# Complex Projective 4-Space

Complex Projective 4-Space recently celebrated its first birthday, and I was surprised to learn it was that young. I’ve been reading since January or so, and I guess I just assumed it had been around longer. It’s written by Adam P. Goucher, a former mathematical olympian (if that’s a term), and it features a wide variety of interesting mathematical topics generally accessible to motivated people with an advanced undergraduate understanding of math.

A visualization of the complex function (z/2)^7-1, with the 7 Antarcticas showing where the roots of the function are. Image: Adam P. Goucher.

In the blog’s about page, Goucher defines complex projective 4-space mathematically and explains why he used it as the blog name. “Informally, however, ‘complex projective 4-space’ was used in a joint Anglo-Hungarian IMO training camp to refer to a mythical world inhabited by unimaginable beasts. On reflection, these ideas are more similar than one might imagine: complex projective 4-space is indeed inhabited by such impossible-to-visualise objects as polychora, Klein bottles and an embedding of the E8 lattice. Hence, it seemed like a reasonable title for a blog concerned with interesting aspects of mathematics.” (links added by me)

Goucher has done some research on Conway’s Game of Life, so there are a few specialized posts on that topic, including a “breaking news” post about a stable π/2 reflector and some other recent developments. From September to November 2012, Goucher posted approximately a chapter a week from the draft of his book Mathematical Olympiad Dark Arts, which can be found under the MODA tag on the blog. “In this volume, I have attempted to amass an arsenal of the more obscure and interesting techniques for problem solving,” Goucher writes in the preface. Never having been involved in mathematical olympiad-type competitions myself, it’s interesting to see what sorts of topics and tricks are included.

Goucher also publishes a roughly weekly cipher on CP4. I’m not into ciphers, so I’ve never played with any of them, but if you like that sort of frustration, you can check out the cipher archive on the blog.

Some of my favorite posts from CP4 have been:

Affine spaces over F3, an interesting way of looking at the card game Set.

Analyzing Escher. One of the pictures in question in which water appears to be flowing downhill in a perpetual loop. “The fact that this can be drawn but not built is a consequence of the non-invertability of the projective transformation used to convert a three-dimensional scene into a two-dimensional photograph. It has a non-trivial kernel, which means that many points (indeed, infinitely many) in the three-dimensional space are mapped to the same point in the photograph, causing a loss of information that can conveniently be exploited to yield impossible drawings.”

Field With One Element, a post about several mathematical objects that may or may not exist, such as a certain class of primes and aperiodic monotiles.

Influential Mathematicians, “randomly selected examples of influential mathematicians” including Hypatia and Emmy Noether.

Visualising complex functions uses a map of the Earth to help illustrate the zeros and poles of complex functions. Antarcticas are zeros, and the North Pole is a pole. The illustrations, such as the one at the top of this post, are very entertaining.

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