Stereographic Projection 3D-Printed Physical Model (updated)

every Valentines Day ♥ ♥ ♥ is the day to post the cardioid curve, because cardioid means heart-shaped.

more about Cardioid at Cardioid

if you search the web, you'll find 3D equations for 3D heart ♥ (that is a polynomial of 3rd degree who's level set is like a heart-shaped foil balloon)

what does bagua, taichi mean? Bagua, the trigrams, as binary system ☰ ☱ ☲ ☳ ☴ ☵ ☶ ☷

see bottom Art:〈The Grandmaster〉 The Opera Fight Scene; 一代宗師，八卦掌 📺

this is Chinese metaphysics.

exercise: look at South Korean flag. It has 4 of the 8 trigrams. Find a description of why those 4.

co-inventor of calculus, Gottfried Leibniz, also as the father of computer, said that “hexagrams a base for claiming the universality of the binary numeral system.”. Why?

Great Math Software (old collection)

Graphing Software for Microsoft Windows (update on plotting software for Microsoft Windows)

see http://mathquill.com/. It's cool and all, but note the syntax. Quote:

MathQuill renders LaTeX math … try \sqrt x, try \sin\theta

when you want square root or sin etc, why cannot we simply write `Sqrt[x^2+y^2]`

and let that be the syntax? why the backslash syntax soup?

the damage of TeX, like unix, is deep and pervasive. It's free, like cig given to children, washed people's brain. It's so rooted that people are now saying the web should abolish MathML and let TeX be the standard.

on a separate note, learned this: Manuel de Codage «The Manuel de Codage (abbr. MdC) is a standard system for the computer-encoding of transliterations of Egyptian hieroglyphic texts.»〔☛ Unicode: Egyptian Hieroglyph Characters〕

comment at https://plus.google.com/+XahLee/posts/69Pf7DGpr9F

9 Tools to Display Math on Web (updated)

Math Typesetting, Mathematica, MathML (updated)

How Mathematica does Unicode? (updated)

The Geometric Significance of Complex Conjugate (oldie but goodie)

complex numbers, or complex analysis, is one of the most beautiful math. If you don't know complex numbers, and you love geometry, it is essential you get to know it.

updated:

If you want to understand Einstein's theory of relativity, you must understand mobius transformation. To understand mobius transformation, you must first understand Riemann sphere, complex numbers, geometric inversion. Video and explanation at Stereographic Projection 3D-Printed Physical Model

yay, my site made it to AMS blog. 〔Astroid as Catacaustic of Deltoid By John Baez. @ blogs.ams.org…〕

it's written by the redoubtable mathematician John Baez. Baez is great, in that he writes serious math for any math undergraduate to appreciate, as opposed to many math popularizing authors who write for the laymen.

i did my curves project Visual Dictionary of Special Plane Curves mostly in 1994 〜 1997, almost 2 decades ago, while i was a college student. I never seen the proof of how Deltoid's Catacaustic is a Astroid. I recall trying to, but it was too difficult for me back then. I haven't done much math since.

Do you know a proof of how Deltoid's Catacaustic is a Astroid? Post to John's g+ post. Thanks.

Equiangular Spiral, also known as log spiral, has the property that the angle of tangent to center is constant.

Mathematician Jacob Bernoulli (1654 〜 1705) requested this spiral be engraved on his tombstone with the epitaph:

Though changed I rise unchanged

more pics and properties at Equiangular Spiral

hi, am a mathematician. When the subject of math comes up in conversation, the usual response i get is a somewhat uneasy utterance of “i am never good at math”.

you know what? that's right, you are a idiot, period.

