ITERATION

 

22. Sierpinski triangle

  1. Draw triangle ABC
  2. Construct midpoints DEF  of the  sides
  3. Construct setments DE, EF, FD.
  4. Select points ABC
  5. Go to the TRANSFORM menu:  Iterate.
  6. For each Point A, B, C in  turn, click  on  D, E, F as the point to map to.
  7. Click on  the  Display box and  increase iterations to 4, then to  5.
  8. Click iterate.

 

6B:  SIERPINSKI:

 

  1. Construct a triangle with midpoint triangle as before.  Select the vertices DEF and  hit  Control P to color the interior of triangle DEF.
  2. Select points ABC as above, but map  AàA; BàF;  Cà D.
  3. Click on the  STRUCTURE button, then “add new map.”  Map AàF; BàB;  Cà E.
  4. Finally, click STRUCTURE and “add new map” and map AàD; BàE;  Cà C.
  5. Click display and increase the number of iterations  to  5. 
  6. Iterate. 

 

 

 

 

 

23.  More fractal pictures

 

 

 

 

 

24.  Quadrilateral Iteration

  1. Draw four points ABIH as vertices of your quadrilateral.
  2. Construct the four sides of the quadrilateral.
  3. With the sides still selected, construct the midpoints DCFE
  4. Construct segments CE and DF joining opposite midpoints.
  5. Construct the intersection G of CE  and  DF.

  1. Select points DGEH.  CONSTRUCT  “polygon  interior.”  (Change the color if you like.)  Make sure you have selected the labels for your points with the A tool.
  2. Select points ABIH in that order, and TRANSFORM menu Iteration
  3. For each beginning point, select a new target point for map 1:  NOTE that you select the target point by clicking on it.  A new starting point will then light up.                       A àA              Bà C              Ià G               HàD                                                  
  4. Click  the structure button in the dialogue  box, click new map.
  5. For this new map send:                                                                                                 AàC               BàB               IàF                 Hà G
  6. Click  the structure button in the dialogue  box, click new map.
  7. For this new map send:                                                                                                 AàG               BàF               IàI                  Hà E
  8. In the dialogue box, click display and increase iterations. 
  9. Do this one more time so the number of iterations is 5.
  10. Click:  iterate. 
  11. Play with your picture.  Move vertices  around, and past each other.Try removing the colorings if you like.

 

***MAKE THIS CONSTRUCTION INTO A TOOL by selecting the entire construction, and clicking the >> double arrow box  on the  left and click “create new tool.”***

 

25. CRAB:  This wonderful picture is care of David Henderson. 

 

Given an irregular quadrilateral ABCD, show how to put a quadrilateral similar to ABCD on side BC (see quadrilateral crab picture).

 

Different starting shapes give different flavors:

 

 

 

By making the similar quadrilaterals in both directions, you “tile” the plane with crabs.

 

          

 

 

 

26. SPIRALS

Use the transformations tools to draw some kind of a spiral, a raying out picture like a flower, a bunch of concentric figures. 

 

A. Rotate and Dilate

  1. Draw a segment AD
  2. Draw circle with center A and point  C on the circle.
  3. CONSTRUCT  point B on the circle 
  4. Select D, A, B and TRANSFORM   Mark Angle
  5. Select segments AB and AD in that order and TRANSFORM Mark  Ratio
  6. Draw point E.  TRANSFORM    Mark Center
  7. Draw point F.  

  1. TRANSFORM;  rotate point F (select “by marked angle”.)
  2. TRANSFORM this point F’ by “marked  ratio” to get point F’’
  3. HIDE the intermediary point F’
  4. Select point F.  TRANSFORM iterate  and map F to F’
  5. Click Display and increase number of iterations.  Use the shift + key to increase the  number to 30.
  6. Click iterate. 

Moving point C in and out controls the ratio.  Moving point B controls the angle.

 

  1. Draw segment FF’’.  It will help you keep track of what  is happening.   Play with the angle and ratio.
  2. Select point F.  iterate as before, increasing the number to 30.  You should now see segments iterated as well.  Play with the angle and ratio.

 

Measure angle BAD and  ratio AB/AD.

 

QUESTIONS

What is happening around 144 degrees?  Why?  What is the relation between the patterns near 144 and near 72 and 216? What other angles give interesting patterns? 

B. SPIRAL and geometric series

 

When you control the angle t and ratio m of a geometric spiral, you are controling a complex geometric series.  You have to think of plane numbers each being a rotation and magnification <t,m> of the unit vector. 

 

Where does the series converge to? 

The convergence point L  is also the fixed point of the transformation

Z---> <t,m> Z + GH.

 

L is the vertex of a triangle with base GH, vertex angle L = rotation angle t, and ratio of sides LH/LG= m.  (THIS REQUIRES PROOF:!)

 

Empirically: Drag point L around until the angle and ratio equal the angle and ratio of the geometric series. 

 

For a position of L, and for an initial length GH,  you can get a series converging to L.

The circle through L, G, H is the locus of convergence points for the given angle t when we let the ratio m vary from 0 to 1. 

 

 

QUESTION:  Given segment GH, for which points L  in the plane can you find a series which converges?

 

C. NAUTILUS SPIRAL

 

 

28. MANDELBROT SET ORBITS

 

This sketch allows you to move point  C around to see the  orbits of iterating C in the parameter space.  You can get a real sense of the connection between the angles and the lobes on the  border of the Mandelbrot set.  When orbits stabilize and when they escape.