Dimension of a sphere?

Finding dimension of sphere has always been a bit confusing. If I see a sphere it feels like 3D instinctively. Is that so ?

After this post we will be crystal clear on dimension of a sphere.

This is an extension to the post : What is dimension of space? Please take a look at that post before continuing on this further.

Keep in mind definition of dimension:

In mathematics, dimension of a space is informally defined as the minimum number of coordinates, values, numbers required to specify an exact position in that space.

NOTE: Will use sphere, sphere space, hollow sphere space interchangeably all are same. If I don't specify solid sphere explicitly its a hollow sphere.

Now lets go step by step to find the dimension of a hollow sphere.

Dimension of hollow sphere

To locate any point on the orange circle we need angle w.r.t. reference point say theta1 for point A, -theta2 or 360-theta2 for point B. We need just one value(angle) to refer a point on the circle.

Now lets try to define a point G lying on a random green circle.

Lets move angle thetaG horizontally w.r.t. reference blue line such that

Lets move angle alphaG vertically w.r.t. reference orange circle plane such that.

So we need 2 values one horizontal angle(ThetaG) and one vertical angle(alphaG) to define the position of G.

Lets take another random point M(M is toward you) and try defining its position.

Move thetaM angle horizontally w.r.t. reference blue line

Move alphaM angle vertically w.r.t. reference orange circle plane

Again we used 2 values one horizontal angle(ThetaM) and one vertical angle(alphaM) to locate M.

In the same way we can pick any point on the sphere(Ex. L,K,Q). Using 2 values horizontal angle and vertical angle we can define their positions. So the dimension of a hollow sphere space is 2.

Dimension of solid sphere

Now lets take a solid sphere. If you think in terms of hollow spheres. Its the collection of multiple hollow sphere of different radius.

Its the collection of multiple hollow sphere each of different radius. Now we know to define a point in a hollow sphere we need 2 values. As there are multiple hollow sphere each of different radius. We need to specify the radius to locate the hollow sphere then 2 values i.e horizontal and vertical angle to locate a specific point in that hollow sphere. So we need 3 values: radius, horizontal angle, vertical angle to locate a specific point in a solid sphere. Solid sphere is a 3Dimentional space.

Note: Be vary of sequence of values you use to locate a point. (ThetaM,alphaM) first value(thetaM) specifying horizontal angle and second value(alphaM) specifying the vertical angle. Because the same point can be referred as (alphaM, thetaM) where first value specifies the vertical angle and second value specifies the horizontal angle.

In case of any doubt or queries or any suggestions please leave a comment below. Would love to help.