Stereographic Projection 3D-Printed Physical Model (updated)

#math When an attractive girl flips her wet hair, the water stream forms a Fibonacci spiral. Math Mysticism: is Hurricane Shape a Fibonaci Spiral?

“Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland”

FLATLAND: A Romance of Many Dimensions. The best book to understand higher dimensions.

reading math is just immense pure pleasure. Some people like video games, some movies, some drink bars… these are considered pleasure. But really, nothing beats reading just math. The purity, the beauty, the depth, and the austerity.

O, math, my true love, but how i have alienated thee, and you being quite difficult.

2 decades ago, i know about gamma function, which is a extention of the factorial to all positive real numbers (to say the least). i didn't understand how it is done. i just knew it's something advanced. but today, reading about it, i can understand it! Gamma function must be thanks to, my study of complex analysis sometimes in 2006!

just remembered, that i had a dream few days ago, that the plane curves website at St Andrews University (Scotland) http://www-history.mcs.st-and.ac.uk/Curves/Curves.html converted their Java curve applet to modern JavaScript. Which is something i've been planning to do for my Visual Dictionary of Special Plane Curves site.

my old rival beats me!

added a hyperbolic tiling interactive software to Great software for Tilings, Patterns, Symmetry

Found this on math overflow, a pure hogwash http://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213 YOU know? the kinda pleasing and meaningless things to say, that everyone loves to hear. Which id wrote that? Then i noticed, the name Bill Thurston. Bill Thurston? The William Thurston, geometry god of the century who died few years ago? Indeed.

〔Groups, Tilings, and Finite State Automata By William P Thurston. @ http://timo.jolivet.free.fr/docs/ThurstonLectNotes.pdf〕

the story of looking up math. so today, i want to lookup ultra-filter. “…an ultrafilter on a poset P is a maximal filter on P, …” so, i have to lookup filter: “filter is a special subset of a partially ordered set.” but don't remember what's “partially ordered set” so, lookup it is: “A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.” ok, but i want to lookup “total order” to see if that's something i recall: “a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X which is transitive, antisymmetric, and total.”

great, i remember having worked out this. That's total satisfaction. (actually, i discovered the properties of the Equivalence relation. When around 1997 i was writing code to decide if 2 polygons in 3d are equivalent, and discovered inconsistancies in my code, namely a==b, b==c, yet my code a≠c. And, i discovered, to great happiness, that to define equivalence is equivalent to partition of a set.)

well, that's a bit excursion. But i was writing about my typical trip to math these days. When trying to understand one thing, involves some 10 or 20 other Wikipedia articles.

by the way, also, some'd suggest math books instead. But no, i prefer dictionary style learning, esp for math. I prefer, the cold, logical, senseless, definition. The human touch of “motivation”, i want after the fact. (side note: sometimes, the human touch are often mis-leading, and there are many different takes, depending on the author. The human touch is often necessary though. But, i've been thinking, it is possible to do without entirely, because, math (defined as the essence of something), is how things are. And, to some degree (perhaps 100%), you really just need to know that gist, anything else is fluff, and possibly even harmful. But why do we have the need for the human touch? i gather possibly it's pure habit. As in, new thinking usually happens with newer generation (as is, only older generation have problems with imaginary number, whereas newer generation who are taught its definitions directly never have this problem and moves on)) (the gist of this thought is that, what happens, when people learned math ONLY by their logical definitions and never the story behind it.)

(side note: xah's edu corner: linguistics: by the way, in math lingo, often you'll encounter the phrase “Motivation” as a section title. It is a idiom among math texts. The context is that, math becomes so abstract, that just definining something seems out of the blue. So, one needs to provide a context, so that readers can see how the definition came to be. And that, is often called “Motivation”, which is kinda a math idiom. I can't help but finding it funny, when reading math papers (the “formal” type), you encounter a section titled “Motivation” pro forma)

after writing all these, i haven't understood ultra-filter yet. Now, back to procrastination…

comment at https://plus.google.com/+XahLee/posts/hxUPxeYyN59

#math #education

POV-Ray ashtray, cubic spline POV-Ray: Surface of Revolution and Prisms

Math Prizes and Nobel Ignobility (repost)

Maryam Mirzakhani (a beautiful girl) won the Fields medal yesterday. Read this post by John Baez on the math she did. https://plus.google.com/117663015413546257905/posts/UTt2WeAcPEY

Geometry: How to Order the Edges of a Cube?. see the g+ link for discussion.

Logical Operators, Truth Table, Unicode (minor update)

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