MATHEMATICS AND ORIGAMI

Jesús de la Peña Hernández

Conoids are warped (not developable) ruled surfaces (with straight generatrices). One surface is

developable when the planes tangent to it along all the points of any generatrix mix in one only tangent

plane (recall the plane tangent to a cone along, of course, of one of its generatrices).

Conversely, in a warped surface, the tangent planes to it in the different points of a generatrix,

vary: they revolve from one position to another along the generatrix.

See in Fig. 5 how two tangent planes to the conoid in any generatrix are different depending of

which extremity of the generatrix segment we consider as point of contact. Similar remark is applica-

ble to Fig. 6.

It should be made clear that the conoid in Fig. 7 is also a warped ruled surface, and therefore

Fig. 9 is but the folding plan that produces a virtual surface by gathering some of the paper.

A TWISTED COLUMN (SALOMONICA)

To close the examples of virtual surfaces, we shall study this interesting and beautiful form af-

ter N. Nagata.

And we ́ll do it beginning from the end. CAD produces the generation of the solid fig-

ure as follows:

1

2

3

4

5

6

7

8

Z

X

Y

O

7

9

8

O

Y

6

8

7

5

1

4

3

(^2) X
Z
C E
B
O
D
F
A
10