# Xah Math Blog

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

Math: Density Plots of Trig Expressions. old. repost.

Valentine's Day Gift for a Platonic Relationship: Regular Polyhedrons Chocolates

### Kolam pattern

Kolam (Tamil-கோலம்) is a form of drawing that is drawn by using rice flour/chalk/chalk powder/white rock powder often using naturally/synthetically colored powders in Tamil Nadu, Karnataka, Andhra Pradhesh, Kerala and some parts of Goa, Maharashtra, Indonesia, Malaysia, Thailand and a few other Asian countries. A Kolam is a geometrical line drawing composed of curved loops, drawn around a grid pattern of dots. In South India, it is widely practised by female Hindu family members in front of their houses.[1] Kolams are regionally known by different names in India, Raangolee in Maharashtra, Aripan in Mithila , Hase and Raongoli in Kannada in Karnataka, Muggulu in Andhra Pradhesh, Golam in Kerala etc.,[2] More complex Kolams are drawn and colors are often added during holiday occasions and special events.

[2017-01-12 Kolam]

many more, see https://twitter.com/mariawirth1/status/810098967761616897

logician Raymond Smullyan is 97 now. ☹

Raymond Smullyan is famous among armature mathematicians for his many books on logic, presented as intriguing logic puzzles.

## Knights and Knaves

On a island, all inhabitants are either knights, who always tell the truth, or knaves, who always lie.

John and Bill are residents of the island.

John says: We are both knaves.

Who is what?

here's his books Raymond Smullyan books

### Truly Amazing Euler, Greatest Mathematician. A Intro.

truly amazing Euler, greatest mathematician. Here's his story told by William Dunham, 2008. Save 1 hour to watch.

### Best Introduction to Graph Theory

the best introduction to graph theory is this one.

if you don't know the basics, get it, read it in a week, 1 hour a day. Once you start reading, you can't put it down.

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.

### Why is Mandelbrot Set Cardioid Shaped?

more about cardioid here: Cardioid

### 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) Buy at 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 and 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