on a microwave, you press 2 0 0 for 120 seconds, but 1 2 0 for 80 seconds, yet 9 0 for 90 seconds.
can you establish the rules for microwave number system or microwave arithmetic?
see also: Teacup Recursion Animation
lousy. page deleted. search YouTube or web instead.
see also: Teacup Recursion Animation
a project by a old friend at Wolfram Research. 〈The State of the Unit: The Kilogram〉 http://www.kickstarter.com/projects/52746223/the-state-of-the-unit-the-kilogram-documentary-fil?ref=live
read this, and you'll understand everything about complex numbers, without hearing a word of “complex”. Coordinate Transformation
Math Symbols in Unicode. Now, hover your mouse over to see the full symbol name.
a blog by Glenn Wilcox with lots beautiful math-related pictures generated in Rhinoceros 3D @ http://www.generativedesigncomputing.net/
O, math, my true love, but how i have alienated thee, and you being quite difficult.
on a microwave, you press 2 0 0 for 120 seconds, but 1 2 0 for 80 seconds, yet 9 0 for 90 seconds.
can you establish the rules for microwave number system or microwave arithmetic?
thx to weizhoupan
can you identify what function is this?
it looks like a Mobius Transformation. But what's the exact parameters and what's the input image? (assume each ball is just a dot.)
Mobius Transformation is a function of the form f[z] = (a z + b) / (c z + d) and a d − b c ≠ 0. The variables are all complex numbers.
For a simple geometric explanation of complex numbers, see: Isometries: Translation, Rotation, Reflection and Mandelbrot Set Explained (no complex number needed).
“recursion” — a central theme in this song. Björk — Bachelorette
here's recursion bonanza.
5-Point Conics (needs Java installed.)
Lojban is a constructed language, design based on predicate logic. Some tutorial here, with voice recording: Xah's lojban Tutorial.
logic; linguistics
“The more there is of mine, the less there is of yours.”
That's the concept of Zero-sum game. However, in real world, most things isn't zero-sum game. Both parties actually benefit. e.g. when you give something you no longer want to someone, a value is created “out of nothing”.
Mathematician Robert J Vanderbei made a comment at Geometry: Gallery of Surfaces. He wrote a surface viewer using latest tech WebGL. It's excellent. Check it out at〔 http://www.princeton.edu/~rvdb/WebGL/minsurf.html 〕 . You will need to use a broweser that supports WebGL, such as Google Chrome.
I can never remember what “xor” do. But now i got a easy way to remember it. xor returns true if the values of two sides differ.
See also: Logical Operators, Truth Table, Unicode
updated links: Great software for Tilings, Patterns, Symmetry
A drawing by my friend Hop David. I ♥ his artworks. He's a math-orinted illustrator. Check out his gallery and books on his website: Hop's Gallery.
this tweet is going around:
Dear math, i'm sick and tired of finding your x, just accept the fact that she is gone.
It needs a prelude:
dear girls, i had a fight with my x, but i think she loves me still. help me find x^2+1=5. —Math
3 wonderful articles on cycloids. By mathematician physicist John Baez.
It mentioned my site a lot. Thanks John.
William Thurston died. (1946 〜 2012). Of melanoma, skin cancer.
I think i first heard of him thru Jeff Week's book 《The Shapes of Space》. (➲ Reading Notes on 《The Shapes of Space》) I realized that he's one of the world's top mathematician. Though, his work is beyond my understanding.
Memories of Kurt Gödel By Rudy Rucker. @ Source www.rudyrucker.com
Discovered a Unicode char ℇ
character: ℇ (codepoint 8455, #o20407, #x2107) name: EULER CONSTANT old-name: EULERS
This is a bit strange. It's not clear whether it meant the base of the natural logarithm or Euler–Mascheroni constant.
The symbol for the base of the natural logarithm is traditionally ℯ (SCRIPT SMALL E), while the symbol for Euler–Mascheroni constant is γ (GREEK SMALL LETTER GAMMA).
