Generalization
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Generalization through problem solving. Part I. Coloring and folding regular solids. Gergely Wintsche Mathematics Teaching and Didactic Center Faculty of Science Eötvös Loránd University, Budapest.

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

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Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

The possibleframes of thecube

Part I / 11 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringtheoppositefaces

Allframesarecolored

Part I / 13 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthematchingvertices

Please fillthesamecolor of the matchingvertices of a cube. (Youcanusenumbersinstead of colorsifyouwish.)

Forexample:

Part I / 14 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthematchingvertices

Allverticesarecolored.

Part I / 15 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthefaces of thecubewith (exactly) twocolors

Part I / 17 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthefaces of thecubewith (exactly) sixcolors

Wewanttocolorthefaces of a cube. Howmanydifferentcolorarrangementsexistwithexactlysixcolors?

Part I / 18 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthefaces of thecubewith (exactly) sixcolors

Letuscolor a face of thecubewithred and fix itasthebase of it.

Part I / 19 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthecube

Coloringthefaces of thecubewith (exactly) sixcolors

Thereare 5 possibilitiesforthecolor of theoppositeface.

Letussayit is green.

Part I / 20 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringthetetrahedron

Coloringthefaces of thetetrahedron with (exactly) four colors

Wewanttocolorthefaces of a regular tetrahedron. Howmanydifferentcolorarrangementsexistwithexactlyfour colors?

Part I / 22 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Coloringtheoctahedron

Coloringthefaces of theoctahedron (exactly) eightcolors

Wewanttocolorthefaces of a regular octahedron. Howmanydifferentcolorarrangementsexistwithexactlyeightcolors?

Part I / 24 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Symmetry

Symmetry

Howmanyrotationsymmetry has theregulartetrahedron?

Part I / 29 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Symmetry

Symmetry

Howmanyrotationsymmetryhas thecube?

Part I / 31 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Symmetry

Symmetry

Letuscontinuewiththecube. Wecanrotateitaround 3 axes (they go throughthemidpoints of theoppositefaces 3·4 = 12 differentways.

Part I / 32 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Symmetry

Symmetry

Letus go back tothetruncatedicosahedronthewellknownsoccer ball.

Howmanyrotationsymmetry has thissolid?

Part I / 35 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Summa

Summarize

Part I / 37 – Coloring and foldingregularsolids,

Gergely Wintsche


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

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


Cme1 2 201 2 0 7 02 rzesz w poland gergely wintsche

Outlook

The case of cube

Ingeneralsense, coloringoptionswithncolors:

Coloringthecubewith

Part I / 41 – Coloring and foldingregularsolids,

Gergely Wintsche


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