Xah's Math Blog Archive 2010-01 〜 2010-05

crop circle Diessenhofen 2008-07-15-s
img src

Geometric Design thru Crop Circle.

violin neck scroll volute
The neck of a unfinished violin. img src.

For more, see: Spirals in Nature.

sphere two-point islamic pattern-s
Sphere with two-point pattern. img src For more info, see: Decorative Pattern on Sphere.

Martin Gardner (1914 〜 2010) Remembrance

Converting Math Problem Into a Question in Formal Language (Some thoughts)

Learned of the Chinese mathematician Fan Chung (金芳蓉). Wife of Ronald Graham. Remember, Ronald is the mathematician who co-authored with Knuth the book Concrete Mathematics (amazon).

In recent years, i learned quite a lot high powered mathematicians who are Chinese. Here's some list of Chinese mathematicians:

Wikipedia actually has a list: Category:Chinese mathematicians.

Chuu-Lian Terng is wife of Richard Palais. Richard i first met online in 1997, and them both in person in 2004, and has been personal friends since.

Here's some mathematicians that i admire. Typically, it more has to do with their subjects. Geometry, discrete math, combinatorics.

Stephen i met in 1995.

There are quite a few more but off-hand these comes to mind.

There are so many mathematicians today, that one hardly know much of them. Is there a list that lists the top one thousand mathematicians?

A mathematician, named Phlexicon, contacted me today, about a error in one of my proof in my learning note of Introduction to Real Projective Plane .

We chatted on Skype for about 40 min. He showed me, how my version of the proof on Sylvester-Gallai Theorem was wrong. Though, it's been 13 years since i wrote the proof, so, i couldn't seriously understand it without spending few days reviewing the stuff. Though, he convinced me he's right. I remember, when i was studying it, the proof given in the text seems a bit complex and convoluted, and i thought to myself that it should be done with a simple induction proof. After i did a proof, i wondered a bit why the problem went without a proof for some 40 years. This is back in 1996, and the internet only started, without all the blog and feedback and social networks, and i didn't put my notes online until 2004, and didn't show it to anyone.

He pointed out the Wikipedia article: Sylvester–Gallai theorem. That is a wealth of info.

Phlexicon also showed me a interesting elementary geometry problem. You might try to show it to your kids in highschool. Here's the problem:

Suppose there's a pyramid (as in Egyptian pyramid, with a square bottom), such that each of the triangle faces are Equilateral triangle. Let's call this pyramid p4. Now, let's say there's a regular tetrahedron, which is also a pyramid but with the base being a equilateral triangle. Let's call this p3. The question is, what is the ratio of volume of p3 and p4. (The length of the edge of p3 and p4 are the same.) You are to solve this problem by insight, and you are not allowed to use algebraic formulas.

Phlexicon said that this problem can be solved by insight, with 2 key realizations. I thought about it for 20 min yesterday but haven't seen it yet.

knot tiling-mag-d4
Computer generated weaving as decorative art. img src

Knitting, Chinese Knots, Braid Theory.

Cleaning up a blog a wrote few years ago: What is the Difference of Symbolic Logic System, Hilbert's Formalism, Russell's Logicism, Axiomatic System?.

Grigori Perelman and Money. (will you decline 1 million for some personal pride?)

aperiodic tile
A single tile that tile only aperiodically.

Whether a single tile exists that tiles only aperiodically is a unsolved problem. This paper seems to solve it, or partially, this question.

See: An aperiodic hexagonal tile , By Joshua E S Socolar, Joan M. Taylor. arxiv.org 1003.4279v1.pdf

via http://www.mathpuzzle.com/.

See also, some of my tiling studies:

In the past few years, i discovered quite a few math formula editors that are not based on TeX/LaTeX. See bottom of: The TeX Pestilence (the problems of TeX/LaTeX).

There's a painting exhibition in San Francisco, hosted by Science Fiction in San Francisco (SF in SF), featuring the works of mathematician and science fiction writer Rudy Rucker.

Rudy is famous for his books such as “The Fourth Dimension” (1984), Infinity and the Mind (1995), and latest non-fiction on cellular automata: The Life Box, The Seshell, and The Soul (2005). amazon

The painting exhibition will be on from April 9 (Friday) to May 22 (Saturday), at Variety Preview Room in San Francisco. (582 Market Street, San Francisco, CA. (415) 781-3893) (View Map)

You're invited to an opening night party on Friday, April 9, from 6 to 9 pm.

In the closing event on Saturday May 22, from 6 to 10 pm, Rudy will read with author Michael Shea.

Paintings and prints will be for sale at the show during the opening and closing events, or online from Rudy's paintings page.

Wrote a little explanation on the status of my Visual Dictionary Of Special Plane Curves project. Here: Special Plane Curves: What's New.

Bird Flight V Formation (recreational math problem)

Discovered a fun math program (via mathpuzzle.com) called MagicTile. This software lets you play Rubik's cube but represented thru a Stereographic Projection. See: Great Software For 2D Visualization of Geometry.

In preparing to learn throughly about rotations, its representation, quaternions, computational techniques, i thought about what is a rotation? In a plane, you rotate thru a point. In 3D, you rotate around a axis. How about higher dimensions? What is the gist of rotation? A moment of thought leads to isometry with one invariant point, but that does not rule out Point reflection. Wikipedia has something to say about it Rotation (mathematics), will have to read later.

Reading Notes on Nicolas Bourbaki.

Learning Notes Of Symmetric Space and Differential Geometry Topics (learning notes)

Richard Palais and Bob Palais has some article about rotations for computer graphics, supposedly better than Quaternions. Here's the article: New Algorithms For Implementing And Interpolating Rotations , by Bob Palais and Richard Palais. transvection_for_rotations.pdf.

Bob also has some interactive demo written in Flash, here: Source www.math.utah.edu.

Fabrice Bellard, using a PC, Computed π to about 2.7 trillion places, claimed to be the latest world record. (previous records are made by super computers that costs millions.) He's home page is at http://bellard.org/, which details this among other things. A highly accomplished C programer. Probably the world's top 100 or even 10.

What's personally interesting is that he also created a Emacs-like editor: http://bellard.org/qemacs/. Undoubtedly he was a emacs user, but got frustrated with emacs's inherent inability of opening large files, or files with long lines, for dealing with π digits.

His other accomplishments include: FFmpeg (for processing multimedia data (⁖ audion and video)), QEMU (cpu emulator).

Euler Angles And Gimbol Lock

Am starting this math blog, of any thing that comes to my mind about math. This blog is branched off from my main blog Xah Lee's Blog, so it is more subject focused.

For ≈500+ pages related to math on my website since 1997, see: Xah Math.

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