Mathematical Experiment 12

Construct the line segments joining
(cos(t),0) with (0,sin(t)) as t ranges
over [0,2p].

> x:=cos(t);
> y:=sin(t);
> m:=[[x,0],[0,y]];
> t:=n*Pi/50;
> plot([m$n=1..100],color=green,
scaling=constrained,axes=none);

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Construct the velocity vector field
along a constant motion around a circle.

Construct the line segments joining
 [cos(t),sin(t)] with [cos(2t),sin(2t)]
 as t ranges over [0,2p].

Construct the line segments
joining [cos(t),sin(t)] with
[cos(3t),sin(3t)] as t ranges
over [0,2p].

Construct this graph associated with the logistic equation

x ' = ax(1-x)
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with a=3.7
> x:=0.7;
> a:=3.7;
> y:=a*x*(1-x);
> m:=[ ];
> for k to 100 do
m:=[op(m),[x,x],[x,y]]:
x:=y:y:=a*x*(1-x): od:

> plot(m,scaling=constrained);
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Draw 20 concentric circles as thus:

Construct the circles with center at (cos(t),sin(t)) passing through the point (1,0) with t ranging over [0,2p].

Construct the circles with center at (cos(t),sin(t)) and tangent to the y-axis with t ranging over [0,2p].

Construct the circles with center at (cos(t),sin(t)) and and passing
through (2cos(t)/3+cos(2t)/3,2sin(t)/3+sin(2t)/3)
 with t ranging over [0,2p].

Construct this figure:

Construct this pattern:

Construct the reflections of a light ray inside a square:

> m:=1.7123:x:=0:y:=0:v:=[ [x,y] ]:

> for k to 100 do xx:=floor(x)+1:yy:=floor(y)+1: if (yy-y)>m*(xx-x) then x:=xx:y:=m*x else y:=yy:x:=y/m: fi: v:=[op(v),[1-abs(x-2*floor(x/2)-1),1-abs(m*x-2*floor(m*x/2)-1)]] od:

> plot(v,scaling=constrained);
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