It was argued at "Are there mathematical objects that have been proved to exist but cannot be described in words?" that there are mathematical objects that we can't describe in words because there are only countably many ways to arrange words and letters.
My follow-up question: If we describe mathematical objects by drawings (subsets of $\mathbb{R}^2$), could we then describe every mathematical object?
The answer is essentially the same as for the question you link. The only difference is that instead of arguing by countability you need to step up the size of the elements you care about two levels. Assuming a "drawing" is any subset of $\mathbb{R}^2$ then there are exactly $2^\mathfrak{c}=2^{|\mathbb{R}|}$ many distinct drawings.
Now we just need to look at the subsets of the powerset of $\mathbb{R}$. That is $\mathcal{P}(\mathcal{P}(\mathbb{R}))$ which has size $2^{2^\mathfrak{c}}$ which is strictly bigger then $2^{\mathfrak{c}}$.
edit As an aside another problem is that if you are thinking of drawings in the real world then you're actually much worse off. Since any real world drawing will actually be finite as far as I understand current physics since you have some finite minimal distance and only finitely many graphite atoms to make the drawing out of. All together you are no better off than when using words.
