Files
uni/year2/semester1/logseq-stuff/pages/Convex Polyhedra.md

7.6 KiB

  • #MA284 - Discrete Mathematics
  • Previous Topic: Definitions & Planar Graphs
  • Next Topic: Colouring Graphs; Eulerian & Hamiltonian Graphs
  • Relevant Slides: MA284-Week09.pdf
  • Non-Planar Graphs

    • Most graphs do not have a planar representation, however, it takes some work to prove that a graph is non-planar.
      • To do this, we can use Euler's formula for planar graphs to prove that they are not planar.
    • Theorem: K_{5} is not planar (Theorem 4.3.1 in Textbook)

      • The proof is by contradiction.
      • So assume that K_5 is planar. Then, the graph must satisfy Euler's formula for planar graphs.
        • K_5 has 5 vertices & 10 edges, so we get 5-10 + f =2, which says that if the graph is drawn without any edges crossing, there would be f = 7 faces.
      • Now consider how many edges surround each face. Each face must be surrounded by at least 3 edges.
        • Let B be the total number of boundaries around all the faces in the graph.
          • Thus, we have that 3f \leq B, but also B = 2e, as each edge is used as a boundary exactly twice.
            • Putting this together, we get 3f \leq B, but this is impossible, since we have already determined that f = 7 and e = 10, and 21 \nleq 20.
              • This is a contradiction, so K_5 is not planar.
      • Q.E.D.
    • Theorem: K_{3,3} is not planar (Theorem 4.2.2 in Textbook)

      • Please read the proof in the textbook (or edit this later).
      • The proof for K_{3,3} is somewhat similar to K_5, but it also uses the fact that a bipartite graph has no 3-edge cycles.
    • To understand the importance of K_5 & K_{3,3}, we first need that the concept of homeomorphic graphs.
      • Recall that a graph G_1 is a subgraph of G if it can be obtained by deleting some vertices and / or edges of G.
    • Homeomorphic Graphs

      • What is a subdivision of an edge? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:15:33.031Z card-last-score:: 1
        • A subdivision of an edge is obtained by "adding" a new vertex of degree 2 to the middle of the edge.
      • What is a subdivision of a graph? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:52:43.154Z card-last-score:: 1
        • A subdivision of a graph is obtained by subdividing one or more of its edges.
      • What is the smoothing of a pair of edges? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T15:52:16.893Z card-last-score:: 1
        • The smoothing of the pair of edges \{a,b\} & \{b,c\}, where b is a vertex of degree 2, means to remove these two edges, and add \{a,c\}.
      • What does it mean if two graphs are homeomorphic? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:25:55.182Z card-last-score:: 1
        • Two graphs G_1, G_2 are homeomorphic if there is some subdivision of G_1 that is isomorphic to some subdivision of G_2.
      • Kuratowski's Theorem #card

        card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:26:18.688Z card-last-score:: 1
        • What is Kuratowski's Theorem? #card card-last-interval:: 3.69 card-repeats:: 2 card-ease-factor:: 2.46 card-next-schedule:: 2022-11-22T10:36:43.656Z card-last-reviewed:: 2022-11-18T18:36:43.657Z card-last-score:: 5
          • A graph is planar if and only if it does not contain a subgraph that is homeomorphic to K_5 or K_{3,3}.
        • What this really means is that every non-planar graph has some smoothing that contains a copy of K_5 or K_{3,3} somewhere inside it.
  • Polyhedra

    • What is a polyhedron? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:25:09.000Z card-last-score:: 1
      • A polyhedron is a geometric solid made up of flat polygonal faces joined at edges & vertices.
    • What is a convex polyhedron? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:22:24.900Z card-last-score:: 1
      • A convex polyhedron is one where any line segment connecting two points on the interior of the polyhedron must be entirely contained inside the polyhedron.
    • A remarkable, and important fact, is that every convex polyhedron can be porjected onto the plane without edges crossing.
      • image.png
      • image.png
    • Now that we know that every convex polyhedron can be represented as a planar graph, we can apply Euler's formula.
      • Euler's Formula for Polyhedra #card

        card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:17:27.666Z card-last-score:: 1
        • If a convex polyhedron has v vertices, e edges, & f faces, then
          • v - e+ f =2
        • image.png
      • The Handshaking Lemma #card

        card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:22:43.283Z card-last-score:: 1
        • The sum of the vertex degrees is 2|E|:
          • Let G = (V,E) be a graph, with vertices V = v_1, v_2, \cdots, v_n.
          • Let \text{deg}(v_i) be the "degree of $v+i$". Then
            • \text{deg}(v_1) + \text{deg}(v_2) + \cdots + \text{deg}(v_n) = 2|E|
    • Example

      • Show that there is no convex polyhedron with 11 vertices, all of degree 3. background-color:: green
      • If such a convex polyhedron existed, we could draw its graph.
        • Each vertex has degree 3, so
          • 2|E| = \sum d(v_i) = 3 \times 11
          • \therefore |E| = \frac{33}{2} \text{, which is impossible}
  • The Platonic Solids

    • What is a Regular Polyhedron? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T20:25:24.689Z card-last-score:: 1
      • A polyhedron is called regular if:
        • All its faces are identical, regular polygons.
        • All its vertices have the same degree.
    • What are the Platonic Solids? #card card-last-interval:: -1 card-repeats:: 1 card-ease-factor:: 2.5 card-next-schedule:: 2022-11-15T00:00:00.000Z card-last-reviewed:: 2022-11-14T16:16:02.569Z card-last-score:: 1
      • The convex regular polyhedra are also called the Platonic Solids.
    • There are exactly 5 regular polyhedra. This fact can be proven using Euler's formula.
      • For full details, see the proof in the textbook.
      • Here is the basic idea:
        • Consider a regular polyhedron with f triangular faces.
          • So 2e - 3f.
        • Suppose that every vertex has degree k.
          • So 2e - vk. Also, v - e +f =2. So
            • e = \frac{3f}{2}, v = \frac{3f}{k}, \Rightarrow \frac{3f}{k} - \frac{3f}{2} + f =2
            • and thus
              • f = \frac{4k}{6-k}
              • f is defined for any k < 6, but undefined for k = 6 , and if k > 6 then f < 0 - no solutions.