Autoreference and magnetization in dynamic geometry

I.3. Autoreference and tracks

Home | I.1 |I.2 | I.3 | I.4 | II.1 | II.2 | II.3 | III.1 | III.2 | III.3 | III.4 | III.5 | IV.1

Yves Martin - University of La Réunion

Laboratory of Computer Science and Mathematics (LIM, EA 2525)

ICTMT9, Metz, July 8, 2009

Usign autorefernce of a tracked point : Sierpinsky's triangle

Let ABC be a triangle and P a point. The sequence (P_n) is defined as P_n+1 = h_X,1/2(P_n) for X = A, B, ou C chosen randomly.

Sierpinsky's triangle is the attractor of these similitudes.

On the next figure, we give a recurrence definition for the coordinates of a point P and we add a movement for a random choice of X between A, B, or C.

abscissa for P : if(go==0;x(P);(alea3==0)*(x(A)+x(P))/2+(alea3==1)*(x(B)+x(P))/2+(alea3==2)*(x(C)+x(P))/2)

with alea3 = floor(random(3)), a random interger 0, 1 or 2

Using autoreference in a counter

As points, expressions can also be autoreferent. A counter can be made and we can use it for example for lines of a spreadsheet in exploration of statistics.

In the next file, we have

nb recursive : if(go==0;0;if(d(M)>0;nb+1;nb))

buff recursive : if(nb==0;0;if(coupe==1;buff+1;buff))

Needle [AB] : A(if(go==0;0;xA), if(go==0;0,5;yA)) and B(if(go==0;1;xA+cos(t)),if(go==0;0,5;yA+sin(t))) with t =random(360), xA and yA =-5+random(10)

cut : abs(floor(y(B))-floor(y(A)))==1

We draw the track of needle [AB], with two colors, blue if cut=0 and red if cut=1.

More about the dynamic approach of Sierpinsky's triangle in the article To suspend time in dynamic geometry (in French)

Home | I.1 |I.2 | I.3 | I.4 | II.1 | II.2 | II.3 | III.1 | III.2 | III.3 | III.4 | III.5 | IV.1