Tag Archives: Euclid

A premature book review and a 60 year history with complex variables in 4 acts

“Visual Differential Geometry and Forms” (VDGF) by Tristan Needham is an incredible book.  Here is a premature review having only been through the first 82 pages of 464 pages of text.

Here’s why.

While mathematicians may try to tie down the visual Gulliver with Lilliputian strands of logic, there is always far more information in visual stimuli than logic can appreciate.  There is no such a thing as a pure visual percept (a la Bertrand Russell), as visual processing begins within the 10 layers of the retina and continues on from there.  Remember: half your brain is involved in processing visual information.  Which is a long winded way of saying that Needham’s visual approach to curvature and other visual constructs is an excellent idea.
Needham loves complex variables and geometry and his book is full of pictures (probably on 50% of the pages).

My history with complex variables goes back over 60 years and occurs in 4 acts.


Act I:  Complex variable course as an undergraduate. Time late 50s.  Instructor Raymond Smullyan a man who, while in this world, was definitely not of it.  He really wasn’t a bad instructor but he appeared to be thinking about something else most of the time.


Act II: Complex variable course at Rocky Mountain College, Billings Montana.  Time early 80s.  The instructor and MIT PhD was excellent.  Unfortunately I can’t remember his name.  I took complex variables again, because I’d been knocked out for probably 30 minutes the previous year and wanted to see if I could still think about the hard stuff.


Act III: 1999 The publication of Needham’s first book — Visual Complex Analysis.  Absolutely unique at the time, full of pictures with a glowing recommendation from Roger Penrose, Needham’s PhD advisor.  I read parts of it, but really didn’t appreciate it.


Act IV 2021 the publication of Needham’s second book, and the subject of this partial review.  Just what I wanted after studying differential geometry with a view to really understanding general relativity, so I could read a classmate’s book on the subject.  Just like VCA, and I got through 82 pages or so, before I realized I should go back and go through the relevant parts (several hundred pages) of VCA again, which is where I am now.  Euclid is all you need for the geometry of VCA, but any extra math you know won’t hurt.


I can’t recommend both strongly enough, particularly if you’ve been studying differential geometry and physics.  There really is a reason for saying “I see it” when you understand something.


Both books are engagingly and informally written, and I can’t recommend them enough (well at least the first 82 pages of VDGF).


Book Review: Tales of Impossibility

Here is a book for anyone who has had high school geometry and likes math.  It is “Tales of Impossibility” by David Richeson.  It’s full of diagrams and is extremely well written.  A bright high school student could go all the way to the end, and would learn a lot of abstract algebra, up to and including complex numbers, irrational numbers and transcendental numbers.  It describes the 2000+ year search for ways to trisect an angle, double the cube, construct any polygon using a compass and straight edge,  and find the area of a circle (squaring the circle), or prove that it was impossible using basic methods.

It took until the late 1800s to finish the job.  Proving that something is impossible is subtle and difficult.    The book is 368 pages long and contains 40 pages of notes and references, but it is definitely not turgid.

There is a huge amount of historical detail about each of the great figures who worked on the problems starting with Euclid and going on through the the Greek geometers, Fermat, Descartes.

The battles about what could be considered kosher in math occurred every step of the way and is well covered.  Could algebra be used to solve a geometric problem?  Was a negative number a number. What about an imaginary number, or an irrational one?   Was something you could draw using a marked ruler (neusis) really a geometric figure?

If you look at nothing else, have a look at how Descartes was able to multiply and divide the length of various lines, using nothing more that Euclid’s geometry (but apparently no one had figured it out before).

The ultimate impossibility proofs involved abstract algebra, so we meet Viete and Descartes, Galois, Hermite etc. etc.  So it might help if some high school algebra was on board.

For the right smart high school kid, this book is perfect.  For the cognoscenti or even for nonCognoscenti with a lifelong interest in math (such as me) there is a lot to learn.  The proofs of all the geometric statements are all well laid out, and now it’s time for me to go through the book a second time and follow closely.