MATHEMATICS AND ORIGAMI

Mathematics and Origami

We overlap now Fig. 3 to the figure just obtained, in order to check how the rounded

extremities of the cardioid lie in a very particular way on the respective chords 1,2; 2,4; etc.

We have to prove now that the folding process after Fig. 4 is equivalent to the geometry

in Fig. 3. To obviate a tangle of lines we outline in Fig. 5 only the elements of Fig. 4 related to

ω = 45º and ω = 292,5º.

We shall focus our reasoning in ω = 45º ; alike result would be reached for any other

value of ω. Being the proportions of the segments as shown in Fig. 6, with the marked angles in

O and A ́ measuring 45º, B ́ will coincide with X, bearing the consequence ofB ́ 4 = 2 ×B ́ 2 and

A ́B ́= 2 r.

7

1

2

3

4

5

(^76)
9 8
10
11
12
13
14
15
16
17
18
19
20
22 23
24
25
26
27
28
21
Fig. 7 is a cardioid got after chord
folding, for a circumference divided in 28
equal parts.
13.6.3 NEPHROID
Fig. 1 shows that curve whose shape re-
minds that of a kidney. It has been obtained like
Fig. 7 in point 13.6.2, but applying to chords and
foldings a pitch of i → 3i.
5
A ́
B ́
P O
2
(^43)
10
11
12
13
A ́ ́
B ́ ́
A ́
O
4
B ́
2
X
6
2r
r
3r
2r
1
1
28
27
From Fig. 5 we can
also deduce that a car-
dioid is as well an epi-
cycloid generated by a
circumference with a
diameter equal to the
directrix circumferen-
ce ́s (that having O as
its center).