Xah Math Blog Archive 2013-01

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 Triangular 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. At http://blogs.ams.org/visualinsight/2013/11/15/astroid-as-catacaustic-of-deltoid/ , accessed on 2013-11-16 ]

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 to 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 to 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 and Eigenvectors

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

57color coded

Algorithmic Mathematical Art

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

Abuse of Math Notation

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. Best Scientific Calculators


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

The TeX Pestilence (Why TeX/LaTeX Sucks) (updated)

one of my hero. Willard Van Orman Quine (1908 to 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 (For example, 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 and 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 and 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, and Technology By Lorenz Lachauer. At http://eat-a-bug.blogspot.com/ , accessed on 2013-08-20 ]

JavaScript D3 Rotating Voronoi Demo

Sum of Angles of Triangle on Girl's Chest (math humor)

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

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

dodecahedron tits 1280
dodecahedron tits

Math: Symmetry, Flatland, by Ian Stewart

A new book. [Symmetry: A Very Short Introduction By Ian Stewart. Buy at 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:

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)

wikipedia curvature

The Problems of Traditional Math Notation

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

Math Tiling Automata Morph

recursion yesterday reading todays paper
«Woman spotted yesterday reading today's paper»

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

Rotating Snakes. cats can also see the rotating snakes visual illusion. See video: Visual Illusions 📺

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 At http://www.generativedesigncomputing.net/

Teacup Recursion Animation

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

true false ambigram
“the right way to answer true and false questions”
math rubiks cube tattoo 170353
math rubiks cube tattoo

Microwave Arithmetic

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?

linear algebra comics

linear algebra anime
“线性代数”. Linear algebra tutorial, in the form of comics, in Japan. How wonderful.

thx to weizhoupan

math comics x2 x3 higher power 2008-06-04
by C. Burke (used with permission)

Doyle spiral + Mobius transformation

math art marbles spirals-2
“Doyle spiral + Mobius transform” by fdecomite image source on . 2800×2100 ©

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 bonanza: Teacup Recursion Animation

5-Point Conics (needs Java installed.)

Logic Based Human Language: Lojban

lojban translated
xkcd comics on lojban. (http://xkcd.com/191/)

Lojban is a constructed language, design based on predicate logic. Some tutorial here, with voice recording: Xah's lojban Tutorial.

Alice in Wonderland, Zero-sum game

“The more there is of mine, the less there is of yours.”

from Wonderland: Chapter 9: The Mock Turtle's Story

That's the concept of Zero-sum game. However, in real world, most things isn't zero-sum game. Both parties actually benefit. For example, when you give something you no longer want to someone, a value is created “out of nothing”.

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