Xah Math Blog

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

Update Highlights

WolframLang Mathematica packages

get them on github

Ruled Surface, Mathematica Notes

new edition

now on github

now on github. updated 26-years-old code. still in heavy work. but putting on github now.

some random notes on mathematicians

some random notes

one of the great mathematician, vladimir arnold

vladimir arnold 2024-02-21
vladimir arnold 2024-02-21

xtodo waiting for id

finally, published my PlaneCurvePlot to Wolfram Function Repository! this is updating a 30 years old package. After a week of remake. (was called ParaPlot.)

PlaneCurvePlot Xah Lee 2024-02-18
PlaneCurvePlot Xah Lee 2024-02-18
plane curves 2024-02-18
plane curves 2024-02-18

Yutaka Taniyama

one great mathematician Yutaka Taniyama, killed himself suicide.

Yutaka Taniyama 2024-02-15
Yutaka Taniyama 2024-02-15
diophantine geometry 2024-01-17
diophantine geometry 2024-01-17
stephen wolfram graph 2023-12-23 223526
beautiful graphics. from stephen wolfram
stephen wolfram multiway tiling 2023-12-23
stephen wolfram multiway tiling 2023-12-23 [Aggregation and Tiling as Multicomputational Processes By Stephen Wolfram. At https://writings.stephenwolfram.com/2023/11/aggregation-and-tiling-as-multicomputational-processes/ ]
stephen wolfram multiway game 2023-12-23 210457
[Games and Puzzles as Multicomputational Systems By Stephen Wolfram.. At https://writings.stephenwolfram.com/2022/06/games-and-puzzles-as-multicomputational-systems/ ]
stephen wolfram multiway num 2023-12-23 210644
[Multicomputation with Numbers: The Case of Simple Multiway Systems By Stephen Wolfram. At https://www.wolframphysics.org/bulletins/2021/10/multicomputation-with-numbers-the-case-of-simple-multiway-systems/ ]

lots random reading notes

random list of great mathematicians

Shiing-Shen Chern 2023-11-20
Shiing-Shen Chern 2023-11-20

major update.

Carl Friedrich Gauss's Wives

Carl Friedrich Gauss wife 2023-11-16 X33qH Carl Friedrich Gauss wife 2023-11-16 qmmM8
Carl Friedrich Gauss wife 2023-11-16 X33qH
mathmod 2023-11-01 205111
mathmod 2023-11-01 205111
Wolfram Doug Lenat 2023-09-12 S6Hw
Wolfram Doug Lenat 2023-09-12 S6Hw

[Remembering Doug Lenat (1950–2023) and His Quest to Capture the World with Logic By Stephen Wolfram. At https://writings.stephenwolfram.com/2023/09/remembering-doug-lenat-1950-2023-and-his-quest-to-capture-the-world-with-logic/ ]

Wolfram Ada Lovelace 2023-09-12 jNPH
Wolfram Ada Lovelace 2023-09-12 jNPH

[Untangling the Tale of Ada Lovelace By Stephen Wolfram. At https://writings.stephenwolfram.com/2015/12/untangling-the-tale-of-ada-lovelace/ ]

Computational General Relativity: Live with Jonathan Gorard
Streamed live on May 25, 2023
Computational General Relativity Jonathan Gorard 2023-05-29
Computational General Relativity Jonathan Gorard 2023-05-29

here's WolframLang code, in plain text, rarely seen, as opposed to in notebook rendered math formula form. https://github.com/JonathanGorard/Gravitas

this allows you to do the unix tradition line-based diff and git version control. (big brain damage inherited from C and unix.)

but looking at it, i imagine it's auto generated from notebook. Writing math code in Mathematica notebook is just too feature rich and useful instead of coding it in plain text editor.

jonathan gorard gravitas 2023-05-30
jonathan gorard gravitas 2023-05-30

spectacular article.

[How Did We Get Here? The Tangled History of the Second Law of Thermodynamics By Stephen Wolfram. At https://writings.stephenwolfram.com/2023/01/how-did-we-get-here-the-tangled-history-of-the-second-law-of-thermodynamics/ ]

Homotopy Type Theory, Steve Awodey and Michael Warren

liqo12 Steve Awodey n Michael Warren 2023-01-01 rFvGF
liqo12, Steve Awodey n Michael Warren 2023-01-01

xtodo to watch

A visual introduction to the Hopf fibration
Nov 9, 2022

cleaned up my old blogs, and made some into independent pages.

