The curve associated with an edge(u, u) contains no other vertex point and inter- sects no other edge curve, except at its endpoints This scary definition hints at a theoretical problem associated with planar graphs while the idea seems intuitively simple, rigorous arguments about planar graphs require some heavy-duty math. The classic example of the difficulties that can arise is the jordan Curve Theorem. This states that every simple, closed curve separates the plane into two regions, an inside and an outside, like this Up until the late 1800s, mathematicians considered this obvious and implicitly treated it as an axiom. However, in 1887 Jordan pointed out that, in principle, this could be a theorem proved from simpler axioms. Actually nailing down such a proof required more than 20 years of effort. (It turns out that there are some nasty curves that defy simple arguments. Planar graphs come up all the time and are worth learning about, but a several-month diversion into topology isn t in the cards. So when we need an"obvious geometric fact, we'll handle it the old fashioned way: we'll assume it Planar graphs are worthy of study for several reasons. One is rooted in human pyschol ogy: many kinds of information can be presented as a graph(family relations, chemical structures, computer data structures, contact data for study of disease spread, flow of cash in money laundering trials, etc. ) Big graphs are typically incomprehensible messes, but planar graphs are relatively easy for humans to grasp since there are no crisscross- ing edges. Sometimes the advantages of planarity are more concrete; for example, when wires are arranged on a surface, like a circuit board or microchip, crossings require trou blesome three-dimensional structures. When Steve Wozniak designed the disk drive for the early Apple Il computer, he struggled mightly to achieve a nearly planar design For two weeks, he worked late each night to make a satisfactory design. When he was finished, he found that if he moved a connector he could cut down on feedthroughs, making the board more reliable. To make that move, however, he had to start over in his design. This time it only took twenty hours. He then saw another feedthrough that could be eliminated and again started over on his design. The final design was generally recognized by computer engineers as brilliant and was by engineering aesthetics beautiful Noz later said, 'It's something you can only do if you're the engineer and the PC board layout person yourself. That was an artistic layout. The board has virtually no feedthroughs. 2From apple2history org which in turn quotes Fire in the valley by Freiberger and Swaine6 Graph Theory II • The curve associated with an edge (u, v) contains no other vertex point and inter￾sects no other edge curve, except at its endpoints. This scary definition hints at a theoretical problem associated with planar graphs: while the idea seems intuitively simple, rigorous arguments about planar graphs require some heavy­duty math. The classic example of the difficulties that can arise is the Jordan Curve Theorem. This states that every simple, closed curve separates the plane into two regions, an inside and an outside, like this: inside outside Up until the late 1800’s, mathematicians considered this obvious and implicitly treated it as an axiom. However, in 1887 Jordan pointed out that, in principle, this could be a theorem proved from simpler axioms. Actually nailing down such a proof required more than 20 years of effort. (It turns out that there are some nasty curves that defy simple arguments.) Planar graphs come up all the time and are worth learning about, but a several­month diversion into topology isn’t in the cards. So when we need an “obvious” geometric fact, we’ll handle it the old fashioned way: we’ll assume it! Planar graphs are worthy of study for severalreasons. One is rooted in human pyschol￾ogy: many kinds of information can be presented as a graph (family relations, chemical structures, computer data structures, contact data for study of disease spread, flow of cash in money laundering trials, etc.). Big graphs are typically incomprehensible messes, but planar graphs are relatively easy for humans to grasp since there are no crisscross￾ing edges. Sometimes the advantages of planarity are more concrete; for example, when wires are arranged on a surface, like a circuit board or microchip, crossings require trou￾blesome three­dimensional structures. When Steve Wozniak designed the disk drive for the early Apple II computer, he struggled mightly to achieve a nearly planar design: For two weeks, he worked late each night to make a satisfactory design. When he was finished, he found that if he moved a connector he could cut down on feedthroughs, making the board more reliable. To make that move, however, he had to start over in his design. This time it only took twenty hours. He then saw another feedthrough that could be eliminated, and again started over on his design. ”The final design was generally recognized by computer engineers as brilliant and was by engineering aesthetics beautiful. Woz later said, ’It’s something you can only do if you’re the engineer and the PC board layout person yourself. That was an artistic layout. The board has virtually no feedthroughs.’”2 2From apple2history.org which in turn quotes Fire in the Valley by Freiberger and Swaine