Mathematical Experiment 9

Construct various cross-sections of a cone.

When a cube is rotated about its main diagonal, describe the region wiped out.
Construct one-half of the tangent cone of a sphere.

Given three spheres S1, S2, S3, let P1, P2, P3 be the points of contact of one common tangent plane of the spheres, and Q1, Q2, Q3  the points of contact of the other common tangent plane of the spheres. Show that the lines P1P2 and Q1Q2, meet at a point R12, the lines P2P3 and Q2Q3 meet at a point R23, the lines P3P1, Q3Q1meet at a point R31, and that the points R12, R23 and R31lie on a straight line.

Construct a surface which appears as a triangle along the x-axis,

appears as a triangle along the y-axis,

 

apprears as an astroid along the z-axis

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

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m:=[[cos(t),0],[0,sin(t)]];
t:=n*2*Pi/100;
plot([m$n=1..100],color=blue,scaling=constrained,axes=none);

Construct this pattern:

t:=Pi*n/100:x:=cos(t)-1:y:=sin(t):
x1:=sqrt(1-(y-1)^2):
m:=[[x,y],[x,-y],[x1,-y],[x1,y],[x,y]]:
plot([m$n=1..50],scaling=constrained,color=blue,axes=none);