E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 9: Topology1. ObjectiveTopology identifies objects which can b e morphed into each other. We want to explore here, howmany topological types the letters in our alphabet have.2. Warmup with digits 0-91) The numbers 0, 4, 6, 9 are topologically equivalent.04692) The numbers 1, 2, 3, 5, 7 are topologically equivalent.123573) The number 8 is not topologically equivalent to any other digit.83. Now the alphabetClassify the alphabet according to topological equivalence.ABCDEFGHIJKLMNOPQRSTUVWXYZE-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 9: Polyhedra1. ObjectiveWe meet here some regular and semi-regular polyhedra.2. Euler Characteristic1) Compute the Euler Characteristic V − E + F for thecubeVertices V Edges E Faces F2) Cut the corners of the cube. It will produce newfaces and produce octagonal faces from the old faces.Count the Euler characteristic of the new object whichis a semi-regular polyhedron.Vertices V Edges E Faces F3) Start again with the cube, but now cut each ofthe faces into 4 faces by drawing the diagonals in thesquares. If the midpoints are lifted up a bit so that alltriangles become equilateral, the new object is called astellation of the cube. It is an other semi-regular poly-hedron.Vertices V Edges E Faces F3. DualityIf we take a polyhedron and replace every face with a vertex in the middle of the face and everyvertex with a face. We obtain an other polyhedron.4) What is the dual of the cube?5) What is the dual of the octahedron?6) What is the dual of the tetrahedron?E-320: Teaching Math with a Historical Perspective Oliver Knill, 2010Lecture 9: Surfaces1. ObjectiveBy identifying sides of a square we obtain models of compact surfaces: the sphere, the torus, theprojective plane and the Klein bottle. We want to explore here the topology of these spaces,especially the simply conntectedness: can one pull any closed rope in this space to a point? Onlythe sphere is simply connected.2. The torus1) Draw some curves on the torus, which can not be pulled together to a point.2. The projective plane2) Draw a curve on the projective plane, which can not be pulled together to a point.3. The Klein bottle3) Move a letter R around on the Klein bottle, what happens with the letter as it moves over theboundary to the right and appears to the left?R4. The sphere4) Draw a curve on the sphere. Visualize that you can pull it together to a point.5. Euler characteristicIf you have more time: By triangulating the space, we are also able to compute the Euler char-acteristic of these spaces. The Euler characteristic of the sphere is 2, the Euler characteristic ofthe torus is 0, the Euler characteristic of the projective plane is 1, the Euler characteristic of theKlein bottle is 0. Can you show this in the