# What is the meaning of these pictures?

Spotted on a professor's whiteboard on their office door. • It's very nice! – Michael Rozenberg Oct 27 '17 at 14:31
• The meaning of that ? Mathematics makes us smile :-) – ama Oct 27 '17 at 14:40
• I can't figure out why this already has a "close" vote – caverac Oct 27 '17 at 14:43
• @caverac Especially because it’s tagged appropriately – gen-z ready to perish Oct 27 '17 at 16:22
• This is very funny. =) (= – izrik Oct 27 '17 at 16:39

These are some excellent visual puns.

1. $x^{-1}$ is usually read as "$x$-inverse". Accordingly, the other side of the equation has an inverted smiley.
2. $x^2$ is usually read as "$x$-squared".
3. $x^3$ is usually read as "$x$-cubed".
4. Given a set (or surface or whatever it is), $\partial S$ refers to the boundary of $S$. Indeed, the "boundary" of our smiley is the outer circle.
5. In topology, $\pi_1(S)$ refers to the fundamental group of the space $S$. Indeed, a disk with $3$ holes in it has $\pi_1(S) = \Bbb F_3$, which in this context denotes the free group on $3$ generators.
6. Re and Im are used to denote the real and imaginary parts of a complex number. So, $\operatorname{Re}(a + bi) = a$, and $\operatorname{Im}(a + bi) = b$ (where $a$ and $b$ are real numbers). In other words, Im produces "the $i$ part", and Re produces everything but the $i$ part. Correspondingly, $\operatorname{Re}$ takes the smiley and produces "everything but the eye part", and $\operatorname{Im}$ produces "the eye part".
7. $\nabla f$ is usually read as "grad $f$".
8. $\nabla \times f$ is usually read as "curl $f$".
9. The $c$ superscript denotes the set complement.
• Damnit, I had half of them written down already :( I didn't figure out how the fundamental group one worked, though, I focused too much on the black parts, I guess. – Arthur Oct 27 '17 at 14:47
• "The eye part" - and I was trying to figure out how eyes are related to imagination and the mouth to realism ... – Hagen von Eitzen Oct 27 '17 at 16:38
• I don't understand why the set complement includes a cat face? – jwg Oct 27 '17 at 16:39
• @jwg I don't get it myself; maybe the professor just likes cats. Also, I don't think we're meant to interpret that $+$ as having its usual mathematical meaning. – Omnomnomnom Oct 27 '17 at 21:35
• @Omnomnomnom or should $(\text{smiley} + \text{cat})^c$ be equal to $\text{black smiley} \cdot \text{black cat}$?? – jwg Oct 30 '17 at 14:15