Math for a cake My wife likes to decorate birthday cakes. She told me that she will make a math cake for my birthday and I should provide her a "famous math formula" to be written on the top of the cake.
I realized I can name dozens of physics related famous formulas that one could recognize (Maxwell's equations, Newtons laws, Einstein's $E=mc^2$...) but I couldn't name one that would be more "math related". 
Writing some axioms wouldn't work, they take too much space. The famous theorems I know of are not really "a formula" but more like of "statements" that would need some background, or they are not visually appealing (like Fermat's last theorem). (Quests are not math-oriented thus the visual side matters.)
Any ideas what we could put on top of the cake?
 A: Euler's classical formula for convex polyhedra
$$v-e+f=2$$
where $v$ is the number of vertices, $e$ the number of edges and $f$ the number of faces of a convex triagulated polyhedron in $3$-space.
A: At Michael Atiyah's 80th birthday conference, the cake had the Atiyah-Singer index formula:
$$\text{Ind}(D) = \int_{T^*M} \text{ch}(\sigma_D) \text{Todd}(TM \otimes \mathbb{C})$$
I can verify that it made the cake even more delicious.
A: $\mathrm{P}=\mathrm{NP}$ or $\mathrm{P}\neq \mathrm{NP}$, whichever you prefer.
A: In a different vein from the other answers, how about one of the classic visualizations of the proof of the Pythagorean theorem? It's basically just a bunch of triangles and squares rearranged in a couple ways, and would come out nicely with cake decorator colors. And folks might actually recognize it.
A: Gödel's completeness theorem:  A (first order) sentence $\varphi$ is provable from the axioms $\Sigma$ iff  it holds in every model of $\Sigma$:  $$ \Sigma \vdash \varphi \Leftrightarrow \Sigma \vDash \varphi$$
A: Gödels incompleteness theorem in the language of modal logic (where  $\Box\varphi$ means that $\varphi$ is provable  - say in Peano Arithmetic - and $\bot=\lnot \top$ is any false statement): $$\Box \lnot \Box \bot \Rightarrow \Box \bot.$$
A: A geometric one, where the zero can be made a cake (circle) itself
$$ x^2 + y^2 -1 = \Huge \circ $$
A: Maybe just make the cake in the shape of a golden rectangle, and use two colors of icing to show the decomposition into a square and a smaller golden rectangle.
A: How about the snake lemma? It's not a formula, but it could still look great on a cake! Plenty of excellent .tex diagrams here: https://tex.stackexchange.com/questions/3892/how-do-you-draw-the-snake-arrow-for-the-connecting-homomorphism-in-the-snake-l
A: My all-time favourite formula: Stokes theorem
$$\int_{M}\mathrm{d}\omega=\int_{\partial M}\omega$$
A: One which I like much is
$$ \exp \left(\begin{bmatrix} 
. & . & . & . & .\\\ 
1 & . & . & . & . \\\ 
. & 2 & . & . & . \\\ 
. & . & 3 & . & . \\\ 
. & . & . & 4 & . \\\ 
\end{bmatrix} \right)= \begin{bmatrix}
1 & . & . & . & . \\\ 
1 & 1 & . & . & . \\\ 
1 & 2 & 1 & . & . \\\ 
1 & 3 & 3 & 1 & . \\\ 
1 & 4 & 6 & 4 & 1 \\\ 
 \end{bmatrix}$$
It is practically easier and a bit more iconic if we reduce it a bit - although for me it is not so pleasing, because the immediate remembering of the Pascal-triangle comes with the 1-4-6-4-1-row:
$$ \Large  \exp \small \left(\begin{bmatrix} 
. & . & . & . \\\ 
1 & . & . & .  \\\ 
. & 2 & . & .  \\\ 
. & . & 3 & .  \\\ 
\end{bmatrix} \right)= \begin{bmatrix}
1 & . & . & .  \\\ 
1 & 1 & . & .  \\\ 
1 & 2 & 1 & .  \\\ 
1 & 3 & 3 & 1  \\\ 
 \end{bmatrix}$$
With a bit explanation which might be useful for other guests 
http://go.helms-net.de/math/binomial/index-Dateien/image008.png 
A: Not famous, perhaps, but how about
$$\int_0^a f_A(x)dx = \int_a^1 f_A(x)dx = 1/2$$
from Better Ways to Cut a Cake by Brams, Jones, and Klamler?
A: I think the diagram should be several dotted rays emanating from the
same point, arranged so that if you cut along the lines, each piece will
have the same volume of cake and of frosting.  It is an impressive diagram
when the number of pieces is a not too small odd number  such as 5, 7, or
9.
(There is also an interactive n player version.)
Gerhard "Save A Piece For Me" Paseman, 2012.12.29
A: $e^{i \pi} = -1$
A: 196884 = 196883 + 1
A: 22/7. Because a cake is, approximately, a pi(e).
A: How about the Grothendieck-Hirzebruch-Riemann-Roch formula:
ch(f!F) = f*(ch(F)td(Tf))?
A: (comment to D. Pavlov)
I once attempted to bake GRR onto cookies (leavened with hartshorn, naturally).  It didn't turn out too legible, but probably doable with icing.