The Unicode ℇ has name EULER CONSTANT. It's a bit ambiguous, but i suppose it means ℯ. However, i tried several Unicode fonts, all of them has the appearance of mirrored 3, none showed it as ℯ.
lousy. page deleted. search YouTube or web instead.
Nice article:
See also: The Problems of Traditional Math Notation.
complex number simply explained: Geometric Transformation on the Plane
update, with video: Grigori Perelman and Money: Will You Decline 1 Million to Make a Statement?
minor updated. Buy the book: “Curves And Their Properties” by Robert Yates
slight update Cornu Spiral.
Some pictures of Magic Polyhedrons.
hoping to read this book:
but my plate is quite full already….
Thanks to Stanley Rabinowitz of mathpropress.com for donation, for the Visual Dictionary of Special Plane Curves project.
Note that you can purchase the entire project as downloadable file for easy reading on your computer or iPad. See: Buy A Visual Dictionary of Special Plane Curves. I still don't have Mathematica files updated yet, but you'll get free update when available.
Various very interesting mathematical objects.
In particular, it features of solids of constant width.
See also: Visual Dictionary of Special Plane Curves.
Discovered a unicode math symbol that's from Mathematica: “⧴” (RULE-DELAYED). Are there others? (see: Math Symbols in Unicode.)
Update, interactive GeoGebra applet showing Product of Rotations: Product of Rotations.
Reminder. If you like this blog, you might also enjoy my other programer related blogs:
Usually i don't repeat a article in more than one place, even if it's related to both.
“Year on Earth”
There's a extremely simple, nice, web site for creating math notations then you can point a url to. You type LaTeX code, it displays the results as a image on-the-fly immediately as you edit. It also comes with a short perm url that you can point to. You can come back to the url later and edit the expression anytime. The site is at mathurl.com
added to: Tools to Display Math on Web.
Google made a fractal application, based on Google Maps, at juliamap.googlelabs.com.
Though, am rather disappointed. When you zoom in just a few steps, the resolution does not automatically increase accordingly.
Much better are some dedicated fractal apps. See: Great Fractal Softwares.For a basic explanation of the mandelbrot set, see: Mandelbrot Set.
Looks like a great out of print book Tiling and Patterns (1986) By Branko Grunbaüm and G C Shaphard, is being republished by Dover. amazon
I'd say this is best math book i've ever read. (technically, i'd put this book in one of the top 10 i've read, because there are quite a lot good math books on various topics and written with different style for different audiences.)
below is some very old articles i wrote, about the book and other math books.
I worked intensively on tilings for about a year in around ~1997. The most mathematical is: The Discontinuous Groups of Rotation and Translation in the Plane, which are linked a lot. It is basically my own learning notes. All images are generated by Mathematica, a package i wrote: Plane Tiling Mathematica Package. I'll be updating it from Mathematica v3 to v7 in next few days. At the time, i wanted to create a most versatile software that generates any type of tilings, decorative patterns, without human intervention. A sort of AI in the spirit of Douglas Hofstadter. (See: Gödel, Escher, Bach amazon) Of course, i didn't get that far. For gallery, see: Geometric Tilings and Patterns Image Gallery.
All these pages are written in late 1990s. Much update needs to be done, on the html and the writing too.
Robert Dickau has done many nice combinatorial diagrams with Mathematica. For example, here's some i like:
His home page robertdickau.com has many more. You can also get many of his interactive Mathematica files at demonstrations.wolfram.com.
You might also enjoy a combinatoric diagram i did. See: Number Of Ways To Loop n Points. (it was done with Mathematica v3. I'll update the notebook to v7 soon.)
Many of Robert's work are about space filling curves. It is one of the math i learned in 1990 that had a big effect on me. In short, it shows that there are same number of points of one side of a square to the square itself. At the time, it was a shock to mathematicians. See: Space-filling curve.
