Is there such thing as a "3-dimensional surface"?

Solution 1:

I have seen some authors who use "surface" as an equivalent of "manifolds". 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."

Solution 2:

Although as a mathematician I would never use that terminology, your target audience is probably non-mathematicians. Non-mathematicians do understand "three-dimensional surface" better than "embedding of a surface in three dimensions". I think I would leave the text as it is, except replace "multiple number of" by "many".