# Xah Math Blog

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

### Peg solitaire

It's, BEAUTIFUL!

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/〕

### Interactive Geometry of Plane Curves

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.

- 5-Point Conics
- Archimedean spiral
- Archimedes Circle
- Inversion of Archimedes's spiral
- Astroid Construction
- Astroid Construction
- Astroid's Pedal Curve
- Astroid Roll
- Astroid Trammel

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.

been working on Geometry: Plane Tiling Mathematica Package

for illustration at the Nature of Associative Property of Algebra

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

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

- 〔Biases between consecutive primes By Terence Tao. @ https://terrytao.wordpress.com/2016/03/14/biases-between-consecutive-primes/〕
- 〔Unexpected biases in the distribution of consecutive primes By Robert J Lemke Oliver And Kannan Soundararajan. @ http://arxiv.org/abs/1603.03720〕

### 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

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

〈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.

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

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

〔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.

〔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.

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/〕

http://modto.com/integral-house-by-shim-sutcliffe-architects/

〔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/〕

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.

- https://plus.google.com/117663015413546257905/posts/dttufb7MCk2
- http://blogs.ams.org/visualinsight/2015/09/15/mcgee-graph/
- http://mathoverflow.net/questions/215211/what-algebraic-structures-are-related-to-the-mcgee-graph
- McGee graph

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

the Nature of Associative Property of Algebra (minor update)

〔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)