Turning a torus inside out.
Added more symbols to Emacs xmsi-mode for Math Symbols Input.
∅ ƒ Ø ⌘ ⌥ ‸ ⇞ ⇟ ↖ ↘ ⌫ ⌦ ⎋ ⏏ ↶ ↷ ⇧ ⌤ ⇄ ⇤ ⇥ ↹ ↵ ↩ ⏎ ⌧ ⌨ ␣ ⌶ ⎗ ⎘ ⎙ ⌚ ⌛ ✂ ✄ ✉ ✍ □ ○ ◇ ■ ● ◆.
The symbol “ƒ” has unicode name “LATIN SMALL LETTER F WITH HOOK”, and its old unicode name is “LATIN SMALL LETTER SCRIPT F”.
It's a popular symbol on the mac in the 1990s. Often, a folder's name will end with it. You can type it on the Mac by pressing 【Opt+f】 See: Mac OS Roman. In math, that symbol is sometimes used to stand for “function”.
Many other symbols above are computer key symbols. They are used by Apple. (See: A Short Survey Of Keyboard Shortcut Notations)
Added a bunch symbols to Emacs xmsi-mode for Math Symbols Input.
◀ ▶ ▲ ▼ ◁ ▷ △ ▽ ☉ ☼ ☾ ☽ ☿ ♁ ♄ ♅ ♆ ♇ ♃ ♂ ♀ ☄. The symbols includes the 9 planets. Note that the male sign ♂ and female sign ♀ is also Mars and Venus, and in Unicode, there's just one char for them.
Also some interesting readings. See: Astronomical symbol.
Also, discovered this article about the pseudo-science book Worlds in Collision, which was a best-seller in 1950s. But today's people probably all forgot about it. By analogy, you should know that many concepts, ideas, practices, we have today in math, sciences, in programing, are probably garbage. In particular, i could personal say this is so with regards to math notations, computer syntaxes, and many computing practices (e.g. Design Patterns, eXtreme Programing, which were hot in early 2000s but today pretty much waned. See: Why Software Suck) See also: Men Are From Mars, Women Are From Venus.
Also added the del operator Nabla symbol ∇. The partial derivative symbol ∂.
If you use Emacs and type a lot math symbols, you may be happy to note that i wroe a minor mode for inputting math symbols. See: Emacs xmsi-mode for Math Symbols Input. For why i created it, see: Designing a Math Symbols Input System.
Discovered another citation to my Wallpaper groups: The 17 Wallpaper Groups work.
Chapter 1 Symmetry Groups Mathematics in Art and Architecture GEK1518 Helmer Aslaksen Department of Mathematics National University of Singapore aslaksen@math.nus.edu.sg www.math.nus.edu.sg/aslaksen/ Symmetry and Patterns
At Source www.math.nus.edu.sg.
More links:
See also: XahLee.org Site Awards and Recognitions.
How Mathematica does Unicode? (tutorial)
Tools to Display Math on Web (tips)
some links to xahlee.org math:
plants vs zombie last boss.
It's my greatest sorrow to have found that my friend Robby Villegas has died last month. Robby Villegas Died.
Here's a linear algebra notes i wrote in 1998, using Mathematica version 3, with much math typesetting. linearAlgebraNotes.nb. Mathematica 7 convert it to pdf very well. See: linearAlgebraNotes.pdf. Surprisingly, when converting to html, it actually generated a valid html with valid CSS. linearAlgebraNotes.html But further, it also exported to XML+MathML well: linearAlgebraNotes.xml (can be viewed with Firefox 3.6.12). Very well done!
