Königsberg Bridges

1. Königsberg Bridges

2. Leonard Euler Study calculus Develop graph theory Major foci of Euler's work

3. 15 April 1707 – 18 September 1783 Swiss mathematician and physicist. Discoveries Infinitesimal Calculus Graph Theory Introduced much of the modern mathematical terminology and notation for mathematical analysis , ex. expression of functions as sums of infinitely many terms Worked in mechanics, fluid dynamics, optics, and astronomy

8. Is it possible to draw a continuous curve that passes through each of the ten edges (line segments) exactly once? A curve that passes through a vertex is not allowed.

10. Create a graph with four vertices, one for each of the four regions including the outside region, D

11. Is it possible to draw a continuous curve that passes through each of the edges exactly once?

12. Graph Graph -the whole diagram is called a graph. Not this Degree - the number of edges that lead to a vertex is called the degree Vertex - a point is called a vertex (plural vertices) Edge - a line is called an edge. Simple Path - route around a graph that visits every vertex once Euler Path -route around a graph that visits every edge once

16. Graphs are among the most ubiquitous models of both • natural and • human-made structures. • They can be used to model many types of relations and process dynamics in • physical, • biological and • social systems. • Many problems of practical interest can be represented by graphs.

18. Form a planar graph from a polyhedron place a • light source near one face of the polyhedron, and a • plane on the other side.

19. Euler’s Formula The function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians.

20. Three-dimensional visualization of Euler's formula in the complex plane

21. Internet

22. Airlines

23. Internet Routes Put to color 2005

24. Genetic composition of baker’s yeast

25. Topology

26. Möbius Strip …have only one surface and one edge Type of object studied in topology

27. Paul Bunyan and the Hard Rock Mine Conveyor Belt Michigan is awash in Paul Bunyan tales of his logging exploits, but perhaps less well known are his ventures into mining and his extensive mathematical knowledge. While on his western tour, Paul developed a deep mine and was extracting high quality ore. Our story begins when the mine was a horizontal shaft about a quarter of a mile back into the mountains and had a fairly high volume of ore coming out the wind- and Babe (the blue ox)-powered main conveyor belt four feet wide. The mine branched at that point, but the best ore was found straight ahead and the mine shaft had developed such that a longer but narrower conveyor belt would increase efficiency. Specifically it was decided that a conveyor belt twice as long and half as wide as the original would be the best solution. Paul got out his sharp ax and proceeded to slice the conveyor belt down the middle. When he finished they installed the new converyor belt without having to cut it anymore or even splice it, much to everyone's surprise. Paul explained that he had put the half twist in it so it would wear evenly on both sides. A few years went by. Mining operations continued. Although the best ore vein curved slightly and narrowed down, they could still improve operations with a narrower, but longer conveyor. As Paul got out his new chain saw, he sent his trusty assistant to town to get a couple splicing kits. His assistant bulked having seen the results of the prior modification. He went anyway and was surprised when he came back to find the kits were not only needed, but the two resulting rings were linked together.

28. Trefoil knot

29. DNA Folding And Protein Folding

31. Sort the alphabet Homeomorphismclasses are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K is almost too short to see), one hole one tail, and no holes four tails. Homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.

32. Map of London

35. Shoreline Lake Of The Woods

43. A mathematician named KleinThought the Möbius strip divine.Said he, "If you glueThe edges of twoYou'll get a bottle like mine."

47. Hamiltonian Circuit path that ends where it starts It visits each vertex once without touching any vertex more than once.

48. Cut a Mobius Strip in half Cut in half again

49. A mathematician confidedThat a Möbius strip is one-sided, And you'll get quite a laugh If you cut one in half,for it stays in one piece when divided.