FACT OF LIFE❕

The term magma for this kind of structure was introduced by Nicolas Bourbaki. The term groupoid is an older, but still commonly used alternative which was introduced by Øystein Ore.

why did the Bourbaki guys introduce the term magma? it seems to me groupoid is a better term. Was it introduced to avoid confusion due to the many slightly different definitions of groupoid?

this is why sine, cosine, are called circular functions

all the trig function, {sin, cos, tan, asin, acos, atan}, can be defined using just one of them, sin.

so, there is really just one function: sine.

you can see why it's so important in math, because the nature of it is that it's the height when you sweep a circle with constant speed.

in other words, anything that rotates in a constant speed, sine is in it.

earth rotates, moon rotates, so came sine. Then, wheels are invented, more sine.

read more at Sine Curve

added the complete list. Refresh page. Math Font, Unicode, Gothic Letters, Double Struck, 𝔄 𝔅 ℭ, 𝔸 𝔹 ℂ

math gothic font. Can you identify the following letters? 𝔅 𝔙

math demystification: If you hear “stochastic process”, you can safely replace it with “random process”

Algorithmic Mathematical Art ₂ The gist here is to distill a math art into its algorithmic essence. By recursion or some encoding (such as math equation)

my concern in life is math and women. But since they are both difficult, my activity is mostly reduced to visualization aspect.

In many fields of mathematics, morphism refers to a structure-preserving mapping from one mathematical structure to another. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms are functions; in linear algebra, linear transformations; in group theory, group homomorphisms; in topology, continuous functions, and so on.

In category theory, morphism is a broadly similar idea, but somewhat more abstract: the mathematical objects involved need not be sets, and the relationship between them may be something more general than a map.

The study of morphisms and of the structures (called objects) over which they are defined, is central to category theory. Much of the terminology of morphisms, as well as the intuition underlying them, comes from concrete categories, where the objects are simply sets with some additional structure, and morphisms are structure-preserving functions. In category theory, morphisms are sometimes also called arrows.

here's a excerpt from Wikipedia Laplace transform

The Laplace transform is a widely used integral transform with many applications in physics and engineering. Denoted ℒ{f} (or alternatively ℒ{f[t]} ), it is a linear operator of a function f(t) with a real argument t (t ≥ 0) that transforms f(t) to a function F(s) with complex argument s.

mathematicans, can you stop using the term “operator” and use “function” instead? Because, you are not talking about syntax, you are talking about meaning.

the reason that people write “linear operator” instead of “linear function” is due to lose unspoken convention. This convention, came about, is probably because: it gives them something concrete to speak of, namely the symbol shown on paper.

see also What's Function, What's Operator?

recently, finally learned what Quadratic form is.

quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,

4*x^2 + 2*x*y - 3*y^2is a quadratic form in the variables x and y.

(and Homogeneous polynomial is a polynomial whose nonzero terms all have the same degree. For example, `x^5 + 2 x^3 y^2 + 9 x y^4`

is a homogeneous polynomial of degree 5, in two variables; the sum of the exponents in each term is always 5.)

you hear about quadratic form all the time. Though, i don't know why is it significant. Is there something about them that makes them fundamental? (as in “given structure X, it is the {least, most} something?”) Or is it something happens to be a calculational convenience?

“homeomorphism” (aka “topological isomorphism”, “bicontinuous function”) is a continuous function between topological spaces that has a continuous inverse function.

In topology, two continuous functions from one topological space to another are called homotopic if one can be “continuously deformed” into the other, such a deformation being called a homotopy between the two functions.

is there a homotopy that maps identity in the complex plane to 1/conjugate[z]? or sin[z]?

i think there is. It's obvious, that if the 2 spaces are topologically equivalent, there always is, the question is to find the homotopy. In my case, just use the idea of Geometric Inversion. Let p be the point in domain and p' in range, then just smoothly swap them by gradually narrowing their distance.

on its own page. The Beauty of Roots

depth of the universe lies here.

I must say, this is the most amazing math i've read in perhaps 5 years.

the pics are beautiful, but that's not it. The real beauty is the math, and Baez's explanation of it. (there are lots of pretty pictures of math on the net, most are shallow and tired, even doing harm to math)

this article, will take you days to study. Nay, weeks. Nay, months. Well, can't be wrong to say years or a few lifetime.

btw, it took 4 days of Mathematica to compute these roots.

the full resolution image on the site, is 94 M bytes. Some image viewer on linux have problem opening it.

the meaning of the word variety is varied. There's variety show, then there's algebraic variety. The variegation and etymology is fantastic, not to mention manifold.

show your calculator to Euler and Gauss. What's the Latest ＆ Greatest in Calculators?

