Re: Pascal's Triangle

*To*: mathgroup at smc.vnet.net*Subject*: [mg37356] Re: Pascal's Triangle*From*: "Borut L" <gollum at email.si>*Date*: Fri, 25 Oct 2002 02:46:40 -0400 (EDT)*References*: <ap86hf$554$1@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Helo, I should thank you for this pleasent mind excercise, though not tough, it was just basic geometry. $TextStyle={FontFamily->Verdana,FontSize->10}; (* first create a basic object, a colored triangle with text inscribed, note that the height of the triangle is a unit *) Triangle[j_,i_]:= Module[ { x0=(i-.5 j)2/Sqrt[3],y0=-j,a=2/Sqrt[3] }, { If[EvenQ[Binomial[j,i]],Hue[0,.5,1],Hue[.55,.5,1]], Polygon[{{x0,y0},{x0-.5 a,y0-1},{x0+.5 a,y0-1}}], GrayLevel[0],Text[ToString[Binomial[j,i]],{x0,y0-.5},{0,1}] } ] (* the following command will show the agglomerate, note that it doesn't draw each of the triangles having no text, instead it draws a big triagle as a background and puts the numbered ones onto it *) ShowBigTriangle[jmax_]:= Show[ Graphics[ Prepend[ Table[Triangle[j,i],{j,0,jmax},{i,0,j}], {Hue[0,.5,1], Polygon[{{0,0},{-(jmax+1)/Sqrt[3],-(jmax+1)},{(jmax+1)/Sqrt[3],-(jmax+1)}}]} ] ] ,AspectRatio->Automatic ] ShowBigTriangle[33]//Timing (1.3 sec on my P2-350, how's your timming?) p.s. : According to S. Wolfram and his new kind of science, the solution to your problem is just a simple program. He ilustrates this in 2nd Chapter. It may be worthy to take a peak. Bye, Borut "Al Mannon" <almannon at attbi.com> wrote in message news:ap86hf$554$1 at smc.vnet.net... | I want to write a program that will create an array of equilateral triangles | such that in the first row there is 1 triangle, in the second row there is 3 | triangles, in the third row there will be 5 triangles...in the nth row there | will be 2n - 1 triangles. Putting all of these triangles together and | calling the first row, row 0, we would have a large triangle with n + 1 rows | along the side of the large triangle and 2n - 1 columns along the base of | the triangle. | | This would be phase one of the project. | | Phase 2: Fill in the binomial coefficients into the triangle. | Phase 3: Color all odd numbered triangles blue. | Phase 4: Color all even numbered triangles red. | Phase 5: Any triangle that shares an edge with a red triangle, color red. | | The result is Zierpinski's Triangle! I have done this by hand for a triangle | with 16 rows. Needless to say the work was tedious. The result, however, is | quite satisfying and remarkable. I would like to be able to use this as a | tool to teach some of the other derivations that are possible from Pascal's | triangle other than binomial coefficients and combinations. Therefore it | would be beneficial to be able to reproduce this work at will. | | Since my Mathematica programming skills are practically nil, any help would | be appreciated. | | I can be reached directly by electronic mail by deleting "the" in the | following: | | althemannon at attbi.com | |

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