Xah's Math Blog

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

Math Prizes and Nobel Ignobility (repost)

Math Politics: Simon Plouffe and nth Digit Formula of π (repost)

math of Maryam Mirzakhani, hyperbolic geometry

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.

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

Moravian star
Miura fold

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's Difference Set of a Group

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?

Great Math Software (old collection)

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

stereographic projection by Henry Segerman 2013-12-12 95487
Stereographic projection by Henry Segerman

Stereographic Projection 3D-Printed Physical Model (updated)

「1+2+3+4+... = -1/12」, Ramanujan's Sum, String Theory, Quantum Mechanics 📺

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.

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

John Baez on Octonion

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)

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.


tri game2
Black's turn. Who wins?

Go Board Game on Hexagonal and Other Grids

stereographic projection by Henry Segerman 2013-12-12
stereographic projection by Henry Segerman

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 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.

equi spiral def
A example of equiangular spiral with angle 80°.

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.


Mathematica 7 deltoid notebook Xah 2013-10-22
Mathematica 7 notebook, deltoid

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

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

On the Naming of Eigenvector and the Igon Value Problem

the Nature of Linear Transformation


What's Eigenvalues & Eigenvectors

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

57color coded

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)

Hilbert curve
Space-Filling Curve
clock secondhand gear mechanism
clock secondhand gear mechanism

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

a Riemann surface. From http://3d-xplormath.org/

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.

The Importance of Terminology's Quality In Computer Languages

Math Writing Style: Use of the Term “Linear Operator” vs “Linear Function”

What is Quadratic Form in Math?

laplace transform formula
a typical snippet of math. This one is from from Wikipedia Laplace transform

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.


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]?

Complex Inversion

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.

Twin Prime Conjecture Breakthru: Yitang Zhang

grid inversion3

The Geometric Significance of Complex Conjugate (updated)

The Significance of Complex Numbers: Frobenius Theorem

the Nature of Associative Property of Algebra


understanding complex numbers

polynomial roots crops

The Beauty of Roots

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)

The TeX Pestilence (the problems of TeX/LaTeX) (updated)

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

math formula MathJax

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.)

Math Formula For Beauty

logic & linguistics. The Logical Levels of Interpretation

karcher je st
Karcher JE Saddle Tower

Dead Reckoning in a Non-Orientable 3D Space

Geometric Dissection, Mechanical Iris, Expandable Table Furniture 📺 (updated)

GeoGebra No Longer FSF Free Software

GeoGebra was open source (GPL) for about 10 years (up to version 4.0), but since version 4.2, now only for non-commercial use. This is bate & switch, but the problem is really open source. When it gets big, it needs funding, but nobody wants to pay.

Here's quote from Wikipedia GeoGebra on its licensing:

Most parts of the GeoGebra program are licensed under GPL, making them free software. However some parts, including the Windows and Mac installers, have a license which forbids commercial use and are therefore not free software.[9][10][11] In practice, this means that non-commercial use by teachers and students is always free of charge, while commercial users may need to pay license fees. For details see the GeoGebra license description.[9]

Since July 2010 the Debian GNU/Linux distribution offers a free version of GeoGebra 4.0 in which all un-free parts of the program were removed or replaced by free software. This version may be used for commercial purposes without paying licensing fees. However, starting with version 4.2 since December 2012, the license is changed to be more restrictive[11] so that GeoGebra cannot be included in Debian GNU/Linux any longer. On the other hand, the software can still be downloaded from its official download page free of charge for many platforms (including Debian as well).

GeoGebra is a Java Applet. But since Apple Apple killed Flash in 2010, as well as not including Java, Java applet is pretty much dead (it doesn't run on any Apple iOS nor Google Android phone/tablet).

GeoGebra just came up with a tablet version from a KickStarter project that got funding of $12k from 310 backers. 〔http://www.kickstarter.com/projects/geogebra/geogebra-for-the-ipad/posts/488589〕.

60 Plane Curves GeoGebra Files

polyhedra dices
Regular Polyhedrons Dice and Platonic Solids Chocolate Dice
math lamp net shadow
Math net shadow lamp. Beauty of projective geometry at work.

Projective Geometry Java Applets

Some old projective geometry interactive applets. Requires Java installed.

23 invarienceOfHarmSet 2.8
harmonic set

The main file is Intro to Real Projective Plane

following are Java applets.

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…

JavaScript D3 Rotating Voronoi Demo

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