# Xah Math Blog

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

〔 Drawing fractal Droste images By Roy Wiggins. At http://roy.red/fractal-droste-images-.html 〕

### Groups and Their Graphs

i learned group theory from this book, in 1997. Get it.

For coders, if you don't know math, the most important and basic of math, is group theory. (other than basic calculus) learn it to start to get into serious math.

group theory, is something you can learn the basics in 1 hour. And you can spend 5 years getting phd specialization. It is also, one of the most simple and beautiful theory, and the most important, practical, widely used.

this book i used back in 1992, and loved it.

get old edition, as newer edition of math text book don't add much. are basically scams to get you buy new.

Why Are Textbook So Expensive?

i also like Abstract Analysis, Andrew Gleason

old article The Problems of Traditional Math Notation

hit hackernews https://news.ycombinator.com/item?id=15631151

2 great JavaScript for visualization

### Automatic differentiation and differentiation without limits

from ~2017-10-17 https://plus.google.com/+XahLee/posts/hzPVxNEWbe1

comment at https://plus.google.com/+XahLee/posts/49vT1MXY3Fn

### Spivak's Calculus on Manifolds, why is Wikipedia linking to stolen pdf?

is Spivak's Calculus on Manifolds in public domain now? why's Wikipedia linking to pdf?

### 3dxm, worst math software, but great for visualization of manifold

3dxm (aka 3d explore math), by my professor friends Richard Palais and Karcher Hermann, the worst software possible in the universe.

painful to use beyond comprehension

however, it is one of the best for visualizing manifold.

to see what the software can do, see

- http://virtualmathmuseum.org/Surface/gallery_m.html
- http://virtualmathmuseum.org/Surface/gallery_o.html

you can download 3dxm at http://3d-xplormath.org/index.html

### sagemath, and the damnation of the open source

sagemath. sad. broken website. missing pictures. https://wiki.sagemath.org/art?action=show&redirect=pics

download pages looks like 1999. http://www.sagemath.org/download.html

the damnation of the open source

https://plus.google.com/+XahLee/posts/aNMAmKVvVYF

i heard his story of going commercial. (even retweeted it last year on twitter)

i've heard of sagemath for 10 years. I thought it's the great open source math program. Even recommend to clueless newbies. Never used it though. But only started to do math again now, so i tried to download for the first time.

and, the first impression, of its website and download page, is very sad. it reminds me all the pain of the open source.

first i go to go the website. Check out the pic demo. Sadly, 80% of pictures don't show.

i went to download page, to download anyway. I was expecting one single download button as software sites are today, which it'll just download whatever version of my os. But no, i got this wtf list of mirror sites of 1999. Quite confusing, and Virus is on my mind. And, the Mac download page has a app and non-app version, with explanation about which to download. That's one complete screwup.

i haven't tried to launch it yet. But so far, it's not encouraging. At this point, it's probably a correct guess, that if you need math lightly, just goto wolframalpha. And if you use math heavily, just $300 to Mathematica.

i wish sagemath is good, i hope it is. But open source, i think it's a sad pipe dream failure.

i hope he's successful though.

### Riemannian Geometry and Mathematical Physics

Got a gift from John Baez:

GAUGE FIELDS, KNOTS AND GRAVITY by John Baez, Javier P Muniain, 1994.

excellent book. the book is really about the math of physics. More specifically, Riemannian Geometry.

the book is fast easy reading!

i think i start right at chapter 4 on differential forms.

then resolve the mysteries of stoke's theorem, exterior differential forms, cohomology, lie group, then,

in part 2 will be lots goodies for me. bundles and connection, homology. ... chern classes in chapter 4, ... and more differential geometry goodies in Part 3.

differential geometer Richard S Palais' Books and Papers http://vmm.math.uci.edu/PalaisPapers/

### The Life and Mathematics of Shiing-Shen Chern

one of the greatest differential geometer of the century.

The Life and Mathematics of Shiing-Shen Chern

Just read this. Soul touching.

to read bio of Chern, is to also have a glimpse of modern history of China, thru the tumultuous times of war.

