CME1 2 , 201 2 .0 7.02. – Rzeszów , Poland Gergely Wintsche

1. Generalization throughproblemsolving Part I. Coloring and folding regular solids Gergely Wintsche Mathematics Teaching and Didactic Center Faculty of Science Eötvös Loránd University, Budapest CME12, 2012.07.02.– Rzeszów, Poland Gergely Wintsche

2. Outline • 1. Introduction – aroundtheword • 2. Coloringthecube • The frames of thecube • The case of twocolors • The case of sixcolors • The case of the rest • 3. Coloringthetetrahedron • 4. Coloringtheoctahedron • 5. The commonpoints • 6. The football Part I / 2 – Coloring and foldingregularsolids Gergely Wintsche

3. Introduction – Aroundtheword The question Please write down in a few words what do you think if you hear generalization. What is your first impression? How frequent was this phrase used in the school? (I am satisfied with Hungarian but I appreciate if you write it in English.) Part I / 3 – Coloring and foldingregularsolids, Gergely Wintsche

4. Introduction – Aroundtheword The answers –firststudent ”Generalization is when we have facts about something, which is trueand we make assumptions about other things with the same properties. Like 4 and 6 is divisible by 2 we can generalize thisinformation to even number divisibility. In school we didn’t use thisphrase very much because everybody else just wanted to survive mathclass so we didn’t get into things like this.” Part I / 4 – Coloring and foldingregularsolids, Gergely Wintsche

5. Introduction – Aroundtheword The answers –secondstudent ”To catch the meaning of the problem. Undress every uselessinformation, what has noeffect on the solution of the problem.Generally hard task, but interesting, we have tounderstand theproblem completely, not enough to see the next step but all of them. Not generally used in schools.” Part I / 5 – Coloring and foldingregularsolids, Gergely Wintsche

6. Introduction – Aroundtheword The answers –thirdstudent ”To prove something for n instead of specific number. I have made upmy mind aboutsome mathematical meaning first but after it aboutother averagethings as well. In this meaning we use it in schools very frequently atleastweekly.” Part I / 6 – Coloring and foldingregularsolids, Gergely Wintsche

7. Introduction – Aroundtheword The answers –wiki http://en.wikipedia.org/wiki/Generalization ”... A generalization (or generalisation) of aconcept is an extension ofthe concept to less-specific criteria. It is a foundational element of logicand humanreasoning.[citation needed] Generalizations posit theexistence of a domain or set of elements, as well as one or morecommon characteristics shared by those elements. As such, it is theessential basis of all valid deductive inferences. The process ofverification is necessary to determine whether a generalization holdstrue for any given situation...” Part I / 7 – Coloring and foldingregularsolids, Gergely Wintsche

8. Introduction – Aroundtheword The answers – wiki Example: ”... A polygon is a generalization of a3-sided triangle, a 4-sided quadrilateral, and so on to n sides. Ahypercube is a generalization of a 2-dimensional square, a3-dimensional cube, and so on to n dimensions...” Part I / 8 – Coloring and foldingregularsolids, Gergely Wintsche

9. Introduction – Aroundtheword The answers – Marriam-Websterdictionary • Definition of GENERALIZATION • the act or process of generalizing • a general statement, law, principle, or proposition • the act or process whereby a learned response is made to astimulus similar to but not identical with the conditioned stimulus • orsome extra words • a statement about a group of people or things that is based ononly a few people or things in that group • the act or process of forming opinions that are based on a smallamountof information Part I / 9 – Coloring and foldingregularsolids, Gergely Wintsche

10. Coloringthecube The frame of thecube Before we color anything please draw the possible frames of a cube. Forexample: Part I / 10 – Coloring and foldingregularsolids, Gergely Wintsche

11. Coloringthecube The possibleframes of thecube Part I / 11 – Coloring and foldingregularsolids, Gergely Wintsche

12. Coloringthecube Coloringtheoppositefaces Please color the opposite faces of a cube with the same color. Let ususe the color red, green and white (or anything else). Forexample: Part I / 12 – Coloring and foldingregularsolids, Gergely Wintsche

13. Coloringthecube Coloringtheoppositefaces Allframesarecolored Part I / 13 – Coloring and foldingregularsolids, Gergely Wintsche

