Xah's Math Blog

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

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 (the problems of TeX/LaTeX)

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 (the problems of TeX/LaTeX) (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

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”

Math Terminology and Naming of Things

Hermitian matrix

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.

Symmetric matrix

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)

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

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

pig icosahedron DSC01119
pigs playing icosahedron. 〔photo by Timothy Gowersamazon

old goodie. State of Theorem Proving Systems 2008

sphere lamp projection Wolfram Research
sphere lamp projection, Wolfram Research
Always tampon infinity logo
infinity ∞ found
polyhedron n nekos
Math in Second Life, page 2

Xah's Math Blog Archive 2013-11 – 2013-12

Bezier Curve (minor update)

Xah's Math Blog Archive 2013-10 – 2013-10

Xah's Math Blog Archive 2013-09 – 2013-09


in this video, interview of John Conway, Conway talked about this problem and his experience with Wu-Yi Hsiang and also Thomas Hales

John Conway - The Game of Life and Set Theory

starting on 6:40 to 25:00 or so.

John Baez the other day, when reading about your hyperreal article, i find myself more interested in another topic the surreal numbers and end up watching this interview of John Conway. In this Conway video he rather rambled on many his life's tales, but i actually got a very good intro to surreal numbers. (which made following up on Wikipedia fruitful). again, thank you John.

#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, how i have alienated thee, and you being quite difficult.

2 decades ago, i know about gamma function, which is a extension 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!

plane curves, java applet obsolete, need JavaScript

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!

hyperplay 2014-09-13

added a hyperbolic tiling interactive software to Great software for Tilings, Patterns, Symmetry

Alexander Grothendieck = Obi-Wan

Math, Algebra: On the Phraseology of X Over K

William Thurston on Tiling & Automata

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

learning math, and math idiom “motivation”

a 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…

#math #education

comment at https://plus.google.com/+XahLee/posts/hxUPxeYyN59