The gist here is to distill a math art into its algorithmic essence. By recursion or some encoding (such as math equation)

wow, finally i understood something that bugged me for 10+ years: why is that whenever i read about category theory, there is never clear/formal definition.

category theory univalent 2021-08-27
category theory univalent 2021-08-27

Mathematicians Interview

xtodo 2021-08-01

Information Theory, Inference, and Learning Algorithms by David J C MacKay

Homotopy Type Theory

Geometry: Transformation of the Plane

Axiomatization of the Computational Universe

Spectacular. First, read this [The Empirical Metamathematics of Euclid and Beyond By Stephen Wolfram. At https://writings.stephenwolfram.com/2020/09/the-empirical-metamathematics-of-euclid-and-beyond/ ] then

emil post recursion theory 2021-02-19
emil post recursion theory 2021-02-19

then watch

Wolfram Physics Project: Axiomatization of the Computational Universe Tuesday, Feb. 16, 2021
Streamed live on Feb 16, 2021
logic critical pair 2021-03-09
Critical pair lemma
univalent foundations 2021-02-22
Univalent foundations

[After 100 Years, Can We Finally Crack Post's Problem of Tag? A Story of Computational Irreducibility, and More By Stephen Wolfram. At https://writings.stephenwolfram.com/2021/03/after-100-years-can-we-finally-crack-posts-problem-of-tag-a-story-of-computational-irreducibility-and-more/ ]

The article looks at Emil Post's tag system, essentially concludes that it's like cellular automata or 3n+1 problem, and the view that most things are simply computation. (plus lots math nuggets and short bio of Emil)

Great Math Board Game Software updated

Freed Go. This guy now works for tesla. Amazing. When you compile math programs, since 1997, over the years i update them, and discover where these people went.

Real Life Pretty Math

this is spectacular. Stephen Wolfram's personal stories of the greatest mathematicians, physicists, technologists of past 100 years

History of Science and Technology Q and A (Jan. 27, 2021)
by Wolfram

[Who Was Ramanujan? By Stephen Wolfram. At https://writings.stephenwolfram.com/2016/04/who-was-ramanujan/ ]

xtodo read

[Droste Effect with Mathematica By Jon Mcloone. At http://blog.wolfram.com/2009/04/24/droste-effect-with-mathematica/ ]

[The 2011 Mathematica One-Liner Competition By Christopher Carlson. At http://blog.wolfram.com/2011/12/01/the-2011-mathematica-one-liner-competition/ ]

parquet transform tiling. old links.

What Is Perspective Drawing


Plane Curves Books

Requirements for a Visualization System

some random page, minor edit

linear programing

Learned linear programing back in 1992 in college. Never encountered it since. But, about 2 years ago, when i was thinking about finding the optimal keyboard shortcut layout, it hit me, the problem is linear programing.

linear programing 2020 qrrpn
Linear programming

Emacs vs vim, Compute Keybinding Efficiency

topology of figure 8 on a stick

Mathematician Grigori Perelman, Decline 1 Million Prize

3dxm math museum 2020-06-09 jqjfm
spectacular plane curves in JavaScript, by mathematician Hermann Karcher http://virtualmathmuseum.org/Curves/index.html

random article

math says no 6z5nk
math says no
HyperRogue 2020-04-12 pyg66-s247x253
Great Software for Cellular Automata

good math book. What Is Mathematics? by Herbert Robbins , Richard Courant Buy at amazon

Math Mysticism: is Hurricane Shape a Fibonaci Spiral?

Fibonaci is one of the undying myth in math.

Lorenz Attractor Malin Christersson 2019-12-31 jn9pq-s281x222
Great Software for Dynamical Systems
Alan Bundy mathematician 2019-12-31 vdpz9
Alan Bundy mathematician 2019-12-31
What if Current Foundations of Mathematics are Inconsistent? | Vladimir Voevodsky

Fluid Simulation, added to Great Software for Cellular Automata

is there a name for a function that's dot product of more than 2 n-dimensional vectors?

for example , 3 vectors of 2D, it would be:

([a1,a2] , [b1,b2], [c1,c2] ) → (a1 * b1 * c1 + a2 * b2 * c2)
polyhedron 2019-11-14 sffwz
polyhedron 2019-11-14

Great Software For 2D Visualization of Geometry

God Created The Integers jn4vc
[God Created The Integers: The Mathematical Breakthroughs that Changed History By Stephen Hawking. At Buy at amazon ]

