IC 418 spans 0.3 light-years across. IC 418 IC 418, also known as Spirograph Nebula, is a planetary nebula in the Milky Way Galaxy. The name derives from the intricate pattern of the nebula, which resembles a pattern which can be created using the Spirograph, a toy that produces geometric patterns (specifically, hypotrochoids and epitrochoids) on paper. Only a few million years ago, IC 418 was probably a common red giant star. Since running out of nuclear fuel, though, the outer envelope has begun expanding outward leaving a hot remnant core destined to become a white dwarf star. IC 418 lies about
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