What is the relation between hypocycloids and ideals in polynomial rings as alluded to in Arnold's text on teaching mathematics? - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T04:48:19Z http://mathoverflow.net/feeds/question/62543 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/62543/what-is-the-relation-between-hypocycloids-and-ideals-in-polynomial-rings-as-allud What is the relation between hypocycloids and ideals in polynomial rings as alluded to in Arnold's text on teaching mathematics? thei 2011-04-21T15:18:46Z 2011-04-21T16:49:51Z <p>While browsing through this site, I came upon the text of Arnold: "On teaching mathematics".</p> <p><a href="http://pauli.uni-muenster.de/~munsteg/arnold.html" rel="nofollow">http://pauli.uni-muenster.de/~munsteg/arnold.html</a></p> <p>containing the phrase</p> <blockquote> <p>... it can be said that a hypocycloid is as inexhaustible as an ideal in a polynomial ring. But teaching ideals to students who have never seen a hypocycloid is as ridiculous as teaching addition of fractions to children who have never cut (at least mentally) a cake or an apple into equal parts.</p> </blockquote> <p>So, here is my question:</p> <blockquote> <p>What <em>is</em> the relation between the hypocycloid and ideals?</p> </blockquote> <p>Edited to add in view of the first comment:</p> <p>The hypocycloid is an algebraic curve and the polynomials that vanish on this curve form an ideal. But is there anything about the hypocycloid that motivates the question of regarding vanishing polynomials (which is clearly the case for cutting cakes and adding fractions or some of the other mathematical/physical examples in the text).</p>