surface, the shape formed can be a circle, a single point, or nothing at all. Special circles called great circles happen when the flat surface goes through the center of the sphere.

In navigation, a special path called a loxodrome is used. This path keeps the same angle compared to north no matter where you are. These paths look like straight lines on certain maps. There are also paths that go around the poles in a spiral shape.

Another interesting shape on a sphere is called a Clelia curve. This curve moves in a way that its position around the sphere and its distance from the top are connected in a simple way. One famous example is Viviani's curve.

Spherical conics are shapes on a sphere that are similar to conic sections, which are curves you can find on flat surfaces. They can be described in a few different ways, such as where a special cone meets the sphere.

When a sphere meets another shape, like a cylinder, the lines they make together can be more complex than simple circles. This happens when the equations that describe both shapes are solved together.

Generalizations

Ellipsoids

An ellipsoid is like a sphere that has been stretched or squished in one or more directions. It is made by changing the shape of a sphere using special math rules. An ellipsoid relates to a sphere in the same way that an ellipse relates to a circle.

Dimensionality

Main article: n-sphere

Spheres can also exist in spaces with more than three dimensions. For any whole number n, an n-sphere, written as _S_‍ⁿ, is a set of points in (n + 1)-dimensional space that are all the same distance r from a central point. Here are some examples:

_S_‍⁰: a 0-sphere is just two points
_S_‍¹: a 1-sphere is a circle
_S_‍²: a 2-sphere is a regular sphere
_S_‍³: a 3-sphere is a sphere in four-dimensional space

Spheres in more than three dimensions are sometimes called hyperspheres.

The ordinary sphere we know is a 2-sphere because it is a flat surface in three-dimensional space.

Metric spaces

Main article: Metric space

In a special kind of space called a metric space, a sphere is made by picking a center point and a distance r, then finding all points that are exactly distance r from the center.

If the center point is a special point called the origin, we don’t need to mention it in the rules. The same goes for the distance if we always use one unit.

Even a big sphere might sometimes have no points at all, depending on the space. For example, in a space made of whole numbers, a sphere only exists if its radius squared can be made from adding up squares of whole numbers.

In some special geometries, shapes like an octahedron can act like a sphere, and a cube can also be thought of as a sphere.

History

The ancient Greeks were very interested in studying spheres. Euclid wrote about spheres in his famous work, Elements, describing their properties and how to fit special shapes inside them. However, he did not give formulas for the area or volume of a sphere.

Later, Archimedes figured out the exact formulas for the area and volume of a sphere using a clever method. He also discovered that among all solids with the same surface area, the sphere holds the most volume.