[Math] Is there such thing as a “3-dimensional surface”

definitionsurfacesterminology

The reason I'm asking this question: I work at the National Museum of Mathematics and, amidst my sundry duties (which generally have nothing to do with the exhibits), I do have the authority to alter some of the text in the exhibit descriptions, being the only on-site, full-time, PhD-holding mathematician who works there. When I see things that I know to be wrong, I seek to fix them (though it can take a long time to get to it). I thought it would be interesting to, in a case where I'm not sure, look for a consensus among the community in this way.

The usage of the phrase in question refers to a surface embedded in $3$-dimensional space. In particular, an exhibit in which one can view $2$-dimensional solutions to equations in $3$ variables is advertised as: "Bring formulas to life by exploring the multiple number of unusual three-dimensional surfaces they can create."

My opinion is that a surface is, by definition, 2-dimensional, and that the only reason someone would use the phrase "3-dimensional surface" is if they are not familiar with proper mathematical nomenclature. However, I want to know what the community thinks.

Update

Now that I no longer work at MoMath, I can say that I was reprimanded by the Director for this post, who also refused to consider any change to the text in question. For context, a colleague found a typo in someone's last name in other exhibit info, and fixing that was also rejected. People should know that that Museum needs a lot of help, politically (the math errors are the tip of the iceberg). I did my part but more people from the math community should look into this and speak up.

Best Answer

I have seen some authors who use "surface" as an equivalent of "manifolds".(Zorich, _Mathematical Analysis_) Personally I don't like it when someone refers to an $n$-dimensional manifold ($n\neq2$) as a surface; it is contradictory to intuition. But I wouldn't say it is wrong.

However, in your case, I do think it is wrong to confuse a "3-dimensional surface" with a "2-dimensional surface embedded in $\mathbb R^3$". I'd suggest changing it to "Bring formulas to life by exploring the multiple number of unusual surfaces they can create in three-dimensional space". (The audience probably wouldn't notice a thing.)

Situation 1: the description reads "...3-d surfaces..."

general public: "cool."

mathematician: "bad terminology."

Situation 2: the description reads "...surfaces...in 3-d space"

general public: "cool."

mathematician: "cool and rigorous."

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