Introduction to Graph Theory Quotes

“A theorem is no more proved by logic and computation than a sonnet is written by grammar and rhetoric, or than a sonata is composed by harmony and counterpoint, or a picture painted by balance and perspective. Logic and computation, grammar and rhetoric, harmony and counterpoint, balance and perspective, can be seen in the work after it is created, but these forms are, in the final analysis, parasitic on, they have no existence apart from, the creativity of the work itself. Thus the relation of logic to mathematics is seen to be that of an applied science to its pure ground, and all applied science is seen as drawing sustenance from a process of creation with which it can combine to give structure, but which it cannot appropriate.”
― Richard J. Trudeau, Introduction to Graph Theory

“Geometric diagrams are to geometers what board and pieces are to chessmasters: visual aids, helpful but not indispensable.”
― Richard J. Trudeau, Introduction to Graph Theory

“It appears then that the essence of chess is its abstract structure. Names and shapes of pieces, colors of squares, whether the “squares” are in fact square, even the physical existence of board and pieces, are all irrelevant. What is relevant is the number and geometric arrangement of the “squares”, the number of types of piece and the number of pieces of each type, the quantitative-geometric power of each piece, etc. Everything else is a visual aid or a fairy tale.”
― Richard J. Trudeau, Introduction to Graph Theory

“I use logic all the time in mathematics, and it seems to yield “correct” results, but in mathematics “correct” by and large means “logical”, so I’m back where I started. I can’t defend logic because I can’t remove my glasses.”
― Richard J. Trudeau, Introduction to Graph Theory

“Map coloring problem. Find the smallest number m such that the faces of every planar graph can be colored with m or fewer colors in such a way that faces sharing a border have different colors.”
― Richard J. Trudeau, Introduction to Graph Theory

“A statement that mathematicians believe but cannot as yet prove is called a “conjecture”.”
― Richard J. Trudeau, Introduction to Graph Theory

“A statement that mathematicians can prove is called a “theorem”, or sometimes a “lemma” or “corollary” if it bears a certain relationship to another theorem.”
― Richard J. Trudeau, Introduction to Graph Theory

“The Pythagoreans were mystics, and believed that the tetrahedron, cube, octahedron, and icosahedron respectively underlay the structure of the four elements of Greek science: fire, earth, air, and water. The dodecahedron they identified with the universe as a whole. Plato was quite taken by all this and spent some time in his dialogue Timaeus (named after the Pythagorean who is the chief interlocutor) discussing the connection between the five regular polyhedra and the structure of the universe. For this reason the regular polyhedra came to be known as the “platonic solids”.”
― Richard J. Trudeau, Introduction to Graph Theory

“The Pythagoreans were mystics, and believed that the tetrahedron, cube, octahedron, and icosahedron respectively underlay the structure of the four elements of Greek science: fire, earth, air, and water. The dodecahedron they identified with the universe as a whole.”
― Richard J. Trudeau, Introduction to Graph Theory

“There has been a steady escalation of conditions since our first discussions in Chapter 2. Then we talked about plain old graphs. Subsequently we restricted our attention to planar graphs, then to planar connected graphs, then to planar connected graphs with each edge bordering two faces (polygonal graphs), and now to planar connected regular graphs with each edge bordering two faces and all faces bounded by the same number of edges (platonic graphs).”
― Richard J. Trudeau, Introduction to Graph Theory

“Algebra is another branch of mathematics; it studies sets on which there have been defined things called “operations”. An operation on a set is a rule whereby two or more elements of the set can be combined to form another element of the set. High school algebra is the algebra of one specific set, the set of real numbers, and four specific operations defined on that set, addition, subtraction, multiplication, and division. High school algebra is only the tip of the algebraic iceberg.”
― Richard J. Trudeau, Introduction to Graph Theory

“A topologist enters a coffee shop, orders coffee and a doughnut, and is served. Preoccupied with topological theorems, he takes a bite out of his coffee cup and has to finish his thoughts in a nearby emergency ward. His mistake is somewhat understandable as a doughnut and coffee cup are topologically equivalent”
― Richard J. Trudeau, Introduction to Graph Theory

“Topology] is a purely qualitative subject where quantity is banned. In it two figures are always equivalent if it is possible to pass from one to the other by a continuous deformation, whose mathematical law can be of any sort whatsoever as long as continuity is respected.”
― Richard J. Trudeau, Introduction to Graph Theory

“It has been said that geometry is the art of applying good reasoning to bad diagrams.”
― Richard J. Trudeau, Introduction to Graph Theory

“Leonhard Euler (pronounced “oiler”, 1707–1783) is judged by all to have been the most productive, and by many to have been the best, mathematician of modern times. He was Swiss, but spent much of his life in Russia because he had a big family and Catherine the Great offered him a lot of money. His paper “The Seven Bridges of Königsberg” (1736), which we will discuss in Chapter 8, is the earliest known work on the theory of graphs. The theorem now known as Euler’s Formula was proved by Euler in 1752. It is one of the classic theorems of elementary mathematics and plays a central role in the next three chapters of this book.”
― Richard J. Trudeau, Introduction to Graph Theory

“you should not allow yourself to be convinced by repeated failures,”
― Richard J. Trudeau, Introduction to Graph Theory

“The village barber shaves those and only those men who live in the village and do not shave themselves. The village barber is a man and he lives in the village. Consider the question “Who shaves the barber?”
― Richard J. Trudeau, Introduction to Graph Theory

“In particular the rules of logic tell us how to create, from the opening arrangement (the list of axioms), new arrangements (called “theorems”).”
― Richard J. Trudeau, Introduction to Graph Theory

“Four Color Conjecture is one of the most famous unsolved problems in mathematics.”
― Richard J. Trudeau, Introduction to Graph Theory

“But the pure mathematician’s goal is to have a good time, not to be efficient; and machine-building is a lot more fun than problem-solving.”
― Richard J. Trudeau, Introduction to Graph Theory

“Definition 20. If some new vertices of degree 2 are added to some of the edges of a graph G, the resulting graph H is called an expansion of G.”
― Richard J. Trudeau, Introduction to Graph Theory