the lemniscate of bernoulli

The cool way to write infinityBy Jonathan O’Connell

Bernoulli was born in The Netherlands in 1700.

Part of a predominate math family.

Worked in areas of statistics, physics, and mathmatics.

Developed fluid dynamics equations.

1 2OP QQ

Cartesian:

Polar:

Parametric:

2 2 cos 2r a

2

2

cos

1 sinsin cos

1 sin

a tx

ta t t

yt

2 2 2 2 2( ) ( )x y x y

OC = CA = OP’ is parallel to

HF’ OQ = PP’ 2QC = OH 2PQ = HH’ Then is

similar to Now 2PC = OH’

= a

12

2a

'H HOPQC

My cool geogebra file

COQ Let be and OQ be r

Segment OH =

OA = sinOA

2a

sin 2 sinOA a

It’s a trig identity!

2 2 2 2' 'r HH OH OH 2 2 22 sina a

2 cos 2a

0.2

0.4

0.6

0.8

1

30

210

60

240

90

270

120

300

150

330

180 0

Matlab generated polar graph

2 2 cos 2r a

cos

sin

x r

y r

2

2

2

cos 2

cos cos 2

sin cos 2

r a

x a

y a

4 4

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

x axis

yaxi

s

r2=cos(2*theta) on [-pi/4,pi/4] and [3*pi/4,5*pi/4]

Matlab generated graph

=

Delicious math

http://en.wikipedia.org/wiki/Daniel_Bernoulli

Lee, Xah. "Lemniscate of Bernoulli." Visual Dictionary Of Special Plane Curves. 23 Nov. 2008 <http://xahlee.org/specialplanecurves_dir/lemniscateofbernoulli _dir/lemniscateofbernoulli.html>.

A Book of Curves by E. H. Lockwood