       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,y0=-j,a=2/Sqrt
},
{
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,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,-(jmax+1)},{(jmax+1)/Sqrt,-(jmax+1)}}]}
]
]
,AspectRatio->Automatic
]

ShowBigTriangle//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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