1 1 Nytt Munchmuseum: et SKRIK etter kombinatorikk og geometri!! Geir Dahl Matematisk inst., Universitetet i Oslo Faglig-pedagogisk dag, UiO, 2013
2 Museum Cover Comments First, let me proclaim how much I like the cover design! It s beautiful. Math folks will appreciate the way the left and right boundaries of the title and sub-title taper downward in agreement with the perspective of the ceiling of the art gallery. I like the preponderance of blues and greens throughout and the sharp yellow sub-title and flourish. As explained below, the location of the mathematical expressions on the floor of the gallery scene is an excellent How to Guard Cover (but I assume Comments unintentional) idea. I d like to address two items. The first concerns the mathematical expressions. The First, let me proclaim how much I like the cover design! It s beautiful. Math folks will second is about script font for the main title and author name. appreciate theart way left and Gallery right boundaries of the title and sub-title taper downward in agreement 1. Mathematical with perspective expressions. ofhere the ceiling are correct of the versions art gallery. for the I like twothe incorrect preponderance mathematical andexpressions greens throughout in the pdf and file theof sharp the cover yellowsent sub-title to theand author flourish. on 29As May, explained 2009: below, of blues the location of the mathematical expressions on the floor of the gallery scene is an excellent (a) and other Discrete (but I assume unintentional) idea. 3w 1 I d like to addressmathematical two items. The first gg(w) concerns = the mathematical expressions. The 7 second is about the script font for the main title and author name. The denominator Adventures ( 7") is centered. Also, note that the enclosing symbols are not 1. Mathematical brackets; expressions. they lack a top Here strut. are These correctmathematical versions for the symbols two incorrect represent mathemati- floor cal expressions function." So in the it spdf entirely file ofapt thethat cover they sent appear to theright author on onthe 29floor May, of 2009: the artistic scene! (a) (b) 3w 1 w gg(w) = w gg(w) The denominator ( 7") is centered. Also, note that the enclosing symbols are not There is a factor of 2 in front of the expression on the right. Again, the enclosing brackets; they lack a top strut. These mathematical symbols represent the floor symbols are not brackets. function." So it s entirely apt that they appear right on the floor of the artistic I likescene! the idea of using the formula in (a) on the cover a lot. (It is a formula I discovered.) The (b) string of inequalities in (b) is less effective, I think. From a mathematical point of view, it would be better to use the following w expression w gg(w) 2 instead: 3 3 w There is a factor of 2 in front ofg(w) the expression = on the right. Again, the enclosing 3 symbols are not brackets. (There is only one g" in this formula.) This is the art gallery formula," so it ties in Inicely like the with idea the ofcover usingart. themy formula preference in (a) on would the cover be to adisplay lot. (Itthe is atwo formula formulas I discovered.) as: The string of inequalities in (b) is less w effective, I think. 3w From 1 a mathematical point of view, it would be better tog(w) use the = following gg(w) expression = instead: 3 7 mighty is geometry, joined with art, resistless w 2. Script font. The script font is eye-catching, g(w) = but I don t think it is legible enough. For 3 instance, it looks like my name is spelled incorrectly; the h" in my last name look like (There a b." Could is only T. we one choose g" in S. amichael less this decorative formula.) This font? is One the option art gallery is to use formula," a larger, so non-italic it ties in nicely version with of the the simple cover art. font My currently preference used would for the be sub-title. to display the two formulas as: w Another option is to choose a more legible calligraphic 3w font 1 g(w) = gg(w) = for the title and author name. I have attached a 2-page Microsoft 3 Word file with 7 four examples of such fonts. The fonts in the attachment resemble the handwriting an artist would use to sign a mighty 2. Script is