Xah Math Blog

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

toroidal map in real life
toroidal map in real life. Who'd thought?

Magic Polyhedrons (updated)

Great Software for Plane Geometry (updated)

obsolete math software, 1990s to 2005

revisiting and updating my page on list of great math software, mostly from 1990s. Many, are from hobbyists, truly great programs. Times flies. People get old. Many are not updated. Many websites gone. And many Java applets, as relics of tech progress.

It's like, right now you are hot into programing, machine learning, this or that. 10 or 20 years later, you might be something else, or puffed off. You look into the ebb and flow of each human animal life activity as cellular automata, it'd be something.

for nostalgia, here's list of obsolete math software. 1990s to ~2005. In Mac classic, Java, Flash. Old Math Software

Great software for Tilings, Patterns, Symmetry (major update)

preorder, partial order, total order

The original concept of totally ordered set or order, still dominant today, obeys a bunch of rules:

  1. reflexivity: x≤x
  2. transitivity: x≤y and y≤z imply x≤z
  3. antisymmetry: if x≤y and y≤x then x=y
  4. trichotomy: for all x,y we either have x≤y or y≤x.

The real numbers with the usual ≤ obeys all these. Then people discovered many situations where rule 4 does not apply. If only rules 1-3 hold they called it a partially ordered set or poset. Then people discovered many situations where rule 3 does not hold either! If only rules 1-2 hold they called it a preordered set or preorder.

Category theory teaches us that preorders are the fundamental thing: see Lecture 3. But we backed our way into this concept, so it has an awkward name. Fong and Spivak try to remedy this by calling them posets, but that's gonna confuse everyone even more! If they wanted to save the day they should have made up a beautiful brand new term.

2018-03-29 by John Baez from https://forum.azimuthproject.org/discussion/comment/16083/#Comment_16083

Top Ten Math Books for Math Haters

category theory online course

So, am taking a online course of category theory, lead by famous mathematician and theoretical physicist John Baez.

the home page is at https://forum.azimuthproject.org/discussion/1717/welcome-to-the-applied-category-theory-course

the text book is: https://arxiv.org/abs/1803.05316

local mirror category_theory_brendan_fong_david_spivak_2018-03.pdf

Great Math Software

some great math related JavaScript site added.

Using Neural Net to Create New Knitting Patterns

Quite amazing.


by Janelle Shane

Tilings and Patterns Book

random sequence, not mathematically defined

it's interesting, that the concept of random number, or random sequence, is actually undefined in math, because it is impossible to define mathematically.

The concept of a random sequence is essential in probability theory and statistics. The concept generally relies on the notion of a sequence of random variables and many statistical discussions begin with the words “let X1,…,Xn be independent random variables…”. Yet as D. H. Lehmer stated in 1951: “A random sequence is a vague notion… in which each term is unpredictable to the uninitiated and whose digits pass a certain number of tests traditional with statisticians”.[1]

Axiomatic probability theory deliberately avoids a definition of a random sequence.[2] Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement “let us consider a random sequence” an abuse of language.[3]

[2018-03-19 Random sequence]

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Philosophy of Math: Logicism, Formalism, Intuitionism, and Category Theory

the 3 philosophy of math of 1900s are: ① logicism. Math is pure derivation of logic. ② Hilbert's formalism. Math is just bunch of fomulaic symbols. No meaning. ③ intuitionism. Math is mind's construction.

these 3 schools are the basis of foundations of math. Each has its problems. logic became formal (i.e. symbolic) logic, which is basically formalism. They in turn, became constructivism (a variation of intuitionism), so it can run by computer.

comment at https://plus.google.com/+XahLee/posts/LZJPcj4KGRB

here's my comment to mathematician john baez:

thanks for correction.

i've always wondered what's the difference of logicism and formalism. Back in 1990s, I was introduced to them by Russell. (my top 3 fav author) Tried to read about them now and then. In 2000s, wikipedia is still not great, but in past year, i read a lot math again, and i seem to get some understanding. Is my characterization roughly correct?

i never cared about intuitionism, until now, and now i very much appreciate constructivism. As i understand, it is what mechanical manipulation can actually build.

PS few days ago i tried to read wikipedia about category theory again. Again, came away as incomprehensible abstraction. Meant to write a rant about it… but here's a gist…

so i spend half hour thinking, what does abstraction mean? After all, numbers 1 2 3 ... are abstraction to begin with. But then we have equations, such as 2*x+3 = 4, which is abstraction of description of math problems. Then, abstract algebra and 1800s's math, are the 2nd stage of abstraction. e.g. abstract algebra came from the systematic formal maniputlation of equations. And likewise stuff in algebraic geometry e.g. variety, and differential geometry geometry e.g. manifold, and so on in other branches of math.

