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$$1728 V^5 + F^3 = E^2 \;.$$

Can anyone point me to a concise, modern derivation and explanation of the significance of the icosahedron equation, more modern and concise than Klein's description in his book?

Lectures on the Ikosahedron and the solution of equations of the fifth degree. Felix Klein, 1888.

The equation first appears in Klein on p.62 (Dover edition) as $$T^2 = -H^3 + 1728 f^5 \;,$$ where $f$, $H$, and $T$ are "forms." $H$ represents the "Hessian form," and $T$ the "functional determinant."

I cannot find the equation in Jerry Shurman's 1997 book, Geometry of the Quintic (PDF download), although I admit I have only scanned his book.

Incidentally, here is a (crude—Sorry!) plot of $x^3 + y^5 + z^2 = 0$, equivalent to the above by scaling variables:
Perhaps this equation has been studied in its own right?

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My guess is that "form" means "invariant polynomial with respect to the action of $A_5$ on $\mathbb{C}^2$." – Qiaochu Yuan Jul 14 '13 at 1:44
You don't want to think of that equation as defining a surface, but as a curve in the appropriate weighted projective space. Then it's probably the modular curve $X(5)$ or something similar. – Felipe Voloch Jul 14 '13 at 1:45
up vote 4 down vote accepted

There is this:

and this:

Icosahedral symmetry and the quintic equation R.B. King Department of Chemistry, University of Georgia Athens, Georgia 30602, U.S.A. E.R. Canfield Department of Computer Science, University of Georgia Athens, Georgia 30602, U.S.A., How to Cite or Link Using DOI Permissions & Reprints

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Thanks, Igor! I will check these out... – Joseph O'Rourke Jul 14 '13 at 2:01
Actually I revised my notes linked above a few months ago. I hadn't got round to updating the online PDF but after seeing a little traffic from this question I now have. – Oliver Nash Jul 14 '13 at 20:28
@Oliver Nash: cool! the notes were pretty nice to begin with! – Igor Rivin Jul 14 '13 at 22:54
@Igor Rivin: Thanks! – Oliver Nash Jul 14 '13 at 23:14

The equation is at the bottom of p.61 of Geometry of the Quintic. It is the syzygy on the three degenerate icosahedral forms.

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Try looking into

MSRI Publications -- Volume 35

The Eightfold Way: The Beauty of Klein's Quartic Curve

Edited by Silvio Levy

Cambridge University Press, Cambridge, 1999, x + 331 pp.

ISBN: 0521660661

which is freely available on the web.

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Great suggestion---Thanks! It may be that Noam Elkies's chapter, "The Klein Quartic in Number Theory," is exactly what I seek. – Joseph O'Rourke Jul 14 '13 at 2:27
That's the paper I was about to mention in my answer but then left it out because I was sure you would be able to figure it our for yourself. – Chandan Singh Dalawat Jul 14 '13 at 3:53
This is a gem of a book but it is really about the modular curve X(7) and not the curve X(5) defined by the equation in the question. I'm not sure if the syzygy referred to is derived in any of the papers it contains. – Oliver Nash Jul 14 '13 at 20:40
Thank you, Oliver, I was too quick to assume the the Kline quartic was exactly the equation I detailed. – Joseph O'Rourke Jul 15 '13 at 19:48

I think some of this is also described in McKean and Moll's pretty book on Elliptic Curves.

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