14. Coloringthecube Coloringthematchingvertices Please fillthesamecolor of the matchingvertices of a cube. (Youcanusenumbersinstead of colorsifyouwish.) Forexample: Part I / 14 – Coloring and foldingregularsolids, Gergely Wintsche

15. Coloringthecube Coloringthematchingvertices Allverticesarecolored. Part I / 15 – Coloring and foldingregularsolids, Gergely Wintsche

16. Coloringthecube Coloringthefaces of thecubewith (exactly) twocolors Calculatethenumber of differentcoloringsofthecubewithtwocolors. Two coloringsare distinct if no rotationtransforms one coloringinto the other. Part I / 16 – Coloring and foldingregularsolids, Gergely Wintsche

17. Coloringthecube Coloringthefaces of thecubewith (exactly) twocolors Part I / 17 – Coloring and foldingregularsolids, Gergely Wintsche

18. Coloringthecube Coloringthefaces of thecubewith (exactly) sixcolors Wewanttocolorthefaces of a cube. Howmanydifferentcolorarrangementsexistwithexactlysixcolors? Part I / 18 – Coloring and foldingregularsolids, Gergely Wintsche

19. Coloringthecube Coloringthefaces of thecubewith (exactly) sixcolors Letuscolor a face of thecubewithred and fix itasthebase of it. Part I / 19 – Coloring and foldingregularsolids, Gergely Wintsche

20. Coloringthecube Coloringthefaces of thecubewith (exactly) sixcolors Thereare 5 possibilitiesforthecolor of theoppositeface. Letussayit is green. Part I / 20 – Coloring and foldingregularsolids, Gergely Wintsche

21. Coloringthecube Coloringthefaces of thecubewith (exactly) sixcolors The remainingfourfacesform a beltonthecube. Ifwecolorone of theemptyfacesofthisbeltwithyellowwecanrotatethecubetotaketheyellowface back. Thesethreefaces fix thecubeinthespacesotheremainingthree facesarecolorable 3·2·1=6 differentways. The total number of different colorings are 5·6=30. Part I / 21 – Coloring and foldingregularsolids, Gergely Wintsche

22. Coloringthetetrahedron Coloringthefaces of thetetrahedron with (exactly) four colors Wewanttocolorthefaces of a regular tetrahedron. Howmanydifferentcolorarrangementsexistwithexactlyfour colors? Part I / 22 – Coloring and foldingregularsolids, Gergely Wintsche

23. Coloringthetetrahedron Coloringthefaces of thetetrahedron with (exactly) four colors Letuscolor a face of thetetrahedron withred and fix itasthebase of it. The other three faces are rotation invariant, so there are only 2 differentcolorings. Part I / 23 – Coloring and foldingregularsolids, Gergely Wintsche

24. Coloringtheoctahedron Coloringthefaces of theoctahedron (exactly) eightcolors Wewanttocolorthefaces of a regular octahedron. Howmanydifferentcolorarrangementsexistwithexactlyeightcolors? Part I / 24 – Coloring and foldingregularsolids, Gergely Wintsche

25. Coloringtheoctahedron Coloringthefaces of theoctahedronwith (exactly) four colors Letuscolor a face of theoctahedronwithred and fix itasthebase of it. Wecancolorthe top of thissolidwith 7 colors, letussayit is green. Part I / 25 – Coloring and foldingregularsolids, Gergely Wintsche

26. Coloringtheoctahedron Coloringthefaces of theoctahedronwith (exactly) four colors Letuschoosethethreefaceswith a commonedge of theredface. Wehave differentpossibilities. We had todivideby 3 becauseifwerotatetheoctahedronasweindicateditthenonlythebase and the top remainsunchanged. Part I / 26 – Coloring and foldingregularsolids, Gergely Wintsche

27. Coloringtheoctahedron Coloringthefaces of theoctahedronwith (exactly) four colors The remainingthreefacescancoloredby 3·2·1 = 6 differentways, sothetotalnumber of color Part I / 27 – Coloring and foldingregularsolids, Gergely Wintsche