Then, what's the next level abstraction? I'm thinking, scheme and sheaf etc (which i have no idea what they are), or, the category theory stuff.

but anyway, so i was reading Wikipedia on category theory. Category theory

in other math articles, say, holomorphic function, homotopy, hilbert space, riemann sphere, homomorphism, etc i've recently read, there's a cold definition. Most of the time, i can at least understand the definition, and go on from there. (in the above examples, i also appreciate what they mean, why they are there, etc, except hilbert space.)

But the category theory, it begin with pages of pages of meta description. And am at loss. But perhaps, to appreciate it, one must first have solid understanding of various branches graduate level math?

but then, my first exposure to it is

Conceptual Mathematics: A First Introduction to Categories by F William Lawvere, Stephen Hoel Schanuel. [Buy at amazon]

in 1990s. Which claimed to be written for laymen, and useful even outside of math. I recall, after reading 1 chapter, it's going too slow, and i was feeling, “what's the point?”. I never continued after the 1 chapter. (i don't have the book now. must gave away or something long ago.)

Math, Naturalness of a Field's Conditions

So, in algebra, you have a field. It's a set and 2 functions f and g, of the form f(a,b) and g(a,b), and X nesting properties of f and g (called commutativity, associativity, distributive, invertible, etc).

So, what happens if we have more than 2 functions, 3 parameters each? So, such study is called universal algebra. (i haven't studied, but i wonder what happens there, in general. The mix of nesting of function, i gather, would create more complex concept similar to associativity and distributive, involving 3 functions, but we don't have a name.)

But, WHAT is the fundamental nature, that real number (a field) is this specific X nesting properties? What is it, that real numbers, which we consider as naturally occurring or developed, form this “field” with such specific nesting properties?

am thinking, there must be some logical answer.

to describe my questions further, for example, real number developed because, first we have counting, 1, 2, 3, then naturally we developed 0, then we have rational, which is ratio, e.g. 1/2 as cutting a pie, then we have negative numbers (from, say, I OWE YOU). And from rational we discovered irrational, as in pythagorean. So there, we have real numbers. And, addition came from simple counting. Multiplication can be considered as a short for repeated additions.

so, am guessing, addition, and multiplication (repeated addition) necessitates the commutativity, associativity, distributive, properties?

now, having written this out, it seems obvious and is the answer to my own question.

... because, by looking at the definition of field, i've always thought, it's somehow arbitrary and complicated. I'd be interested, in a systematic approach of studying structures, e.g. a set, with n operation of m-arity, starting with n=1 and m=1. Then, we develop, all possible ways of nesting n such functions of m arity, so that associativity, commutativity (order of arg), distributivity, are just 3 of the possible properties.

but i gather that, universal algebra may began like this, but actually has become a bit something else.

similar situation is group theory

Group, is much simpler than field. But if you look at definition of group, you see that, it seems also arbitrary and complex.

But then, if you look at symmetry, such as symmetry of polyhedra, you see that all the requirement of group is necessary, and no more, no less.

but the question remain, why is the group definition seems arbitrary?

i mean, is there some point of view, so the associativity (a • b) • c = a • (b • c) requirement, dissolved as if it is natural?

again, i really like to see, a combinatorial exploration of all such possible condition of n functions of m arity. (as in universal algebra)

perhaps after seeing that, then one can judge, comparatively, whether the associativity condition is natural.

now, to be sure, function with 1 or 2 arity is actually perhaps the most natural. And 1 or 2 functions is also pretty bare. As opposed to, a structure with function that has 3 arity, or more than 2 functions.

If you just have 1 function, or just function wit 1 arity, than it may become too simple to have interesting things going on.

but again, would like to see a systematic combinatorial list of the conditions that may arise of n functions and m arity.

comment at https://plus.google.com/+XahLee/posts/6tiAGpLiQ5z

Introduction to Probability, by Charles M Grinstead, J Laurie Snell Shoup (free book)

for those of you programers doing big data or AI, understand probability is essential.

Groups: A Path to Geometry

major issues of writing 2D math plotter

when writing 2D math plotter, major issues are

Chinese number names

Quality Free and Legal Math Textbooks https://www.patreon.com/posts/17392806

added a number theory book.

ideal of ring theory

For an arbitrary ring ( R , + , ⋅ ), let (R,+) be its additive group.

