Xah Math Blog

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

The maths of spherical video

Henry Segerman has been doing amazing things lately.

The story of SageMath. http://wstein.org/papers/talks/2016-06-sage-bp/bp.pdf

excellent story, if you want to know about math software, open source, academia.

here's video version.

the origins of SageMath; I am leaving academia to build a company.

Abuse of Math Notation

Why is Mandelbrot Set Cardioid Shaped?

Times Tables, Mandelbrot and the Heart of Mathematics

more about cardioid here: Cardioid

Peg solitaire

solitaire puzzle marbles set
buy! Authentic Models Solitaire Di Venezia 20mm Semi-Precious Marbles


in my youth, i studied this game and lots similar ones. You can search wikipedia for math and books. The name is peg solitaire. Here: Peg solitaire.

You begin with a configuration where there's a hole in the center, while all other are filled with marbles (or called pegs). You can remove a peg by jumping one over it. The goal is to leave just 1 peg on the board.

The Involute of a Cubical Parabola By John Baez. @ https://johncarlosbaez.wordpress.com/2016/03/22/the-involute-of-a-cubical-parabola/

The Capricornoid By John Baez. @ https://johncarlosbaez.wordpress.com/2016/03/06/the-capricornoid/

Free Calculus Textbooks

Why Are Text Book So Expensive?

Geometer's Sketchpad Closed Shop? Now, Free Download!

Interactive Geometry of Plane Curves

real rose

Plane Curves: GeoGebra Files Index

update complete. There are 60 of them. Enjoy!

Interactive Geometry of Plane Curves

following are interactive geometry of plane curves.

They are GeoGebra software. Google no longer supports Java Applets. I'm updating them to html5 so they run in your browser. There are about 50 more to do, i'll be posting them in next few days. Hope you like them.

plane tiling mathematica 2016-05-04 35001
Plane Tiling Mathematica package.

been working on Geometry: Plane Tiling Mathematica Package

math quiz 2 plus 2 times 2

for illustration at the Nature of Associative Property of Algebra

barth sextic mapped to sphere-s

beautiful images of Barth Sextic

〈The Symmetries of Things〉 by John H. Conway (Author), Heidi Burgiel (Author), Chaim Goodman-Strauss (Author) amazon

John H Conway's new old book

math art: islamic pattern metal sphere

cubical two point pattern metal sphere Craig S Kaplan
“cubical two point pattern metal sphere” by Craig S Kaplan. Buy at http://www.shapeways.com/product/Y26YZRL3Y/cubical-two-point-pattern
islamic geometry pattern z
islamic geometry pattern 〈Tending toward the convex〉 〔Tending toward the convex By Craig S Kaplan.. @ http://isohedral.ca/tending-toward-the-convex/

Craig S Kaplan is a expert at mathematical decorative patterns. You can find more of his work on his site at http://isohedral.ca/

Computer Generated Islamic Star Patterns By Craig S Kaplan. @ http://vismath4.tripod.com/kaplan/index.html

visual proofs is such that we have to artfully distribute the errors all over, so that in the limit, the error disappears!

Rectified Truncated Icosahedron and Elongated Square Gyrobicupola

John Baez again, from which i learned about the term “rectification” (of regular polyhedron), and also read what's archimedean solid, and learned about this strange beast Elongated square gyrobicupola.

the Elongated square gyrobicupola is interesting, besides the name, because

It is sometimes considered to be an Archimedean solid, because its faces consist of regular polygons that meet in the same pattern at each of its vertices. However, unlike the rest of the Archimedean solids, it lacks a set of global symmetries that take every vertex to every other vertex.

see Baez's article here: 〔Rectified Truncated Icosahedron By John Baez. @ http://blogs.ams.org/visualinsight/2016/04/01/rectified_truncated_icosahedron/

by the way, i highly recommend this software KaleidoTile. See Great Math Software: Polyhedrons & Polytopes

Find Intersections of Lines. Bentley–Ottmann Algorithm

In computational geometry, the Bentley–Ottmann algorithm is a sweep line algorithm for listing all crossings in a set of line segments. Wikipedia Bentley–Ottmann algorithm

beautiful internactive demo (JavaScript) by Adam Pearce http://bl.ocks.org/1wheel/464141fe9b940153e636

Prime Number's Ending Digit, Probability

Soundararajan was drawn to study consecutive primes after hearing a lecture at Stanford by the mathematician Tadashi Tokieda, of the University of Cambridge, in which he mentioned a counterintuitive property of coin-tossing: If Alice tosses a coin until she sees a head followed by a tail, and Bob tosses a coin until he sees two heads in a row, then on average, Alice will require four tosses while Bob will require six tosses (try this at home!), even though head-tail and head-head have an equal chance of appearing after two coin tosses.

Mathematicians Discover Prime Conspiracy. A previously unnoticed property of prime numbers seems to violate a longstanding assumption about how they behave. By Erica Klarreich. @ https://www.quantamagazine.org/20160313-mathematicians-discover-prime-conspiracy/

see also

Free Good Category Theory Book

Mathematician John Baez, and expert of category theory (and quantum mechanics), has a blog about it his course of category theory, at http://math.ucr.edu/home/baez/qg-winter2016/

in particular, he highly recommend this free book on category theory.

Emily Riehl, Category Theory in Context, 2014. Emily_Riehl__Category_Theory_in_Context_2014.pdf

Google AI AlphaGo vs Lee Sedol

Combinatorics, Algorithm, and Humanity

Spare your arithmetic, never count the turns. Once, and a million!

—Shakespeare, in Cymbeline, Act II. Scene IV.

hard scifi 〈The Three-Body Problem〉 2015 Hugo Award

Three Body Problem  scifi book cover
The Three-Body Problem amazon

〈The Three-Body Problem〉 (三体) is a science fiction novel by the Chinese writer Liu Cixin.

An English translation by Ken Liu was published by Tor Books in 2014. It won the 2015 Hugo Award for Best Novel and was nominated for the 2014 Nebula Award for Best Novel.

read reviews on amazon, at amazon

black holes and gravitational wave, by Stephen Wolfram


Black Hole Tech? By Stephen Wolfram. @ http://blog.stephenwolfram.com/2016/02/black-hole-tech/

enjoyed this article. If for nothing, go look at the 3-body problem animation.

classic text book on algebraic geometry

This is the classic book on algebraic geometry.

Algebraic Geometry Robin Hartshorne cover
Algebraic Geometry by Robin Hartshorne. amazon

though, i recognized the name. He wrote another book:

Geometry Euclid and Beyond  Robin Hartshorne 0387986502
Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics) by Robin Hartshorne. amazon

looking at amazon, apparently he wrote a few geometry texts. interesting is that he wrote one on Projective Geometry 〔Foundations of Projective Geometry amazon

You might find many good geometry texts at Printed References On Plane Curves

… text books are too expensive. I think one can find lots of equivalents, cheap or free.

there's something wrong with pricing of text books in USA. The price tag is exorbitant. And i think the bottom line cause, is greed. No, am not inclined to blame anything or anyone just because they are successful or rich. But in the case of text books, i think indeed it's greed and schemes and made them so expensive, and universities have a lot to do with it.

educational institutions, such as universities, public or privately owned, as big organizations, they are not really doing the good they are supposed to do.

27 Lines on a Cubic Surface

clebsch diagonal cubic
27 Lines on a Cubic Surface – Greg Egan

27 Lines on a Cubic Surface By John Baez. @ http://blogs.ams.org/visualinsight/2016/02/15/27-lines-on-a-cubic-surface/

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. 〔➤see 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. 〔➤see 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)

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