MATHEMATICS AND ORIGAMI

Mathematics and Origami

• H is the altitude of a tetrahedron of side R (see the tetrahedron with vertices OAB in Point
  18.13.2); its value is:

H R 0 , 8164965 R

3

2

= = (see Point 18.2.1.1)

• Therefore, flattening b is:
  b=R−H= 0 , 1835035 R


• Cone ́s semiangle γ is the complement of angle β as calculated in Point 18.2.1.1:
  γ= 90 − 54 , 73561 = 35 , 26439 º


• An angle of 54,73561º is also formed by any of the three circles defining the sphere, and the
  reference plane. The sphere seats on this plane through the dashed circle shown in Fig. 6.


• In order to have the cylinder just reaching the cone surface, the cone ́s altitude a up to the
  upper base of the cylinder will be:

R R

R

a 2 1 , 4142135

tg

= = =

γ

• The radius of the cone ́s exterior base has to be:

R

R

R a tg 1 , 3535534

2

́  =









= + γ

• Cone generatrix is:

R

R

g 2 , 3444232

sen

́

= =

γ

• The developed angle of the conic surface is (Fig. 4):

207 , 8461 º

2 ́ 2 ́ 180

=

×

= =

g

R

radians

g

πR

ε

• Total altitude of cylinder:
  a+H= 2 , 2307101 R


• The circumference length of cylinder base is:
  2 πR= 6 , 2831853 R