Mathematical Experiment 10

Draw this pretty leaf:

w:=1+cos(t)/2:z:=t/6-sin(2*t)/12:x:=w*cos(z): y:=w*sin(z):
plot([x,y,t=0..12*Pi], axes=none, scaling=constrained);
[Maple Plot]
Draw the graphs of the polynomials given by the binomial expansions

m:=[x,binomial(100,k)*x^(100-k)*(1-x)^k,x=0..1]:

plot({m$k=0..100},axes=none);
[Maple Plot]

 
Construct he graph given in polar coordinates by

[Maple Math][0,4[Maple Math]]

plot(cos(7*t/2)+1/4,t=0..4*Pi,coords=polar, axes=none,scaling=constrained);
[Maple Plot]
Construct this interesting drawing:

plot(2-cos(3*t)-cos(31*3*t/32),t=0..64*Pi, coords=polar, numpoints=1000,axes=none, scaling=constrained);

Reference: William F. Rigge, Envelope Rosettes, Amer. Math. Monthly, (1920), p. 152.
[Maple Plot]
Construct this interesting drawing:

plot(2-cos(7*t)-cos(31*7*t/32),t=0..64*Pi, coords=polar, numpoints=1000,axes=none, scaling=constrained);

Reference: William F. Rigge, Envelope Rosettes, Amer. Math. Monthly, (1920), p. 154.
[Maple Plot]
Construct this interesting drawing:

plot(100+t+15*cos(3.05*t), t = 0 .. 200, coords =polar, axes = none,scaling=constrained);
[Maple Plot]
Construct a gif file animating the function
f(x,t) = sin (x + t)
and include the gif in your web page.
Construct the line segments joining (cos(t),0) with (0,sin(t)) as t ranges over [0,2p].

x:=cos(t);
y:=sin(t);
t:=k*Pi/50;
m:=[[x,0],[0,y]];
plot([m$k=1..100],color=green, scaling=constrained,axes=none);
[Maple Plot]
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 this figure: 
Construct this graph associated with the logistic equation:
x ' = ax(1-x)
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);
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);