Visualizing Quaternions

Software to Visualize 3D Geometry

Geometer's Sketchpad, WebSketch

WebSketch, seems to be a new version of the Geometer's Sketchpad. http://geometricfunctions.org/fc/tools/

there are lots of them in past 10 years. see

Town of the Great Math Hermit Alexander Grothendieck

updated at Mathematician, Alexander Grothendieck = Obi-Wan

Notes on the Riemann Hypothesis by Ricardo Pérez-Marco

Notes on the Riemann Hypothesis by Ricardo Pérez-Marco 1707.01770.pdf via https://twitter.com/johncarlosbaez/status/1164016020425543681


[The Octonions By John C Baez. At http://math.ucr.edu/home/baez/octonions/octonions.html ] (Published in Bull. Amer. Math. Soc. 39 (2002), 145-205. Errata in Bull. Amer. Math. Soc. 42 (2005), 213.)

pop cult of Feynman

God so many programer nerds are into Richard Feynman. I despise that guy, for no reason. it's like, any joe will shout Einstein when science is the topic. ok, Feynman is like top 10 physicist. but am not interested in physics. and whatever Feynman's math i have no interest.

Feynman math 2019-08-13 vcvg6
Feynman math 2019-08-13

i despise physicist. Whenever there is good piece of math, physicists ruin it.

#math John Milnor , big mathematician. one of his book is Topology from the Differential Point of View, 1965, John W Milnor


Something that is impredicative, in mathematics, logic and philosophy of mathematics, is a self-referencing definition. Roughly speaking, a definition is impredicative if it invokes (mentions or quantifies over) the set being defined, or (more commonly) another set that contains the thing being defined. There is no generally accepted precise definition of what it means to be predicative or impredicative. Authors have given different but related definitions.

The opposite of impredicativity is predicativity, which essentially entails building stratified (or ramified) theories where quantification over lower levels results in variables of some new type, distinguished from the lower types that the variable ranges over. A prototypical example is intuitionistic type theory, which retains ramification so as to discard impredicativity.

Russell's paradox is a famous example of an impredicative construction—namely the set of all sets that do not contain themselves. The paradox is that such a set cannot exist: If it would exist, the question could be asked whether it contains itself or not — if it does then by definition it should not, and if it does not then by definition it should.

The greatest lower bound of a set X, glb(X), also has an impredicative definition: y = glb(X) if and only if for all elements x of X, y is less than or equal to x, and any z less than or equal to all elements of X is less than or equal to y. This definition quantifies over the set (potentially infinite, depending on the order in question) whose members are the lower bounds of X, one of which being the glb itself. Hence predicativism would reject this definition.[1]

2019-08-09 from Predicativism

am surprised, that the definition of infimum (or, e.g. shortest person in a room) is impredicative. and seems there is no predicative definition of them. and this seems to reduce the power of type theory drastically.

omniscience of math

god, please grand me omniscience of math.

while riding bike to pay rent yesterday, i thought about what'd happen if am omniscience of math. First, there are 6 Clay math price, each $1M reward. but as math omniscient, $6 millions is like 6 pennies on a dirty street.

with math omniscience, you now know the secret that's worth more than entire Google ($136 billion revenue in 2018). And you can break any secret message of any nation. Whatever US military spends in research in a decade, your knowledge is worth greater than that per second.

math omniscience also means, you are now the greatest mathematician. What you know is more than all mathematicians combined, dead or alive, infinitely times more. Quantum mechanics, cosmology. You KNOW the mysteries of the universe.

Math omniscience in a man. This is when, i wonder, if catastrophe might happen. You are now inhuman. You may no longer desire to eat.

stability of minimal surface

this is amazing! read the thread, by Daniel Piker: https://twitter.com/KangarooPhysics/status/1136306166349357058

that's amazing cuz it shows many popular minimal surfaces in math are not stable. I think it's rarely talked about. And he created a software that simulate soap film minimizing surface area.

And his blog is spectacular ( https://spacesymmetrystructure.wordpress.com ) i've known since 2011.

i think the stability issue is studied in stability theory, while minimal surfaces are differential geometry. These 2 are separate branches. Typically not studied together or at all both. Thus when we learn about minimal surfaces and soap film, talking about stability distracts.

a #geometry question i had for long. Given a bunch of points (that forms a surface), what's the (efficient) algorithm to triangulate them? any name i can search for?