Maze ＆ Math in Video Games (old essay. Added YouTube videos)

one of my hero. Willard Van Orman Quine (1908 〜 2000). A logical positivists, a logician.

9 Tools to Display Math on Web (updated)

one of the most fruitful thing g+ has ever done for me since its beginning is discovery of John Baez.

He's a mathematician, and also a well-known writer (even before blog days) He writes a lot, but even when writing research level math, he made it easy for undergrad to understand. And, takes the time to write the interesting aspect, and answer and discuss with your comments/questions. (thus, comments on his post/blog are often very high quality as well) I see that he also sometimes write non-math related things, that touches on history, art, linguistics, all in a very appetizing way with quality/rare photos (and yet not the trite, mundane, beaten-horse types you find daily from social networks). Incredible!

you can read his bio on wikipedia and also follow links to his blogs. John Baez

i'm learning lots stuff from John C B. Lots thoughts hard to summarize nicely.

For one thing, related to SEO, is that it solidifies the idea that in order to get more readers, one should really focus on readers — so-called “engagement”. For example, say, instead of writing 4 posts per day, write just 1 and put the time of the 3 into that 1, to include quality image/illustration, answer questions, iron-out hand-waving. In other worlds, this is really the road for professional blogger. (you might not want to have lots readers, or shudder from the idea of wanting to be “popular”. But if you write publicly, more readers is positive in psychological and practical and philosophical ways. “Readership” defines “authorship”.)

JCB is also pulling me back into math. Such a black hole of pure beauty. The depth of which tantamount the very question of existence and universe.

JCB also sets a good example of doing good in a solid way. (as opposed to the countless shallow and crowd-pleasing blogs, exemplified by the marketing droids of Google of recent years (⁖ Google Science, Google Doodle, Google pro-lgbt, …), and countless fanatical “left-leaning liberal” American slackavitists daily pushing their selfish-opinions in the name of greater good.)

GeoGebra was open source (GPL) but now only for non-commercial use. this is bate ＆ switch, but the problem is really open source. When it gets big, it needs money, but nobody wants to pay. Plane Curves: GeoGebra Files Index

great title, from John Baez

Some old projective geometry interactive applets. Requires Java installed.

The main file is Intro to Real Projective Plane

following are Java applets.

- Projective Geometry: Quadrangle 2.4a
- Projective Geometry: Quadrangle Construction
- Projective Geometry: Definition of Quadrilateral
- Projective Geometry: Desargues's Theorem
- Projective Geometry: Harmonic Points Construction
- Projective Geometry: Harmonic Points
- Projective Geometry: Perspective Triangles

Rewriting them in JavaScript soon.

A great blog from a architect. Always lots of most beautiful math images. Be sure to checkout his past articles. 〔eat-a-bug: Art, Design, Architecture, ＆ Technology By Lorenz Lachauer. @ eat-a-bug.blogspot.com…〕

compass ＆ straightedge in JavaScript. Try it. http://sciencevsmagic.net/geo/

for many other similar software, see Great Software for Plane Geometry

A new book. 〔Symmetry: A Very Short Introduction By Ian Stewart. amazon〕

It cited my work: Discontinuous Groups of Rotation and Translation in the Plane. ☺

Ian Stewart is a well-known mathematician. He has written many books introducing interesting math to the laymen. Including:

- 〔Flatterland: Like Flatland, Only More So By Ian Stewart. amazon〕
- 〔The Annotated Flatland: A Romance of Many Dimensions By Ian Stewart. @ amazon〕

You can also read Flatland here: FLATLAND: A Romance of Many Dimensions

There's also a Chinese translation: 神奇的二维国 (Flatland)

now, you can use this search box: Unicode Search: Find Your Symbols to find all your math symbols.

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/

temp on hold asking for permission

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.

- Rolling Circles and Balls (Part 1) By John Baez. @ Source johncarlosbaez.wordpress.com
- Rolling Circles and Balls (Part 2) By John Baez. @ Source johncarlosbaez.wordpress.com
- Rolling Circles and Balls (Part 1) By John Baez. @ Source johncarlosbaez.wordpress.com

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