Mathematical Experiment 11

Draw the graphs of the first six Chebyshev polynomials in the interval [-1,1].

> m:=[cos(x),cos(n*x),x=0..Pi];

> plot({m$n=1..6},axes=none);

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);

Draw the graphs of the polynomials given by the binomial expansions

> restart;

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

> plot({m$k=0..100},axes=none);
　

Construct this pattern of the "sunflower":

> r:=exp(t);

> m:=64;

> a:=plot([[r*cos(t+2*Pi*k/m),r*sin(t+2*Pi*k/m),t=3..5]$k=1..m],

scaling=constrained,color=red,axes=none):

> b:=plot([[r*cos(t+2*Pi*k/m),-r*sin(t+2*Pi*k/m),t=3..5]$k=1..m],

scaling=constrained,color=blue,axes=none):

> with(plots):

> display(a,b);

Reference: T. Cook: The Curves of Life.
　

Construct he graph given in polar coordinates by

[Maple Math] [0,4p]

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.

Draw this interesting pattern:

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

Draw this interesting pattern:

> plot(100+t+15*cos(3.05*t), t = 0 .. 200, coords =
polar, axes = none,scaling=constrained);