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Barnsley's Fern

The Sierpinski Triangle


The Sierpinski Triangle  (The Sierpinski Gasket)



A simple three part algorithm, which incorporates
randomly selected parts of the algorithm, produces 
a triangular lattice of triangles.  

To thoroughly fill in the Sierpinski gasket, we need 
to supply the algorithm with a [pseudo] infinite database,
which is what the randomness algorithm provides.

Program code is at bottom of this page.


Here are snippets from a couple websites explaining the
algorithm:


http://math.bu.edu/DYSYS/chaos-game/node1.html

One of the most interesting fractals arises from what 
Michael Barnsley has dubbed "The Chaos Game".  The chaos 
game is played as follows:  First pick three points at 
the vertices of a triangle (any triangle works - right, 
equilateral, isosceles, whatever). Color one of the 
vertices red, the second blue, and the third green. 

Next, take a die and color two of the faces red, two 
blue, and two green. Now start with any point in the 
triangle. This point is the  seed for the game. 
(Actually, the seed can be anywhere in the plane, 
even miles away from the triangle.) Then roll the 
die. Depending on what color comes up, move the seed 
half the distance to the appropriately colored vertex. 
That is, if red comes up, move the point half the 
distance to the red vertex. Now begin again, using the 
result of the previous roll as the seed for the next. 
That is, roll the die again and move the new point 
half the distance to the appropriately colored vertex,





https://thatsmaths.com/2014/05/22/the-chaos-game/

Fix three points in the plane, C1, C2 and C3.  For 
definiteness, we take the points C1 = (0, 0), 
C2 = (1, 0) and C3 = (0.5, v3/2), the corners of an 
equilateral triangle.


Pick any point P0 and draw a dot there. This is 
our starting point. At each stage, we denote the 
current point by Pk and call it the game point.


Roll the dice. If n comes up, draw a point half 
way between Pk and Cn. For example, if we roll a 2, 
we pick the point half way between the current 
point Pk and C2. This is the new game point.


Repeat this procedure many times, drawing a new 
point at each step.

===================================================


Here is my Visual BASIC program code:


Private Sub chaosbtn_Click()

ITER = Val(itera.Text)

X1 = 0
Y1 = 0
X2 = 700
Y2 = 0
X3 = 350
Y3 = 606

XS = 280
YS = 490

Randomize Timer

For a = 1 To ITER

RN = Rnd(1)

If RN < 0.3333333 Then GoTo 100
If RN < 0.6666667 Then GoTo 200
GoTo 300

100:
XS = (X1 + XS) / 2
YS = (Y1 + YS) / 2
GoTo 500

200:
XS = (X2 + XS) / 2
YS = (Y2 + YS) / 2
GoTo 500

300:
XS = (X3 + XS) / 2
YS = (Y3 + YS) / 2

500:
TX = XS + 20
TY = -YS + 1050   REM  Transforms the y axis

    viewport.PSet (TX, TY), RGB(0, 0, 0)

Next a

End Sub

============================================================

home


Balloting

Simple trig

Logistic map

The Attractor of Henon

Number doubling

Barnsley's Fern

The Sierpinski Triangle