Geometry and Number Theory on Clovers

Subdividing the Trefoil by Origami

“…e 2005 paper of Cox and Shurman [7] expands the family of divisible curves to include the clover. e -clover is the plane curve de�ned by the polar equation:…”

“…In their paper, they prove that these �rst four curves can be divided into arcs of equal length by origami (paper-folding) construction for certain values of , as follows. eorem 1 (see [7] ere are only �ve Fermat primes known, but more than 4000 Pierpont primes have been found; as of June 2012, the largest known Pierpont prime is 3 ⋅ 2 7033641 + 1, which has 2117338 digits. ( [9]; 16th on the list of largest primes.)…”

“…Note. Practically all of the hard work can be found in [7], to which we refer the reader for the details of our arguments. For detailed information about Galois theory, especially its application to subdividing the lemniscate, see [4].…”

“…From the polar coordinate formula, it is easy to derive the arclength formula in terms of ( [7], p.686):…”

A Wallis Product on Clovers

“…In Section 2 we recall the theory of clover curves introduced in [4] where we realize ̟ m as an arc length on the m-clover. Our proof generalizes the definite integral approach by considering the sequence of definite integrals…”