Spirograph Spirograph Spirograph is a geometric drawing toy that produces mathematical roulette curves of the variety technically known as hypotrochoids and epitrochoids. It was developed by British engineer Denys Fisher and first sold in 1965. The name has been a registered trademark of Hasbro Inc. since 1998 following purchase of the company that had acquired the Denys Fisher company. The Spirograph brand was relaunched worldwide with original product configurations in 2013 by Kahootz Toys. In 1827 English architect and engineer Peter Hubert Desvignes developed his "Speiragraph", a machine to create elaborate spiral drawings, intended to prevent bank note forgeries. The mathematician
Twists of curves Twists of curves In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant. First assume K is a field of characteristic different from 2. Let E be an elliptic curve over K of the form: