I was doing a spot of light reading (crystallography), when the term "convex" polyhedron came up in a a section (very prominently) in conjunction with something else called the "Euler characteristic".
The Wikipedia article (linked above) on the "Euler characteristic" is written in the same vein as the book I'm using... but try as I might, I can't seem to understand it :/
Excerpt from the section (in the Wiki article) in question.
I suppose my inability to fully comprehend the section (relevant excerpt included above) is due to my shortcomings as a high-school student; so following the other Wiki links wasn't much help...and neither was Google.
My questions:
Can someone explain (in a way a high-school student such as myself would understand),
1) What is a "convex" polyhedron?
2) What's this "Euler characteristic" all about? Does it have any sort of physical implications (by "physical", I mean "geometry" of the solid)?
(Including my thoughts on this...in case it helps a potential answerer in tailoring an answer best suited to my needs/understanding)
Looking at the figures/models/diagrams provided in the Wiki article, has led me to believe that "convex" polyhedra must be the same as (or at least, very similar to) the idea of a "regular" polyhedra (identical faces, sides and dihedral angles) that we've learned when we were younger...except the corresponding Wiki article on "convex" polyhedra does not reflect this simplicity; so drawing such an equivalence appears to be wrong.
I have not studied "Topology" (at least, not as a field of mathematics). Obviously, I'm quite capable of looking at 3D object and classifying it as a "cube", a "sphere", some sort of "prism", etc...but my knowledge of "topology" pretty much ends there. I'm sorry.
I was tempted to ask the question ("What's a convex polyhedron?", "What's the Euler characteristic?") separately. However, seeing as they're intimately linked (at least, from the crystallography perspective), I felt it more prudent to put these questions in a single post.
Let's start with a precise definition of a convex polyhedron: I assume you are interested in bounded convex polyhedra (also called convex polytopes). A formal definition of a 3-dimensional convex polytope $P$ is that it is a bounded subset of $R^3$ (the 3-dimensional Euclidean space) which equals the intersection of finitely many closed half-spaces and is not contained in a plane. A closed half-space in $R^3$ can be defined algebraically as the solution set of a linear inequality $$ ax + by + cz + d\ge 0, $$ where $a, b, c, d$ are given numbers (such that $(a,b,c)\ne (0,0,0)$). In other words, $P$ is the solution set of a system of linear inequalities in three variables; these inequalities are called the defining inequalities of $P$.
Each convex polytope $P$ has faces of various dimensions, they are traditionally called vertices, edges and faces. These can be defined in several ways. One definition is that, say, a vertex of $P$ is a point $v$ of $P$ which is the solution set of a system of equations and inequalities, obtained from the defining inequalities by converting some of them into equations. Same for faces of other dimensions.
One verifies that each polytope has only finitely many vertices, edges and faces. The numbers of these are usually denoted $V, E$ and $F$. Euler's formula is $$ V- E + F =2. $$ One way to appreciate this formula is to notice that it helps you to do "numerology" of polytopes. Suppose, for instance, that you have a simple polytope, i.e. where every vertex has three edges coming out of it. ("Generic" polytopes have this property: If you perturb randomly the linear inequalities defining $P$, the new polytope becomes simple.) Given this, you already know that $$ 2E=3V $$ (every vertex defines three oriented edges). Now, plug this into Euler's formula and you get: $$ F= 2 - V + \frac{3}{2} V= 2 + \frac{1}{2} V. $$ I doubt that you would be able to come up with this equation without knowing Euler's formula.
Euler's formula has other applications and leads to some interesting advanced mathematics.