Spirograph a hypocycloid. 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.
Inequality of arithmetic and geometric means numbers in the case . An example of this case is : Averaging two different numbers produces two equal numbers, but the third one is still different. Therefore, we never actually get an inequality involving the geometric mean of three equal numbers. Hence, an additional trick or a modified argument is necessary to turn the above idea into a valid proof for the case . With the arithmetic mean of the non-negative real numbers , the AM–GM statement is equivalent to with equality if and only if for all . For the following proof we apply mathematical induction and only