Is a sphere 3 dimensional

Answer 1

A half sphere is not 3-dimensional. Neither is a sphere. They're both 2-dimensional.

A sphere is a 2-manifold in 3-dimensional space. Moreover, since a sphere essentially has dimension 2, we cannot rightfully call a sphere (or a half sphere) 3-dimensional. It is, however, a subset of 3-dimensional space.

For instance, we can take a 0-dimensional figure (like a point) and drag it so that it takes the form of a line (which has dimension 1). This does not mean that our "dragged point" is suddenly 1-dimensional. It's a subset of one-dimensional space.

Similarly, if we twist or bend a 1-dimensional figure (such as a line) and create a circle, we do not have a 2-dimensional object. A circle is by no means 2-dimensional; it is 1-dimensional just like a line. The difference between a line and a circle is that the former is a 1-manifold in 2-dimensional space, while a line is not. Both, however, are 1-dimensional objects.

At first it may seem difficult to distinguish between an n-dimensional object and a topological manifold existing as a subset of n-dimensions.

In particular, a sphere is a 2-dimensional subset of 3-dimensional space. This premise may be extended to include the "half sphere" which is what the question asked about.

Let's re-visit the example of a line being bent into a circle.

A circle is a 1-dimensional figure placed in 2-space. If we were to infinitely zoom-in on any region along that said circle, we find that it begins to look like a line (again). This is because a circle is one-dimensional.

Similarly, a sphere is a 2-dimensional plane that resembles the "skin" of a ball, it does not resemble the ball itself. A ball, which is dense (i.e. it has length, height, and width), has dimension 3.

By definition, a sphere is defined to have ALL of its points equidistant from one reference point in 3-dimensional space. Namely, this point is denoted by (x, y, z) in 3-dimensional space and represents the coordinates of the sphere's center. Again, the sphere itself is a 2-dimensional manifold in 3D and is thus labeled a "surface".

Surfaces (there are different types) are two-dimensional manifolds like spheres, tori, and paraboloids which are extended out to a third dimension without losing their 2-dimensionality. 3-dimensional objects like cubes, bricks, and cylinders, on the other hand, are solids. There is a vast difference between 3-dimensional solids and 2-dimensional surfaces that are "extended" outward to a third dimension.

Holding a 2-dimensional square object in front of you (like a Poker card) and extending it in the direction of a third dimension (like toward you for instance) so as to create width generates a three dimensional cube.

Now, imagine extending a 3-dimensional object outward to a fourth dimension.

For example, imagine pulling your entire room and 3D environment in the direction of a FOURTH dimension. Your entire 3D existence would be a 3-dimensional manifold in 4-dimensional space. Basically, a subset of 4-space. Similar to what a circle is in our 3D world. You wonder why so many mathematicians go insane?

It doesn't necessarily stop at 4-dimensions. What about 5-dimensions? 12-dimensions? 77-dimensions? 109-dimensions? Etc. Perplexity is not even the word. You should explore fractals. It gets even more complicated (but interesting). Imagine having negative dimensions like (-1)-dimensional space (also known as the nullity space). Imagine dimensions that are not integers such as 2.47778-dimensional space or log(3)/log(2) - dimensional space.

So the answer to your question is no. A sphere is 2-dimensional in 3-dimensional space. A half sphere still 2-dimensional in 3-dimensional space.

A SIMPLER WAY OF ANSWERING THE QUESTION:

Exploring the notion of "dimension" at a more rudimentary level might help us make better sense of it all.

In plain English:

A point = object in 0-dimensional space

A line = object in 1-dimensional space

A square = object in 2-dimensional space

A cube = object in 3-dimensional space

A tesseract = object in 4-dimensional space

4-dimensional space is difficult to fathom for some, so we will only concern ourselves with figures in n-space, where n = 0, 1, 2, and 3.

When looking at a line, there is only one dimension. When dealing with circle, some people are prone to mistakenly assigning two dimensions- length and width. This is not the case, however.

A circle is essentially generated by flexing and bending a line. And like this line, there is only one dimension in a circle. Namely, the radius.

And so we call a circle a 1-dimensional manifold in 2-dimensional space.

A square, on the other hand, is in fact 2-dimensional because it has 4 corners. These corners give squares their width and length. Hence, the 2-dimensions.

Now, imagine a 2-dimensional figure like the square we just talked about. So long as something has length and width, it can be categorized as 2-dimensional.

So... take a randomly cut piece of fabric and wrap it around a Basketball so that it fits nicely. Now, imagine taking the Basketball out from underneath the fabric without ruining the spherical shape of our fabric. So, at this point the fabric "looks" like a ball, but has emptiness underneath. It still has an origin, but within the said emptiness. In the end, our fabric is still a 2-dimensional object... except it's folded or wrapped in such a way that makes it seem as if it is 3-dimensional even though it is actually 2-dimensional. This is why it may be difficult for some to see a sphere as 2-dimensional.

A ball is 3-dimensional, because unlike the fabric with emptiness inside, a ball is dense and has length, height, AND WIDTH. This is why 3 dimensional figures can be placed on a 3-coordinate system. The 3-dimensional "setting" (simply referred to as space) of our above-mentioned fabric can be positioned onto a 3-coordinate system too because it is also 3D (like the ball).

But the wrapped up fabric (also known as a sphere) is still only a 2-dimensional object in 3-dimensional space. This is interesting because some may call the sphere itself 3D. Which brings us to our conclusion:

A sphere is two-dimensional, but it is commonly referred to as a two-dimensional topological manifold in 3-dimensional space (or simply 3-space).

The question, is a half sphere 3 dimensional?

No. Because that would represent half of our fabric. It doesn't change the 2-dimensionality. Even 1/3 of a sphere, 1/4 of a sphere, etc. are NOT 3-dimensional.

NOTE: We could have used something other than fabric, like paper, metal, leather, etc. so long as the inside is not dense (or pressurized). In which case it would be called a ball (which is certainly 3-dimensional).