font. geometry, The script joined font iswith eye-catching, art, resistless but I don t think it is legible enough. For instance, painting, it which looksi think like my is aname goodiseffect spelled forincorrectly; this title. I mthe sure h" the in my JHUlast Press name Artlook Department b." has Could a host we choose of similar a less fonts decorative to choosefont? from. One These option attached is to use fonts a have larger, a rightward non-italic like a version slant, soof perhaps the simple the sub-title font currently wouldused needfor to the be upright sub-title. for contrast. I d like Another to thank option the Art is to Department choose a more very much legiblefor calligraphic the work and fonttalent for the they ve title and devoted author to this project. name. I have T. S. attached Michaela 2-page Microsoft Word file with four examples of such fonts. The fonts in the attachment resemble the handwriting an artist would use to sign a painting, which I think is a good effect for 1 this title. I m sure the JHU Press Art Department has a host of similar fonts to choose from. These attached fonts have a rightward slant, so perhaps the sub-title would need to be upright for contrast. I d like to thank the Art Department very much for the work and talent they ve devoted to this project. T. S. Michael 1 2
3 Museum 3 arkitektur og geometri
4 Kombinatorikk 4 Kombinatorikk ( ) n n (n 1) (n k + 1) = = k k (k 1) 1 n! k!(n k)! = antall k-kombinasjoner i en mengde med n elementer, uordnet utvalg av k elementer Blaise Pascal ( ) For alle heltall n og k med 1 k n 1 holder ( ) ( ) ( ) n n 1 n 1 = + k k k 1
5 Kombinatorikk Pascal s trekant Pascal, Traité du triangle arithmétique, n\k
6 Kombinatorikk 6 Binomialteoremet: (x + y) n = n k=0 ( ) n x n k y k k Fordi: (x + y)(x + y)(x + y)(x + y) = x 4 + 4x 3 y +... Sannsynlighet: inklusjon-eksklusjons prinsippet P(A 1 A 2 ) = P(A 1 ) + P(A 2 ) P(A 1 A 2 ) P(A 1 A n ) = n j=1 P(A j) i<j P(A i A j ) + i<j<k P(A i A j A k ) Men kombinatorikk er mye mer...
7 7 Munch og kombinatorikk: Og spørsmålet er: Hvor mange vakter trenger man for å vokte alle veggene? NB: Hver vakt står stille!! Men kan snu seg. Figure : Typisk L-museum
8 8 Figure : Kreativt museum
9 9 Figure : U-museum Hvorfor må vi ha (minst) 2 vakter?
10 10 S(p) p Figure : U-museum punkt p, synbarhetsområde S(p)
11 11 Lavt budsjett! Figure : Konveks-museum
12 12 Høyt budsjett! n Figure : Kam-museum trenger n vakter pga. spissene har 3n vegger vakter/vegger=1/3 Altså: Hvis arkitektene får herje fritt, kan det bli mange vakter!
13 13??????????????? n Figure : Kam-museum vakter/vegger=1/3 HVOR ILLE KAN DET BLI????
14 14 Theorem For ethvert museum med n vegger vil man trenge høyst n 3 vakter. Skal bevise dette.
15 15 Bevis Figure : Munch-museum, n = 10.
16 16 Bevis I: triangulering Figure : Munch-museum, n = 10.
17 17 Bevis II: en graf Har nå fått en graf fra trianguleringen.
18 18 Bevis II: farvelegging Figure : Munch-museum, n = 10. Nok med 3 farver Fins en farve som er på høyst n 3 punkter
19 Bevis III: Går dette alltid? Jepp: induksjonsbevis Triangulering: Velg et hjørne med vinkel < 180 Lag linje mellom nabohjørnene Gjenta dette. 3-farvelegging av den triangulerte grafen: Velg en diagonal; den deler grafen i to deler Ved induksjon kan hver del farves med 3 farver, og ved omdøping av farver stemmer disse overens på diagonalen. 19
20 20 En variant Nye regler: Hvor mange vakter? Hver vakt står ved en vegg og kan bevege seg langs denne Og kan snu seg. Uløst problem! Andre varianter finnes også.
21 21 Triangulering anvendelse
22 22 Kombinatorikk som forskningsfelt Stor aktivitet. Flere problemstillinger motivert av anvendelser, f.eks. digitale verden, nettverk. Noen områder: grafteori farvelegging partielt ordnede mengder, latticer, ekstremal mengeteori flyt i nettverk, optimering enumerativ kombinatorikk, telleproblemer (0, 1)-matriser kombinatorisk matriseteori, spektral grafteori koder og design Latinske kvadrater...
23 23 Lese mer? M. Aigner, G. Ziegler, Proofs from THE BOOK, Springer, 4.th ed., R.A. Brualdi, Introductory Combinatorics, Prentice-Hall, T.S. Michael, How to Guard an Art Gallery and Other Discrete Mathematical Adventures, The John Hopkins University Press, 2009.