Xah's Math Blog

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

Visual Dictionary of Special Plane Curves now has side panel for easy navigation.

Algorithmic Mathematical Art, page 3 (old. repost)

Google DeepMind AI, AlphaGo to Challenge World Champion?

you may have heard, that Google DeepMind AI, AlphaGo, is doing a human vs machine, challenge a go champion Lee Se-dol.

They said they beat European Champion 5-0, so is now challenging a champion.

I got excited, thought it happened sooner than i expected. Spent some 30 min to read about it. Turns out, that “European champion” is only ranked 2 dan. (Pros are 9 dan)

(go playing is ranked by “dan”, or “level” if you will. Rather a bad system. The chess elo system is much better. But basically, the “dan” is meant to be the number of moves a stronger player can yield to the other for a even game. That is, if you are 4 dan, and i'm 1 dan, that means, if you give me handicap 3 stones (i play 3 moves first, usually on designated spots), then we would have a “even” game. That's the ideal, anyway. In reality, there are professional ranks of dan and armature ranks of dan. Armature 9 dan is something like pro 5 dan.)

so, the AlphaGo human vs machine is rather marketing deception. Am very disappointed and annoyed by the marketing tactics.

this “news”, is rather no news. The go human champion would beat google's AI 5-0.

Ladybug Video Game, Optimal Strategy

What Is Spacetime, Really? By Stephen Wolfram. @ http://blog.stephenwolfram.com/2015/12/what-is-spacetime-really/

see also

the answer to life, universe and everything

got a huge laugh out of this.

the answer to life, universe and everything

and if you are mathematically inclined, see, why 42. 〔42 By John Baez. @ http://math.ucr.edu/home/baez/42.html

Numbering System of Hex Grids


so, i was wondering, in writing a game on hexgonal board, whether there's a standard numbering system, as in chess and go. 〔➤ Go Board Game on Hexagonal and Triangular Grids

also, how do one represent the board in program. I've drawn a lot hex and triangular grids in the past, and know that, the underlying data structure, is rather ah-hoc and ugly. Have always been wondering if there's elegant way to do it. 〔➤ Geometric Tilings and Patterns Image Gallery

then, found this fantastic site, dedicated to this question.

Hexagonal Grids By Amit Patel. @ http://www.redblobgames.com/grids/hexagons/

math tan lines 2010-10-11-sunbather
tan lines. 〔artist: http://www.lukesurl.com/archives/comic/366-baby-got-back

Integral House

James Stewart (mathematician)



Berkeley to fire 'love letter to learning' professor By Rory Carroll. @ http://www.theguardian.com/us-news/2015/oct/17/berkeley-math-professor-alexander-coward-campus-battle

BLOWING THE WHISTLE ON THE UC BERKELEY MATHEMATICS DEPARTMENT By Alexander Coward. @ http://alexandercoward.com/BlowingTheWhistleOnUCBerkeleyMathematics.html

ancient article, still relevant. Google Chrome killed MathML. The TeX Pestilence (Why TeX/LaTeX Sucks)

a great website on map projections, with interactive visualization. 〔Maps By Jason Davies. @ https://www.jasondavies.com/maps/

xah math books 20150919 181118-s
Math and Geometry Books
groups a path to geometry by R P Burn book
〈Groups: A Path to Geometry〉 by R P Burn. amazon

there's this book 〈Groups: A Path to Geometry〉 by R P Burn. amazon I bought in year 2000.

I never got time to read it. When i bought it, i thought, i really want to understand the subject.

today, picking it up, haha, i understand all of it.

not bad for magically reading book in 15 years.

if you are in Mountain View, CA, area, come pick it up if you like.

comment at https://plus.google.com/+XahLee/posts/7Fri5maT94T

McGee graph

golly, each time John Baez posts something about math, 30 minutes will be gone if i just glance over to enjoy but understand nothing. And feeling terribly bad about my incompetence et al. Half a day will be gone if i tried to understand something. And feeling terribly bad about escapism. And by next month, all's forgotten.

here, have a cookie.

anyway, here's graph theory, projective geometry, affine transformation, group theory. All my favorite cookies.

McGee graph 37662
McGee graph

comment at https://plus.google.com/+XahLee/posts/K5fEptxAa1e

the Nature of Associative Property of Algebra (minor update)

Seashell Gallery

The reason why Involute gears turn smoothly. By Hyprodium. @ http://hyrodium.tumblr.com/post/123270340099/the-reason-why-involute-gears-turn-smoothly-fig

see also Involute

John Horton Conway: the world's most charismatic mathematician By Siobhan Roberts. @ http://www.theguardian.com/science/2015/jul/23/john-horton-conway-the-most-charismatic-mathematician-in-the-world


a new book. A biography of John Horton Conway

Genius At Play: The Curious Mind of John Horton Conway By Siobhan Roberts. @ amazon

Chen's theorem: every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). Chen's theorem

A Guide to Plane Algebraic Curves by Keith Kending. @ amazon

Geometry: Requirements for a Visualization System for 2020 (repost)

The Stanford Encyclopedia of Philosophy

The Stanford Encyclopedia of Philosophy is a great thing. It is much better than Wikipedia.

magic polyhedron

the Rubik cube of today have advanced. Now, there's stickerless versions, and better mechanical design.

magic dodecahedron DaYan megaminx 2
Megaminx, stickerless. amazon

see updated Magic Polyhedrons

ℭ = Cardinality of the continuum.

