# Pretzel recipe with a twist

Browsing the geometry section in the local library you come across a flimsy book, seemingly misplaced:

"The Pretzel Cookbook: Five More Patterns Using The Spoke Method"

What's more, it's been ripped in half. Flicking through the remaining three pages, you ask yourself:

1. How are these pretzels formed?
2. What was on the three missing pages?

Page 1:

Page 1 — Contents

Page 2 — The "Five"

Page 3 — The "Minus a Half"

Page 4 — The "One"

Page 5 — The "Two"

Page 6 — The "Minus Five"

Page 2:

Page 3:

Hint

The sequence of the pages isn't important

Hint

You could solve this with drawing equipment

I think the pretzels are formed by drawing plots of

the trochoids $x=r\phi + A\sin(\phi)$, $y=r + A\cos(\phi)$ with $r=1$, $\phi\in [0,4\pi]$ and $A$ the value of the page title. The word 'spoke' gave the solution away for me, we can get these shapes by sticking a light to a bike's spoke, and following this light.

That means the pretzels on the missing pages are:
Page 4:

Page 5:

Page 6:

• more intuitively, you could say that the page title is the distance (in spokes) that the light is placed above the wheel's axle (before cycling for two wheel rotations)
– jsh
Commented Jul 2, 2015 at 12:56