28. Coloringthefootball Coloringthefaces of thetruncatedicosahedron Beforewecoloranythinghowmany and whatkind of faces has thetruncatedicosahedron? It has 32 faces, 12 pentagonswheretheicosahedron’s verticeshad beenoriginally and 20 hexagonswheretheicosahedron’s faces had been. The number of differentcoloringsare … Part I / 28 – Coloring and foldingregularsolids, Gergely Wintsche

29. Symmetry Symmetry Howmanyrotationsymmetry has theregulartetrahedron? Part I / 29 – Coloring and foldingregularsolids, Gergely Wintsche

30. Symmetry Symmetry Wecanrotateitaround 4 axesalltogether4·3 = 12 ways. Ifwedistinguishallfaces of thetetrahedronthenthecoloringnumber is 4! = 24. Butwefound 12 rotationsymmetry, soweget 24 / 12 = 2 differentcolorings. Part I / 30 – Coloring and foldingregularsolids, Gergely Wintsche

31. Symmetry Symmetry Howmanyrotationsymmetryhas thecube? Part I / 31 – Coloring and foldingregularsolids, Gergely Wintsche

32. Symmetry Symmetry Letuscontinuewiththecube. Wecanrotateitaround 3 axes (they go throughthemidpoints of theoppositefaces 3·4 = 12 differentways. Part I / 32 – Coloring and foldingregularsolids, Gergely Wintsche

33. Symmetry Symmetry Thereare 4 more rotationaxis: thediagonals. Itmeans 4·3 = 12 more rotatation. Ifwe sum upthenwegetthe 24 rotation. (Wewillnotprovethattheserotationsgeneratethewholerotationgroupbutcan be checkedeasily.) Ifwedistinguishthefaces of thecubeit is colorablein 6! = 720 differentways. 720 / 24 = 30 Part I / 33 – Coloring and foldingregularsolids, Gergely Wintsche

34. Symmetry Symmetry The rotationsymmtries of theoctahedronareidenticalwiththesymmtries of thecube. Butwehave 8! = 40320 differentwaystocolorthe 8 faces, and 40320 / 24 = 1680 Part I / 34 – Coloring and foldingregularsolids, Gergely Wintsche

35. Symmetry Symmetry Letus go back tothetruncatedicosahedronthewellknownsoccer ball. Howmanyrotationsymmetry has thissolid? Part I / 35 – Coloring and foldingregularsolids, Gergely Wintsche

36. Symmetry Symmetry Wecanmove a pentagon toanyotherpentagon (12 rotation) and wecan spin a pentagon 5 timesaroundits center. Itgives 60 rotations. Ontheotherhandwecanseethehexagonsaswell. Everyhexagoncanmovetoanyother (20 rotation) and wecan spin a hexagon 6 timesaroundits center. Itgives 120 rotations. Is there a problemsomewhere? Part I / 36 – Coloring and foldingregularsolids, Gergely Wintsche

37. Summa Summarize Part I / 37 – Coloring and foldingregularsolids, Gergely Wintsche

38. Outlook Outlook The problembecamesreallyhighlevelifyouask: Howmanydifferentcoloringsexist of a cubewith maximum 3-4-ncolors. The questionsaresolvablebutwewouldneedtheintensiveusage of grouptheory (Burnside-lemma and/or Pólya counting). Part I / 38 – Coloring and foldingregularsolids, Gergely Wintsche

39. Outlook Outlook Let G a finitegroupwhichoperatesontheelements of the X set. Letx X and xgthoseelements of X wherex is fixed byg. The number of orbitsdenotedby | X / G |. Part I / 39 – Coloring and foldingregularsolids, Gergely Wintsche

40. Outlook The case of cube • The rotationsorder: • 1 identityleaves: 36elements of X • 6 pcs. of 90° rotationaround an axethroughthemidpoints of twooppositefaces: 33 • 3 pcs. of 180° rotationaround an axethroughthemidpoints of twooppositefaces: 34 • 8 pcs. of 120° rotationaround an axethroughthediagonal of twooppositevertices: 32 • 6 pcs. of 180° rotationaround an axethroughthemidpoints of twooppositeedges: 33 Part I / 40 – Coloring and foldingregularsolids, Gergely Wintsche

41. Outlook The case of cube Ingeneralsense, coloringoptionswithncolors: Coloringthecubewith Part I / 41 – Coloring and foldingregularsolids, Gergely Wintsche