Xah's Math Blog

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

Xah's Math Blog Archive 2013-06 〜 2013-06

geometry: symmetry of symmetry, problem within problem

Curves and Their Properties by Robert Yates

Architecture: Atomium (Regular Polyhedron Building)

Unicode: Greek Alphabet α β γ δ ε ζ η

Logical Operators, Truth Table, Unicode (minor update)

Math Font, Unicode, Gothic Letters, Double Struck, ℤ ℚ ℝ ℂ ℍ ℜ ℑ ⅇ ⅈ (updated)

Math ＆ Code: the Case for Formal Verification

quite amazing. This is when seemingly useless math becomes useful. ⁖ folding solar panels in spaceship.

Math Valentines Heart = Cardioid

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

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?

Stereographic Projection 3D-Printed Physical Model (updated)

ugh, another idiocy of TeX legacy

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.

• The TeX Pestilence (the problems of TeX/LaTeX)
• The Nature of the Unix Philosophy

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

Vectors and Complex Numbers

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:

• understanding complex numbers
• Coordinate Transformation

Go Board Game on Hexagonal and Other Grids

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

Astroid as Catacaustic of Deltoid

yay, my site made it to AMS blog. 〔Astroid as Catacaustic of Deltoid 2013-11-12 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

have you ever heard people say “i am never good at math…”?

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❕

Math Terminology: Magma vs Groupoid

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?

Why is Sine Called Circular Function

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

What's Eigenvalues ＆ Eigenvectors

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)

What is Riemann Surface? Understanding the Concept Without Math.

Wikipedia Reading Snippets: Understanding Category Theory

Category theory

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.

Abuse of Math Notations by Mathematicians, More Rape Than Rapeseed

homeomorphism, and homotopy for 1/z

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

Homeomorphism

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.

Homotopy

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.

The Geometric Significance of Complex Conjugate (updated)

understanding complex numbers

The Beauty of Roots

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

9 Tools to Display Math on Web (updated)

Incredible Mathematician John Baez

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.)

Dead Reckoning in a Non-Orientable 3D Space

Projective Geometry Java Applets

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.