Reading Serre's letter to Gray , I wonder if now modern expositions of the themes in Klein's book exist. Do you know any?

3$\begingroup$ The link should go to page 550 of the book, presumably. $\endgroup$– Mariano SuárezÁlvarezDec 21 '09 at 14:42

$\begingroup$ Related MO question: Can the unsolvability of quintics be seen in the geometry of the icosahedron?. $\endgroup$– Joseph O'RourkeOct 11 '19 at 12:18
"Geometry of the Quintic" is available for free at my website.
Jerry Shurman

14$\begingroup$ To save other people a few clicks, Jerry's webpage is people.reed.edu/~jerry $\endgroup$ Jun 10 '14 at 10:59
I got interested in this subject last year (2011) and just got round to writing up some notes which I hope may be of use. I also have a python script hosted here, which implements Klein's icosahedral solution of the quintic, as well as a brief summary of what it does here.
The geometry is easy to summarise: using a radical transformation, a quintic can be put in the form $y^5 + 5\alpha y^2 + 5\beta y + \gamma = 0$. The vector of ordered roots of such a quintic lies on the quadric surface $\sum y_i = \sum y_i^2 = 0$ in $\mathbb{P}^4$ and the reduced Galois group $A_5$ acts on the two families of lines in this doublyruled surface by permuting coordinates. The $A_5$ actions on these families, parameterized by $\mathbb{P}^1$, are equivalent to the action of the group of rotations of an icosahedron on its circumsphere and the quintic thus defines a point in the quotients — the icosahedral invariants of a quintic. Inverting either of these quotients (e.g., using the hypergeometric functions given below) is sufficient to allow us solve quintic (in rational functions).
Here's how it looks for a quintic in the simpler form: $$ y^5 + 5y + \gamma = 0 $$ (In fact any quintic can be put in this form using only radical transformations.)
Given such a quintic, set: $$ \nabla = \sqrt{\gamma^4 + 256}\\ Z = \frac{1}{2\cdot 1728}[2\cdot 1728 + 207\gamma^4 + \gamma^8  \gamma^2 (81 + \gamma^4)\nabla]\\ z = \frac{{}_2F_1(\frac{31}{60}, \frac{11}{60}; \frac{6}{5}; Z^{1})} {(1728Z)^{1/5}{}_2F_1(\frac{19}{60}, \frac{1}{60}; \frac{4}{5}; Z^{1})} $$ and: $$ f(z) = z(z^{10} + 11z^5  1)\\ H(z) = (z^{20} + 1) + 228(z^{15}  z^5)  494z^{10}\\ T(z) = (z^{30} + 1) + 522(z^{25}  z^5)  10005(z^{20} + z^{10})\\ B(z) = 1  z  7(z^2  z^3 + z^5 + z^6) + z^7  z^8\\ D(z) = 1 + 2z + 5z^2 + 5z^4  2z^5  z^6 $$ Then: $$ y = \gamma\cdot\frac{f(z)}{H(z)/B(z)}  \frac{7\gamma^2 + 9\nabla}{2\gamma(\gamma^4 + 648)} \cdot\frac{D(z)T(z)}{f(z)^2H(z)/B(z)} $$ is a root.
Replacing $z$ with $e^{2\pi\nu i/5}z$ for $\nu=1, 2, 3, 4$ provides all the other roots.
Even in this rather gross explicit form, the link with regular solids is visible:
The roots of $f, H, T$ are, respectively, the locations of the projection of the vertices, face centres, and edge midpoints of a regular icosahedron onto its circumsphere (once this circumsphere has been identified with the extended complex plane by stereographic projection).
The roots of the last two polynomials, $B, D$ are, respectively, the locations of the vertices and face centres of a regular cube inscribed in the icosahedron.
I covered Klein's "Lectures on the Icosahedron" in a modern way in my doctoral thesis:
Elliptic Curves and Icosahedral Galois Representations, Stanford University (1999) http://www.math.purdue.edu/~egoins/notes/thesis.pdf
A much shorter and more direct exposition is my publication in IMRN:
Icosahedral $\mathbb Q$Curve Extensions, Mathematical Research Letters 10, 205–217 (2003) http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2003/0010/0002/MRL20030010000200019947.pdf
There is a (german) new edition of Klein's "Vorlesungen über das Ikosaeder ..." by Peter Slodowy (1993) with a large (about 80 pages) section of comments and remarks about new developments.
Chapter 5 of McKean and Moll's "Elliptic Curves" explores the circle of ideas around Ikosaeder.I'm not sure if you'd consider this sufficiently "modern"  it's certainly a contemporary book but it doesn't use, say, schemetheoretic language.

3$\begingroup$ It looks like our identical answers crossed paths! You beat me by one minute, so I'll delete my answer. $\endgroup$ Dec 21 '09 at 19:18

$\begingroup$ lol, you're right  I caught it just before you deleted. Very gallant of you! and hey, in math, every minute counts. :) $\endgroup$ Dec 21 '09 at 19:58

2$\begingroup$ As an aside to everyone else  McKean and Moll's book is really beautiful! I read large chunks of it as an undergraduate, and I still go back to it periodically. $\endgroup$ Dec 21 '09 at 20:12
In Glimpses of algebra and geometry by Gabor Toth, chapter 25 is devoted to Klein's main result.
There is the outrageously expensive "Geometry of the quintic" by Jerry Shurman, which discusses both Klein and DoyleMcMullen approaches (and then some more).


$\begingroup$ since it's softcover with horrible print quality  yes it is. $\endgroup$ Dec 22 '09 at 5:07

4$\begingroup$ See Jerry Shurman's answer for a link to a free copy ;) $\endgroup$ Jan 12 '11 at 8:33

1$\begingroup$ Re: Dr. Shello  yes, Shurman remarked a few months ago on MO he made the text publicly available. This answer predates his release. $\endgroup$ Jan 12 '11 at 9:03
In my PH Thesis work http://systembit.es/schwarz.htm
In my papier I have asociatted a Riemann Surface to each Schwarz function triangle. After that ,I´ve got genus and geometric density of spherical tesselation expresions in a new method. Then I see Their Poincare Groups ( of each above Riemann Surfaces) as index normal finite subgroup of Г(2) (Thanks to Modular Function Lambda). Then I calculate signature of these fuchisian Groups. Finally I see there are only nine ( of above Riemann Surfaces) more Dihedrical cases.
I think my idea is a new interpretation of Schwarz triangles , different one to the Famous Schwarz Classification based on 14 Schwarz triangles +Dihedrical cases.
Alfonso García alfonso@systembit.es
You could also take a look at Section 1.6 of Finite Mobius Groups, Immersion of Spheres, and Moduli, by Gabor Toth.