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 SuárezÁlvarez Dec 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'Rourke Oct 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 $\endgroup$ – James Cranch 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.
In Glimpses of algebra and geometry by Gabor Toth, chapter 25 is devoted to Klein's main result.
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. $\endgroup$ – Andy Putman 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. :) $\endgroup$ – Alon Amit Dec 21 '09 at 19:58

1$\begingroup$ 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. $\endgroup$ – Andy Putman Dec 21 '09 at 20:12
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. $\endgroup$ – David Lehavi Dec 22 '09 at 5:07

4$\begingroup$ See Jerry Shurman's answer for a link to a free copy ;) $\endgroup$ – Dr Shello 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. $\endgroup$ – David Lehavi 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.