ℚ = be rational, ℝ = get real.

ℭ = Cardinality of the continuum.

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

Tiling & Patterns, Classic Text Now in Print Again

tiling and patterns
Tiling and Patterns By Branko Grunbaüm, G C Shaphard. amazon

just discovered, this classic, definitive, book on tilings & patterns is in print again, now reprinted by Dover.

see also: Reading Notes on Tilings and Patterns

tiling the plane with 5-fold symmetry tiles

John Baez, gave a intro on the problem of tiling the plane with 5-fold symmetry tiles @ https://plus.google.com/117663015413546257905/posts/5CHdeLRckRm

see also 〔The trouble with five By Craig Kaplan. @ https://plus.maths.org/content/trouble-five

sample of tiling & patterns, see:

See also: Discontinuous Groups of Rotation and Translation in the Plane

Skew But Fair Dice

skew dice

it supposed to be fair dice. Am thinking there's a major difference from the cube. This one, has 2 different sides, like a coin. In a sense, when you throw this dice, it decides which of the side will fall first, then, decide which of the 3 faces will land.

So, one can actually make a 3-sided pyramid, and have 2 of them and glue the bottom together. So, we could make a infinite shape of 6 faced dice that's still fair. Then, if we wiggle the cutting plane like a saw tooth, we could create quite a lot strange looking and still fair 6-faced dice.

comment at https://plus.google.com/+XahLee/posts/L9FP178N6eq

Cycloid (animation update)

polyhedron playground japan 2015-04-06
polyhedron playground Japan . 〔photo from 外薗葉月 https://plus.google.com/105047002579470450869/posts/Lgbe4ApL51b

Unicode Char for Logarithm?

apparently there's a Unicode char for log. . The Unicode name is “U+33D2: SQUARE LOG”. It's Japanese origin.

There's also a “U+33D1: SQUARE LN”

for many more, see http://www.unicode.org/charts/PDF/U3300.pdf

See also: Unicode: Math Symbols ∑ ∞ ∫ π ∈ ℝ² (use the search box there to find Unicode)

(thanks to https://twitter.com/Ryuutei)

English/Chinese Math Terminology 中/英 数学术语 (minor update)

archimedes spiral
Archimedean Spiral (updated animation)
ball shadow
Conic Sections (updated animation)
hypotrochoid spiralgraph
Hypotrochoid (updated animation)

LimaconOfPascal (updated gif animation)

Regular Polyhedron Domes (Geodesic Dome) Photo Gallery

Involute of a circle.

Trochoid (updated gif animation)

Tractrix (updated gif animation)

Conchoid (minor update)

Seashell Gallery: Misc Seashells, Cut-in-Half View

Giant Parabolic-dish Photo Gallery


Epi and Hypotrochoid Animation Gallery

Mathematics of Seashell Shapes

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

Notes on A New Kind of Science (photo of the book added)

How to Lose Interest at Go Board Game Fast

in early 1990s i spend 2 years playing go. At the time i bought the best go program called Nemesis for the Mac. Was able to beat it with 5 stone handicap for the computer. (the software is rated 13 kyu, which means, i'm i think 9kyu at the time)

some said that playing with computers gets you bad habits…

sometimes in 2009 or so, i suddenly find go to be not interesting at all, after i read Stephen Wolfram's “A New Kind of Science”.

the essence is that it's a kinda a cellular automata, and as such, there's no intrinsic math in it, and there are infinite cellular automata. And go players are simply those with great memory and specialized talent and seen lots of cellular automata of the go kind…

see Go Board Game as Cellular Automata

the other thing that has been interesting for me to explore is go on triangular grid or other tiling. Though, it's disappointing that i haven't seen much literature about it at all.

tri game2
Go Board Game on Hexagonal and Triangular Grids

how to lose all interest in go. Go Board Game as Cellular Automata

Go Board Game on Hexagonal and Triangular Grids (repost)

What is Technical Drawing, Descriptive Geometry, Projective Geometry, Linear Algebra (minor update)

Extending the Euclidean Plane: Riemann Sphere and Real Projective Plane

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

Visual Complex Functions: by Elias Wegert

Schmidt Arrangement, Algebra Integer, Gaussian integer, Eisenstein Integer

Stereographic Projection and Geometric Inversion

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.


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

the Nature of Associative Property of Algebra

Go Board Game as Cellular Automata

Table of mathematical symbols by introduction date

History of mathematical notation


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.

Nonlinear programming

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