Stereographic Projection 3D-Printed Physical Model (updated)

Unicode: Math Symbols ∑ ∞ ∫ π ∈ ℝ² (minor update)

Here's what mathematician John Baez says:

You build it by taking the real line in the complex plane and applying all possible transformations

`f[z] ↦ (a*z + b)/(c*z + d)`

where a,b,c,d are Eisenstein integers, meaning numbers of the form

`m + n Sqrt[-3]`

The Eisenstein integers lie on a hexagonal grid!

for more math details, see 〔Schmidt Arrangement By John Baez. @ http://blogs.ams.org/visualinsight/2015/03/01/schmidt-arrangement/〕

Stereographic projection is a special case of sphere inversion.

consider a sphere of diameter d, and plane at bottom. The inversion circle is centered on North pole, with diameter 2*d.

also:

Stereographic Projection (minor update Mathematica notebook. If you have a problem running it, let me know, i'll fix.)

also:

Stereographic Projection (minor update Mathematica notebook. If you have a problem running it, let me know, i'll fix.)

In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.[1] It is the sub-field of Mathematical optimization that deals with problems that are not linear.

how many pieces can you get by cutting a donut 3 times? (you are not allowed to re-arrange the pieces after each cut)

rephrase: Maximal number of regions of dividing a torus with n planes? (n=3 here.)

answer: http://oeis.org/A003600

“group theory” should be called “symmetry theory”

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j.

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. example

{ {1, 7, 3}, {7, 4, 5}, {3, 5, 6}, }

Great Math Board Game Software (minor update)

Great Math Board Game Software (minor update)

sad to know. Alexander Grothendieck died 2 days ago. Alexander Grothendieck

geometric design on sphere. http://taffgoch.deviantart.com/

old goodie. State of Theorem Proving Systems 2008

Bezier Curve (minor update)

#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)