While browsing through this site, I came upon the text of Arnold: "On teaching mathematics".


containing the phrase

... 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.

So, here is my question:

What is the relation between the hypocycloid and ideals?

Edited to add in view of the first comment:

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).

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    $\begingroup$ First, in view of the general style of this text (which I just browsed and found quite interesting/entertaining), I would not look for too exact a meaning in each sentence. Second, there is a clear relation between (algebraic) curves and varieties and ideals, see e.g. here en.wikipedia.org/wiki/Algebraic_variety for a glimpse. I assume the main sentiment to be expressed (also in other passages of the text) is that it doesn't make sense to teach algebraic geometry (and the necessary basics from comm. algebra) to somebody who never saw a nice/interesting curve or surface beforehand. $\endgroup$ – user9072 Apr 21 '11 at 16:05
  • $\begingroup$ Thanks for your comment, I have added something to the question because I suspect (and could be wrong, of course) that the author had a more exact parallel in mind. $\endgroup$ – user11235 Apr 21 '11 at 16:53
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    $\begingroup$ You are welcome. However, my impression, but I might be missing something, is that there is nothing particular about the type curves (hypocyloid) and that they are just mentioned to give an example of for nice/interesting curve (and could be replaced by any other type of interesting curve). Now, why and how the study of curves and surfaces is naturally related to ideals and other notions of commutative algebra, a lot could be said on this. However, I think that this site is not the right place to discuss basic notions of Algebraic Geometry. $\endgroup$ – user9072 Apr 21 '11 at 17:06

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