A subset I is called a two-sided ideal (or simply an ideal) of R if it is an additive subgroup of R that “absorbs multiplication by elements of R.”


digging out my 3 Volumes classic 〈Mathematical Thought From Ancient to Modern Times〉. I haven't read for 20 years. To read about how sin exp were extended to complex plane.

Buy Mathematical Thought From Ancient to Modern Times, Volume 2 by Morris Kline

See also: Math and Geometry Books

origami simulator

Origami Simulator, by Amanda Ghassaei, at http://www.amandaghassaei.com/projects/origami_simulator/

added more logic book. Free Math Textbooks

The sound of space-filling curves: examples

this is incredibly beautiful, not just the novelty, but MUSICALLY!

[The sound of space-filling curves: examples By Herman Haverkort. At http://www.win.tue.nl/~hermanh/doku.php?id=sound_of_space-filling_curves ]

listen to the first one, the hilbert curve (the one “without intro”)

that's the best one. The other ones are not so good.

The author, Herman Haverkort, is a mathematician specializing on algorithms.

Free Math Textbooks (added books on logic)

Complex Function Grapher

see Great Software For 2D Visualization of Geometry

Celtic Knots, Truchet tiles, Combinatorial Patterns

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

Math Animations

Groups and Their Graphs

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

Groups and Their Graphs  Israel Grossman Wilhelm Magnusaff87
[Groups and Their Graphs By Israel Grossman And Wilhelm Magnus. At Buy at amazon ]

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.

Linear Algebra, David C Lay

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



hyperbolic surface lamp cover bachman light 88939
“bachman light”
hyperbolic surface lamp cover
[Artist David Bachman Unlocks the Secret Identities of Everyday Objects By Angela Linneman. At https://www.shapeways.com/blog/archives/35766-artist-david-bachman-unlocks-secret-identities-everyday-objects.html ]

Automatic differentiation and differentiation without limits

Automatic differentiation


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?

Calculus on Manifolds (book)

Would You Missout a Lot in Math If You Don't Know Physics?

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

3dxm jd tower 77177
3dxm jd tower 77177

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

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

Visual Complex Analysis  Tristan Needham 83391
Visual Complex Analysis, Tristan Needham

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


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

islamic pattern 43531
islamic pattern 43531
islamic pattern color 43532
islamic pattern color 43532

see https://plus.google.com/+johncbaez999/posts/DGYEUQ3WG4b

and http://math.ucr.edu/home/baez/diary/october_2017.html

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

Abuse of Math Notation

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.

math 3dxm saddle tower 46125
math 3dxm saddle tower 46125

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

see Homotopy Type Theory

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.

e.g. JS: Array.prototype.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

Free Math Textbooks

Free Calculus Textbooks

Theory of Sets by N. Bourbaki

[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

Helicopter Physics Series - #4 They're Gyroscopes - Smarter Every Day 48

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

Visualize Math with 3D Printing

math 3d printing  Sofie M Langdon 2017 03 04
“Home schooler presents the math she's studied via @henryseg's book and some 3D printing. @maanow Golden section art exhibit” [image source https://twitter.com/timchartier/status/838147872323330049 ]

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

euler's disk

Euler's Disk - Longest Spin Ever

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

Euler's Disk Buy at amazon


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

20 rattlebacks.

rattle back on amazon. Buy at amazon

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

curly dragon curve 02902
Curly dragon curve. [All Plane-Filling Curves By Jeffrey Ventrella. At http://fractalcurves.com/all_curves/ , accessed on 2017-02-14 ]

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

Google AlphaGo Beats World Champions, 60 to 0

tire print snow pattern
tire print in snow. [image source https://plus.google.com/+KazimirMajorinc/posts/7SrScNqV3wp by Kazimir Majorinc, 2017-01-17]

Kolam pattern

800px Kolam Indigo Dyed Cloth
3x3 symmetry 9 goddesses swastika Kolam with a single cycle by Nagata S, each of which corresponds to one of the nine Devi (Goddess) of the Vedic system [image source ]

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

Visualizing the Riemann zeta function and analytic continuation
Missing square puzzle
Missing square puzzle

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.

A Tribute to Euler, by William Dunham, 2008. Talk starts at 1:20.
truncated octahedra chc airport 2016-s
Truncated Octahedrons, Rhombic Triacontahedron, Dodecahedron in Real Life

Best Introduction to Graph Theory

the best introduction to graph theory is this one.

Introduction to Graph Theory  Trudeau
[Introduction to Graph Theory By Richard J Trudeau. @ Buy at amazon ]

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.

If you have a question, put $5 at patreon and message me.