关于ℝ²中边与顶点的关系、多边形边顶点数等价性证明严谨性及相关几何问题的问询

First, love the deep dive into the etymology of "polygon"—that's such a fun way to frame the question! The Greek root "polugōnos" ("many-angled") makes total sense, and it ties directly to why we care about the relationship between vertices (where angles form) and edges.

Let's break down your questions clearly:

1. Do polygons always have the same number of edges and vertices? #

You're absolutely right that for simple polygons—the non-intersecting, closed, connected shapes we typically mean when we say "polygon" in basic geometry—the number of edges always equals the number of vertices.

2. Is your Euler's characteristic proof rigorous? #

Your approach is on the right track, but let's refine it to make it fully solid (since you mentioned no formal graph theory background):

• Euler's formula $V - E + F = 2$ applies to connected planar graphs embedded without crossings, which simple polygons correspond to if we model them as graphs plus the outer infinite face.
• For a single simple polygon, we have $F=2$ (the inner region bounded by the polygon and the outer region). Rearranging gives $V - E + 2 = 2$, so $V=E$—that works!
• That said, there's a more straightforward proof that avoids graph theory entirely: start at any vertex, follow the edges around the polygon. Each edge connects to exactly one new vertex, and when you loop back to the starting point, you've used exactly as many edges as vertices. No fancy formulas needed, just basic traversal logic.

A quick note on rigor: your Euler's formula approach is valid as long as you specify you're dealing with simple polygons. If you allowed self-intersecting "complex polygons" (like a bowtie shape), the relationship breaks down—but those aren't the standard polygons we discuss in most geometric contexts.

3. Other useful relationships between vertices and faces in ℝ²? #

Beyond Euler's formula, here are a few key ones:

• Handshaking Lemma: For any planar graph (or any graph, really), the sum of all vertex degrees equals $2E$. For a simple polygon, each vertex has degree 2, so $2V=2E$ → $V=E$—another way to see the edge-vertex equality!
• Planar Graph Inequalities: For connected planar graphs where every face (including the outer one) is bounded by at least 3 edges, we can derive $E ≤ 3V - 6$. This is useful for proving certain graphs can't be drawn without crossings (like the complete graph on 5 vertices, $K_5$).
• Convex Polygon Diagonals: For a convex $n$-gon, the number of diagonals is $\frac{n(n-3)}{2}$—a direct relationship between vertex count and non-edge connections.

4. Angles at vertices in ℝ² and ℝⁿ? #

• In ℝ²: Yes, every vertex of a simple polygon has an internal angle (and an external angle, if you define it consistently). For convex polygons, the sum of internal angles is $(n-2)×180°$, and the sum of external angles is always 360° regardless of $n$. For concave polygons, the internal angle sum still holds, but some internal angles are greater than 180°, and external angles can be negative if you count them in a consistent rotational direction.
• In ℝⁿ: We generalize angles using solid angles, which measure how much of the surrounding $n$-dimensional space is "occupied" by the polytope (the n-dimensional analog of a polygon) at that vertex. For example, in 3D, the solid angle at a cube's vertex is a quarter of a sphere (since three perpendicular faces meet there). In higher dimensions, we can also define angles between edges meeting at a vertex, and generalize dihedral angles to higher-dimensional "facets"—though these get more abstract than 2D angles. For non-convex polytopes, solid angles can be signed (negative) to account for concave vertices, similar to 2D concave polygons.

备注：内容来源于stack exchange，提问作者user1039946