MATHEMATICS AND ORIGAMI

Mathematics and Origami

13.6 ANOTHER CURVES

Origami deals with conics as the envelopes of their tangents; likewise we shall study

now some other curves, though not conics, under an analogous treatment.

13.6.1 LOGARITHMIC SPIRAL

Its equation in polar co-ordinates is:

ρ=kemω (1)

It is represented in Fig. 1 after a hexagon. We can see in it that the angles grow as an

arithmetic progression of ratio

6

π

whereas the radius vectors do as a geometric progression

with

6

cos

π

as ratio. This correspondence of arithmetic and geometric progressions brings forth

logarithms. Let ́s find out the value of the constants in (1) to conclude that the spiral we get is

actually a logarithmic one. Calling a to the apothem of the hexagon we have:

Vertex rw

1 a

6

5

π

2

6

cos

π

a

6

6

π

3

6

cos^2

π

a

6

7

π

4

6

cos^3

π

a

6

8

π

F ́ F

cd

3

O

cd ́