Planar Graphs, Solids, and Surfaces. Planar Graphs 1/28

Transcription

1 Planar Graphs, Solids, and Surfaces Planar Graphs 1/28

2 Last time we discussed the Four Color Theorem, which says that any map can be colored with at most 4 colors and not have two regions that share a border having the same color. It turns out that this result has something to do with the topology of the plane. If we draw maps on other surfaces, we may need a different number of colors. Planar Graphs 2/28

3 For example, if we draw a map on a torus (a doughnut), we may need up to 7 colors, as the following pictures indicate. Here is an animation of a map on the torus which needs 7 colors. The map on the torus shown above has 7 regions and each region is adjacent to each of the other 6 regions. This is why it needs 7 colors. The next video shows an important application of the torus. Planar Graphs 3/28

4 While it may seem that determining the minimum number of colors to color a map drawn on a torus would be much harder than for maps drawn on a plane, it is just the opposite. The Four Color Theorem for maps on a plane took a huge amount of effort, 61 pages in Scientific American, and lots of computer calculation, the result for the torus can be written down in a few pages, and was proved much earlier. Planar Graphs 4/28

5 Planar Graphs The graphs that arise from a map turn out to have the following property: They can be drawn in such a way that no two edges cross. These are called planar graphs. The pentagon graph we saw the previous class is not a planar graph. This is why it does not represent the graph of a map and why it does not violate the Four Color Theorem. Planar Graphs 5/28

6 Let s consider the following graph with 4 vertices and 6 edges: Planar Graphs 6/28

7 Clicker Question Can we redraw the graph so that the edges do not cross? A Yes B No Planar Graphs 7/28

8 Answer Yes we can redraw it by having one of the diagonal edges drawn outside of the square. Thus, it is a planar graph, even though the original drawing does not indicate so. Planar Graphs 8/28

9 Faces of a Planar Graph For solids, such as those above, there is a reasonable meaning of vertex, edge, and face. For example, the cube has 8 vertices (or corners), 12 edges, and 6 square faces. Planar Graphs 9/28

10 The cube can be drawn as a graph in multiple ways. Here are two such. One is a pretty typical way to draw a cube, and the other is less so. The second graph is a planar graph. One way to view the six faces of the cube in it are to consider the 5 regions inside the graph along with the outside. Planar Graphs 10/28

11 In general, with a planar graph, we can see regions inside the graph, and we say the number of faces of the graph is the number of regions inside plus the outside. In this way a graph that represents a solid will have the same number of faces as the solid. Planar Graphs 11/28

12 Clicker Question How many faces does the Octahedron have? Planar Graphs 12/28

13 Answer There are 8 faces. Looking at the figure on the left, there are 4 faces on the top half, and 4 on the bottom half. Looking at the graph on the right, there are 7 regions inside the graph, and 1 outside, making 8 faces. Planar Graphs 13/28

14 Is there a relationship between the Numbers of Vertices, Edges, and Faces? V E F Solid Tetrahedron Square Octahedron Dodecahedron Icosahedron Stare at this table for a bit and see if you can find any relationship between V, E, F that holds for all of the solids. Planar Graphs 14/28

15 Euler s Formula One of Euler s other contributions to graph theory was the following result about planar graphs: If V is the number of vertices, E the number of edges, and F the number of regions formed by a planar graph, then V E + F = 2 This formula was a key factor in the proof of the Four Color Theorem. Planar Graphs 15/28

16 Platonic Solids The solids in this picture are called Platonic solids, named after the Greek philosopher Plato. These solids are the most regular solids that are built from plane figures. While the Ancient Greeks were well aware of these five shapes, they didn t know if there were any others. Planar Graphs 16/28

17 Some Information from Wikipedia Plato wrote about these solids in which he associated each of the four classical elements (earth, air, water, and fire) with a regular solid. Earth was associated with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. There was intuitive justification for these associations: the heat of fire feels sharp and stabbing (like little tetrahedra). Air is made of the octahedron; its minuscule components are so smooth that one can barely feel it. Water, the icosahedron, flows out of one s hand when picked up, as if it is made of tiny little balls. Planar Graphs 17/28

18 By contrast, a highly nonspherical solid, the hexahedron (cube) represents earth. These clumsy little solids cause dirt to crumble and break when picked up in stark difference to the smooth flow of water. Moreover, the cube s being the only regular solid that tesselates Euclidean space was believed to cause the solidity of the Earth. The fifth Platonic solid, the dodecahedron, Plato obscurely remarks,...the god used for arranging the constellations on the whole heaven. Aristotle added a fifth element, ether, and postulated that the heavens were made of this element, but he had no interest in matching it with Plato s fifth solid. Planar Graphs 18/28

19 In the 16th century, the German astronomer Johannes Kepler attempted to relate the five extraterrestrial planets known at that time to the five Platonic solids. In Mysterium Cosmographicum, published in 1596, Kepler proposed a model of the solar system in which the five solids were set inside one another and separated by a series of inscribed and circumscribed spheres. Kepler proposed that the distance relationships between the six planets known at that time could be understood in terms of the five Platonic solids enclosed within a sphere that represented the orbit of Saturn. The six spheres each corresponded to one of the planets (Mercury, Venus, Earth, Mars, Jupiter, and Saturn). Planar Graphs 19/28

20 In the end, Kepler s original idea had to be abandoned, but out of his research came his three laws of orbital dynamics, the first of which was that the orbits of planets are ellipses rather than circles, changing the course of physics and astronomy. Planar Graphs 20/28

21 How Many Platonic Solids Are There? Euler s formula can help us determine all Platonic solids. The file Platonic Solids.pdf describes how one can see that there are exactly five such solids from the formula and some algebra. Planar Graphs 21/28

22 Houses and Utilities Problem Suppose there are 3 utilities and 3 houses. Each house is to be connected to each utility (by, e.g., a pipe, or wire). Is it possible to do this without having the connections crossed? In this picture, think about the utilities as the vertices in red and the houses in blue. Planar Graphs 22/28

23 Clicker Question Can you see a way to connect each reg vertex with each blue vertex without having any lines crossed? A Yes B No C Not sure Planar Graphs 23/28

24 Here is an attempt to solve the problem. It doesn t quite work. Why not? Planar Graphs 24/28

25 Here is another attempt to solve the problem. It also doesn t quite work. Why not? Planar Graphs 25/28

26 It turns out that no matter how hard you try, it is not possible to draw this graph without crossing edges. Euler s formula is actually a result about the nature of surfaces. We ll investigate this further next week. Planar graphs mean something different if you draw them on a surface other than a piece of paper. Planar Graphs 26/28

27 For example, we can solve the utilities problem if the houses and utilities were drawn on a torus (a doughnut). The following animation shows how each house can be connected to each utility without having the lines cross, providing we do this on the torus. What this means is that what it means to be a planar graph depends on what surface do we draw the graph. The website gives an animation of this graph. Planar Graphs 27/28

28 Next Time We will look further at different surfaces, including the plane and the torus, but consider others. We ll see that Euler s formula says something about surfaces. We ll also look at other properties of surfaces, and see that the properties we discuss are enough to classify all surfaces. Planar Graphs 28/28