# Geometrically, "face" or "surface"?

In my native Danish language we can refer to a 2D side of a geometric object as `flade`. We can be more specific in case of the 2D side being exposed to the outside and say: `overflade`, a word that also typically refers to the total, closed surface of an object. I would translate these terms to:

When dealing with mathematics I have come to realise that the word `face` maybe doesn't exist as a geometric term? My peers prefer to always say `surface` and thus remove the distinction between `flade` and `overflade`. But `flade` has many separate uses in Danish, such as when saying `et bords flade` (`a table's face`), `håndflade` (`the face of the hand`, so `palm of the hand`) and more.

I feel that different dictionaries give different answers to whether the term `face` is used for geometry in English. Some do and some don't include the geometric meaning. So, my question is: does the term `face` exist and should I use it within geometry/mathematics, or should I stick to always say `surface`?

The word "face" is correct in a geometrical context, for example "a cube has six faces, twelve edges, and eight vertices". In this context "surface" often refers to the entire exterior of the solid figure, so the surface area of the cube is six times the area of a single face. The word "surface" is more commonly used in a non-technical context; one would speak of the surface of water or the surface of a table.

• Another distinction is that a face is normally flat while a surface can be curved or have edges. Commented Jul 6, 2022 at 19:23
• Thank you for the answer, Peter. To @StuartF, what you are saying here is actually not aligned with mathematical use of the Danish terms then. Specifically, `en flade` (which would translate to `a face`) can have any shape as long as it is 2D. The special case of a totally flat face is called `en plan flade` or simply `et plan` (translates to `a plane face` or `a plane`). Commented Jul 6, 2022 at 21:18

These things can have slightly different meanings depending on context.

Often, the word "face", in mathematics, refers to the flat sides of polyhedra in 3D space, just as Syntax Junkie says. This echoes Peter's example: "a cube has six faces, twelve edges, and eight vertices". Occasionally, you might hear of someone referring to a "curved face" (e.g. the part of a cylinder that are not the two circular bases might be considered a curved face), but this is rare and non-standard.

If you go deeper into the geometry of convex sets, you can find a more general definition of "face". I won't go into it specifically here (as I am not sure about your mathematical background), but the "six faces, twelve edges, and eight vertices" of the cube turn out to all be faces under this definition, as well as the entire cube itself. The "six faces", as we were previously calling them, would be called "facets" instead.

From what I can tell (I don't speak Dutch), when you use the word "surface", you might be better using the word "boundary". The "surface of a ball", for example, would be understandable, but isn't idiomatic in mathematics. "Boundary" is a perfectly standard term in topology, and I would use (and have used) the phrase "boundary sphere of the ball".

The word "surface" usually refers to the graph of a continuous two-variable real function, or more generally, a manifold of dimension 2. "Surface" is a perfectly sensible noun in mathematics, without having to ask "surface of what?".

All these recommendations aside, it's important to remember the context. What I have said may not be appropriate in specialist fields. If you use "boundary" and "surface" as I have suggested, you will certainly be understood, but terminology can be non-standard, even within the English-speaking mathematics community itself.

I would agree with the defintions you proposed in the question.

One complication is that mathematics (in this case geometry) sometimes uses words in ways that can be different from everyday contexts. Depending on the type of dictionary you are using, you may be getting misleading definitions.

In geometry, a face is a sub-type of surface. A surface is any kind of two-dimensional expanse. While a "face" is a surface specifically in the context of a polyhedra.

To quote the English version of Wikipedia, "In elementary geometry, a face is a polygon on the boundary of a polyhedron." You may find other defintions, but the one from Wikipedia matches my training and experience.

In the context of shapes (such as a cube) it would be correct to refer to one side as a "surface" or a "face." You could also call the entire 2D exterior of the cube a "surface." But when referring to the flat, exposed part of (say) a still lake, that would only be referred to as a "surface." Or the two-dimensional exterior of a sphere would only be called a "surface," not a "face."

• Thank you, SyntaxJunkie. I can hear on you (and on other comments) that although you seem to accept the term `face` geometrically to some degree, it still has a different meaning that my Danish `flade`. In particular, `en flade` could be any shape (it just has to be a 2D extent of continuously connected points), not necessarily flat. A flat `flade` would be a special-case which could be called `en plan flade` or simply `et plan` (`a plane face` or `a plane`). ... Commented Jul 6, 2022 at 21:32
• ... So, in Danish, a `flade` is not a polyhedron which by definition is flat. In this Danish sense, the word `flade` (`face`) is thus the top-level and words like `plan` (`plane`) and `overflade` (`surface`) would be sub-types that are slightly more restricted. If this is not the case in English, then I don't think I should use the word `face` as a direct replacement - it might be a safer choice with `surface` as I have been recommended. Commented Jul 6, 2022 at 21:35

I think they can still be used in very distinct ways.

'Face' is a broader, more abstract term: while it can be synonymous to 'surface', in a narrower sense it can indicate a link between points, and it doesn't tell us a lot about the nature of that link.
(An interesting analogy in this context is the word 'facet': "a particular aspect or feature of something" * — it can similarly be used in a more abstract sense. Both words come through French from the Latin 'facies', which can also mean 'appearance', emphasizing its virtual character.)

A surface is a physical boundary that has specific properties.

What follows is that a face can have a specific kind of surface.

• This answer doesn't explain what the words mean in the context of geometry, which is what the question asked.
– MJD
Commented Jul 6, 2022 at 19:56