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Applications of Graph Theory
Published in Rowan Garnier, John Taylor, Discrete Mathematics, 2020
Over the years, various ‘proofs' of the conjecture were given, but each was subsequently found to be flawed. (‘Flawed' does not mean ‘worthless', however, as many interesting ideas and techniques were developed in several of the unsuccessful attempts at a proof.) Eventually in 1976, Kenneth Appel and Wolfgang Haken completed a proof of the four-colour theorem as it then became known. However, their announcement of the proof caused considerable controversy in the mathematical community because, for the first time in mathematics, they had used a computer in an essential way to check the many hundreds of possible configurations to which the problem had been reduced. The sheer size of the number of computations required made the use of many hours of mainframe computer time essential. Even today, many mathematicians are reluctant to accept the Appel - Haken proof as ‘genuine' because it can only be checked by another computer—to work through the details ‘by hand' is impossible in practice.
Planar Graphs
Published in Kenneth H. Rosen, Graphs, Algorithms, and Optimization, 2005
The original proof of the four-color theorem appeared in Appel and Haken [4] and [5]. An excellent survey article of the Appel-Haken proof is Woodall and Wilson [128]. There are a number of excellent books on the 4-color problem, including Saaty and Kainen [106] and Ore [93]. A very readable history of the 4-color problem can be found in Wilson [127]. A shorter proof was accomplished by Robertson, Sanders, Seymour, and Thomas in [104]. Much of the development of graph theory arose out of attempts to solve the 4-color problem. Algner [2] develops the theory of graphs from this perspective.
Fostering collateral creativity through teaching school mathematics with technology: what do teachers need to know?
Published in International Journal of Mathematical Education in Science and Technology, 2022
The investigative tasks analyzed in this paper deal with the variety of mathematical topics and technological tools. For instance, the Creatures task builds on students’ initial insight of additive partitions of a whole number they could further investigate using spreadsheets. This initial exploration leads to further questioning making use of probabilistic reasoning. Another example of potential rich problem to prompt collateral creativity is the Sieves task which allows students to explore number series using cubes to build towers and notice patterns. Moreover, the use of Wolfram Alpha supports generalizations and further explorations. Wolfram Alpha can also support investigations around the year-numbers eventually engaging students with exploration of prime numbers while asking new questions leading to deeper insights into a number theory. Finally, the Square’s Division task brings students to the investigation of special relationships (e.g. area) in a visual dynamic way by means of dynamic geometry software (e.g. Geometer’s Sketchpad or any other dynamic geometry programme), a problem which has potential to endless explorations from different mathematical perspectives, including one related to Four Color Theorem analyzed in this paper.