Defining Gaussian and Mean Curvature for an architect- non-mathematician I am an architect, and just started using the parametric software "Grasshopper", trying to model and reshape surfaces based on geometric formulas.
That's where I came up with the terms "Mean Curvature" and "Gaussian Curvature",
and have no idea what they mean.
Can someone please tell me in plain English what they mean, so that I can get a general idea?   Like I said before, I'm an architect, so please no formulas at this stage :))))
 A: I'll assume you are considering only 2-dimensional surfaces, since higher dimensional cases shouldn't come up very often in architecture. First, however, we can examine the one dimensional case.
For a smooth curve, at a point $P$ we can draw a circle that best "fits" the local curvature of the curve. We call the radius of that circle the radius of curvature at $P$ $\rho$. The straighter the curve, the larger the radius (a straight line has a radius of infinity everywhere). If we want a parameter that describes "how curved it is" we can define the curvature to be $\kappa=\frac 1\rho$, one devided by the radius of curvature. A straight line has a curvature of zero, and the "sharper" the corner, the higher the curvature. It's important to remember that a curve does not have just one curvature; it has a curvature at each point. The special curves that have the same curvature everywhere are straight lines and circles. As a sign convention we typically define a positive direction and refer to curvature in that direction as positive and curvature in the other direction as negative.

We can think about curvature on a 2 dimensional surface by choosing a point $P$. There are lots of curves that are contained within the surface that pass through $P$. We can restrict ourselves to geodesics, or curves that do not have any curvature in the surface itself. You can think of a geodesic like walking in a straight line on Earth. You seem to never change direction, but your path curves along the surface of the earth. The blue curves are all geodesics passing through the same point.

If you look at all the geodesics that pass through $P$, they generally won't all have the same curvature at $P$. There will, however, be a maximum and minimum among them. We call these the principal curvatures $\kappa_1$ and $\kappa_2$. We typically define a "positive side" of the surface and consider curves that curve toward that side to have positive curvature. If the surface is closed, the inside is positive. If we want a single number to describe "how curved the surface is at $P$", we can take the product of the principle curvatures, which we call the Gauss Curvature, or take the average, which we call the Mean Curvature.
On a sphere of radius $r$, all the geodesics are circles of radius $r$, so they all have curvature $\frac 1r$. Thus, the Gauss curvature is $\frac 1{r^2}$ everywhere and the mean curvature is $\frac 1r$ everywhere.
On a flexed sheet of paper, there will always be a direction for which the geodesics are straight lines, so the minimum curvature at each point is zero. This surface will thus have zero Gauss curvature everywhere, but its mean curvature will vary.
A: Consider a smooth surface $S\subset{\mathbb R}^3$, say the surface of a car roof, or an ordinary egg. This surface is not uniformly curved, like a sphere is; instead we have a "curvature status" that changes from point to point on $S$. At each point $p\in S$ we have a tangential plane $T$. If we consider this plane as $(x,y)$-plane and the normal to this plane as $z$-axis then $S$ appears in the neighborhood of $p$ approximately in the form
$$z=ax^2+2bxy+cy^2$$
with certain coefficients $a$, $b$, $c$. These coefficients together encode the "curvature status" of $S$ at $p$. It turns out that this status can be characterized by two numbers $\kappa_1$, $\kappa_2$ called the principal curvatures of $S$ at $p$. These numbers do not depend on the chosen axes in $T$, but are true geometric quantities associated to $S$. An example: If $S$ is a cylinder or a cone then $\kappa_1=0$ since there is a straight line $\ell\subset S$ going through $p$, and $\kappa_2>0$ depends on the curvature of the base circle of the cylinder.
The quantity $H:={1\over2}(\kappa_1+\kappa_2)$ is called the mean curvature of $S$ at $p$. There is no intuitive characterization of this notion. Suffice it to say that for minimal surfaces (soap films bounded by a closed  wire) the mean curvature is $=0$ at all points.
The quantity $K:=\kappa_1\kappa_2$ is called the Gaussian curvature of $S$ at $p$. It is positive if $S$ near $p$ looks like an egg ($z=x^2+y^2$), and is negative if $S$ near $p$ looks like a saddle ($z=x y)$. Of the two numbers $K$ and $H$ the $K$ is more important by "orders of magnitude", because $K$ is an "intrinsic" quantity associated to $S$: If $S$ is a flexible structure embedded in ${\mathbb R}^3$, but is provided with an "inner feeling of distances", then $H$ will change under
flexions, whereas $K$ will remain the same. As a consequence it is not possible to map $S^2$  in an isometrical way onto the plane even for small countries: The sphere of radius $R$ has constant Gaussian curvature $K={1\over R^2}$, whereas the plane of course has curvature $\equiv0$.