(watch great movie Farewell My Concubine )

and, i learned, a towering figure of differential geometry is Élie Cartan

math things i've learned.

**equivalence problem**. For example, triangle are defined by 3 real numbers, length of 2 of which is less than the other. So, that's the condition. It “generates” all possible triangles. And, any 2 triangle can be decided if they are equivalent (by isometry here), by first reducing the 3 numbers to certain canonical form (such as by scaling so that shortest side is 1), then simply compare the numbers.

For plane curves, it's determined by curvature function. It generates all plane curves. And to determine if 2 plane curves are equivalent by an isometry, you just express a given curve by the curvature function. Then you simply just compare 2 functions, literally.

for space curves, it's 2 functions: curvature and torsion.

for surfaces, it's the 2 fundamental forms.

the question of “equivalence problem”, is to formulate a way, so any 2 geometric object can be so compared, and unify the cases for curves and surfaces.

and Cartan began it Cartan's equivalence method

and today the method is “g structure”. G-structure on a manifold

### omg, how boring is topology?

you'd think you gonna see how coffee cup turns into a donut.

instead, you got a fiat “open set”. From there on, its set of sets, subset, empty set, union of set, intersection of set, complementary set, power set, super set, finer set, coarser set, and it's set all the way!

The Method of Fluxions by Issac Newton https://archive.org/stream/methodoffluxions00newt#page/n3/mode/2up

Hilbert's 16th problem. the relative positions of the branches of real algebraic curves of degree n. Hilbert's sixteenth problem

math notation idiocy https://plus.google.com/+XahLee/posts/hzPVxNEWbe1

Comprehensive Introduction to Differential Geometry By Michael Splvak. Classic. Comprehensive Introduction Differential Geometry

that's 5 volumes heavy expensive book. and, is graduate level.

you wonder, why's such book named “introduction”. It's actually graduate text, at 5 thick heavy volumes. Mathematicians not specialized in differential geometry have lots to learn from it, i assume.

but, “introduction” implies, this 5 volumes book only touches the surface. But then, its got the word “comprehensive” in it?

so, the proper interpretation seems to be, that the topic of differential geometry really is deep, and tied with lots other advanced math. Therefore, even at 5 volumes, and being a comprehensive tome, is, still just a intro.

On several occasions, most prominently in Volume 2, Spivak “translates” the classical language that Gauss or Riemann would be familiar with to the abstract language that a modern differential geometer might use.

2017-10-08 Wikipedia Michael Spivak

that'd be interesting to read.

http://3d-xplormath.org/ for Mac

### software for plotting math surfaces: Surfer

software for plotting math surfaces, especially algebraic surfaces. (Microsoft Windows, MacOS, Linux) https://imaginary.org/program/surfer

For some example of plots, see a friend Jean Constant's blog at https://jcdigitaljournal.wordpress.com/category/01-january-the-surfer-program/page/2/

### Vladimir Voevodsky, Fields Medalist, Dies at 51

### notation for recursion, what about notation for reduce?

been wanting a notation for recursion. Found it. In math, it's called iteration of a function, written as f^(°n). However, that doesn't cover reduce.

the term polytope is from mathematician Alicia Boole Stott. Alicia Boole Stott

### understand the “expansion” operation on polyhedron

understand the “expansion” process on polyhedron. http://VirtualMathMuseum.org/Polyhedra/Icosahedron/index.html

[• How real are real numbers? By Gregory Chaitin. At https://arxiv.org/pdf/math/0411418.pdf , Accessed on 2017-04-14 ] (local copy How_real_are_real_numbers_By_Gregory_Chaitin_f76de.pdf)

### gyroscopic precession

Learned about gyroscopic precession. Not intuitive. Don't know how the physics works out. But this video is amazing

### Visualize Math with 3D Printing

Visualizing Mathematics with 3D Printing. one expensive book Buy at amazon

### euler's disk

#physics euler's disk. haven't seen this before.

Euler's Disk Buy at amazon

### rattleback

this thing, your spin it, and it spins in reverse direction. Due to built up of instability, the rocking.

rattle back on amazon. Buy at amazon

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.

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