MATHEMATICS AND ORIGAMI

Mathematics and Origami

5. OG → OG ; E → E
   Result: EF


6. OI → OI ; F → F
   Result: FH

13.6.2 CARDIOD

A BIT OF HISTORY

Fig. 1 shows the conchoid of a straight line named after Nichomedes. Curve c, in its

two branches, is determined by line r, pole P and equal segments like AB = AB ́.

The Pascal ́s limaçon (Fig. 2) is the conchoid of a circumference c (radius OP = r) with

respect to one of its points P, in such a way that always it is AB = AD < 2r. It also has two

branches.

The CARDIOID derives from the Pascal ́s limaçon, with this condition: AB = AD = 2r.

The curve is symmetric but has only one branch. Its shape reminds that of a heart (Fig. 3).

This Fig. 3 shows the rounded points of the cardioid at the extremities of its radius vec-

tors. Being

OP = r ; AB = AP = A ́B ́ = A ́B ́ ́ = 2r

and taking into account the right triangle PAA ́, the cardioid polar equation will be:

ρ= 2 rcosω+ 2 r

ρ= 2 r()cosω+ 1

We have drawn Fig. 3 by dividing the circumference with radius OA in 8 equal parts,

what means that, as P lies on the circumference, angles ω have these values:

22,5º; 45º; 67,5º; 90º; 112,5º; 135º; 157,5º and 180º

Let ́s see now how to make this construction by paper folding.

For that we shall draw (Fig. 4) a circumference with center O and radius 3r; then divide

it in 16 equal parts numbering them orderly.

Afterwards we fold the chords between points i → 2i in the circumference, as we ́ll see

later. Those folded chords will be the tangents to the cardioid, and this will be the envelope of

them.

P

B ́

B

A

c

r

c

P O

A

B

D

1 2