A note about Mathematica's typesetting. Mma's typesetting capabilities is best on this earth. It was invented with Mathematica version 3 in 1996. It's not just a inert incomprehensible code as in TeX. For example, typing 「1/Sqrt[x^2 + y^3]」, press a button, then it become typeset. Yet, Mma understands it as a live math expression. When writing math, you don't need to learn some inane specialized formatting language. You just type as you code in a computer language. In a programing language, you can define functinons. For example, in javascript 「f = function (x,y) {1/(x^2+y^3)}」. Usually, the function name is limited to as a sequence of ascii chars. In other langs such as Haskell, you can define arbitrary operators. Again, the operator is limited to ascii chars, and basically just this binary form 「a ⊗ b」. In Mathematica, you can define arbitrary functions or operators, including fancy operator of big brackets such as matrix.
Also, whatever complex math expressions are automatically formatted, meaning, automatically wrapped. All these capabilities, i'd attribute to 2 very simple ideas:
Today, most mathematicians and engineers still use the extremely inane TeX/LaTeX. And much of open source offerings are based on TeX. MathML was a great idea, but it's a decade now and it's still practically not usable.
See also:
Greg Kennedy juggling inside a conical enclosure.
For comment and 2 more vids, see: Conical Surface Juggling.
Added a new section to bottom of: The Problems of Traditional Math Notation.
Benoît Mandelbrot, the mathematician who invented fractals, is dead. (1924-2010) He died of cancer. Aged 85.
the mandelbrot set.
The mandelbrot set is truely a amazing curiosity. If you only know highschool math, you can get a good explanation at Mandelbrot Set.
Gravity simulator. http://www.nowykurier.com/toys/gravity/gravity.html.
See also: Great Math Programs.
Torus Autologlyph
For more info and where to buy, see: Geometric Pattern on Sphere and Torus.
Detexify is a tool that lets you draw a math symbol and it shows you the code for LaTeX. The tool is created by Daniel Kirsch. At http://detexify.kirelabs.org/classify.html.
See also: Math Symbols in Unicode.
If you are a emacs user, you can set your emacs up so that any frequently used symbols can be entered by a single shortcut key, or a abbreviation. See: Emacs and Unicode Tips.
Ian Stewart has a new book out.
Read 1/3 of his Flatterland amazon in ~2002.
See also: FLATLAND: A Romance of Many Dimensions, and Flatland: A Introduction (by Xah Lee) for many subsequent books and films on Flatland. It is one of my favorite book, say, in top 5, of all books in my life.
Mathematicians Richard Palais and Hermann Karcher, have released a new version of their math visualization software, the 3DXM. The main change is that it now has button-like interface in place of menus, where each button is a icon of the surface or math subject. This makes it much more attractive, and easier to use. Check it out.
3DXM screenshot.
Note: the new version is for Mac only. For Windows or Linux users, there's always the Java version at the same download location.
A fantastic java applet to draw Voronoi diagram interactively. Very nice.
〈Voro Glide〉@ Source
See also: Great Math Programs.
My friend, professor Richard Palais, co-authored with his son Robert Palais, a new book “Differential Equations, Mechanics, and Computation”. I've been hired to help them update the site. The result is this: ode-math.com. Half of the book is free in PDF files. Also, lots of java applets and animation files are coming.
You can buy the book at Amazon: amazon. However, for some reason, Amazon doesn't have extra copies.
For explanation, see: Celtic Knots, Truchet tiles, Combinatorial Patterns.
The neck of a unfinished violin. Source.
For more, see: Spirals in Nature.
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.
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 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. 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 “aha” realizations. I thought about it for 20 min yesterday but haven't seen it yet.
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 (2010-03-22), 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:
I 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.
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.
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.
Symmetric Space, Transvection, Élie Carton (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 (2008-03-21), by Bob Palais and Richard Palais. transvection_for_rotations.pdf.
Bob also has some interactive demo written in Flash, here: http://www.math.utah.edu/~palais/transvection.html.
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/.
His other accomplishments include: FFmpeg (for processing multimedia data (e.g. audion and video)), QEMU (cpu emulator).
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 published since 2000 on xahlee.org, see: XahLee.org Site Math Index