answer: ball-pivoting algorithm. (thx to Daniel Piker)

readings about real number

Real number 2019-08-04 zxy7f
Real number 2019-08-04

Mathematician Adrien-Marie Legendre

Adrien-Marie Legendre 2019-08-04 55zf8
Adrien-Marie Legendre 2019-08-04

truly despise the jargon injection surjection, created by Bourbaki

injection surjection 2019-08-04 kcbk7
injection surjection 2019-08-04

Wang Tiles, and Video of Hao Wang

shape of space

if you want to understand this http://www.espaces-imaginaires.fr/works/ExpoEspacesImaginaires2.html there's a great book

Reading Notes on “The Shape of Space”

applied group theory. 🤩

Magic Polyhedrons

Computable Number

Constructible Number

splitting field

math. Field GF(4)

one of the smallest non-trivial field

math field gf4 2019-07-26 d8m9f
math field gf4 2019-07-26

Combinatorial Exploration of Algebraic structures

girih tiles 2019-07-24 wpgw3
girih tiles 2019-07-24

Braid Group

cable knitting. Such intricacy, weave within weave.

was wondering if John C. Baez https://twitter.com/johncarlosbaez has written about math aspect of weaving. Searching braid theory baez, indeed! Loop braid group

See also:

braid group 2019-07-23 rg95b
braid group 2019-07-23

Braid theory

todo. read http://www.malinc.se/noneuclidean/en/poincaretiling.php

Great Software For 2D Visualization of Geometry

new app added

A Course In Universal Algebra, Burris, Sankappanavar

added a new text book. graduate level.

3 books i love. Now each on its own page.

Introduction to Graph Theory  Trudeau-s199x314
Graph Theory Book
visual complex analysis 20180309-s217x289
Visual Complex Analysis by Tristan Needham
groups and their graphs  Grossman Magnus zdbqt
Groups and Their Graphs by Israel Grossman and Wilhelm Magnus

incidence structure, projective geometry

Undergraduate Math Degree Courses

truchet tile 2019-07-08 2j73n
truchet tile 2019-07-08

Truchet tiles

logic. free variable

logic. principle of explosion

Euler-Mascheroni constant

Continued fraction

Math, Algebra: on the Phraseology of X Over K, and What's Group Theory?

the supreme mystery of the universe, is math. note, not physics, quantum or blackhole crap.

u can write scifi about blackhole or quantum physics stuff, but u cant for math.

cauchy sequence defines calculus and complete metric space

smooth map

been slacking in the math department. Going to pick up again. Each day, i spend 1 hour reading math, and post whatever. Today, let's learn about “smooth map”.

this is logic, part of proof theory. this is what intent to learn in next 5 years, as opposed to category theory. (i got asked about the latter often, from programer idiots) [ twitter johncarlosbaez ] https://twitter.com/johncarlosbaez/status/1122976661132021760

Peano axioms

Tennenbaum's theorem

Tennenbaum's theorem is a result in mathematical logic that states that no countable nonstandard model of first-order Peano arithmetic (PA) can be recursive (Kaye 1991:153ff).

Compactness theorem

In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent.

Finite model theory

Finite model theory (FMT) is a subarea of model theory (MT). MT is the branch of mathematical logic which deals with the relation between a formal language (syntax) and its interpretations (semantics). FMT is a restriction of MT to interpretations on finite structures, which have a finite universe.

Interpretation (logic)

An interpretation is an assignment of meaning to the symbols of a formal language. Many formal languages used in mathematics, logic, and theoretical computer science are defined in solely syntactic terms, and as such do not have any meaning until they are given some interpretation. The general study of interpretations of formal languages is called formal semantics.

The most commonly studied formal logics are propositional logic, predicate logic and their modal analogs, and for these there are standard ways of presenting an interpretation. In these contexts an interpretation is a function that provides the extension of symbols and strings of symbols of an object language. For example, an interpretation function could take the predicate T (for “tall”) and assign it the extension {a} (for “Abraham Lincoln”). Note that all our interpretation does is assign the extension {a} to the non-logical constant T, and does not make a claim about whether T is to stand for tall and ‘a’ for Abraham Lincoln. Nor does logical interpretation have anything to say about logical connectives like ‘and’, ‘or’ and ‘not'. Though we may take these symbols to stand for certain things or concepts, this is not determined by the interpretation function.

An interpretation often (but not always) provides a way to determine the truth values of sentences in a language. If a given interpretation assigns the value True to a sentence or theory, the interpretation is called a model of that sentence or theory.

Lindström's theorem

In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic[1] (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.[2]

Modal logic

model logic 2019-04-29 h8zj7
model logic 2019-04-29

these shapes form hex stars. #geometry https://twitter.com/CGTNOfficial/status/1115445476537450496

[Too good to be Truchet By Colin Beveridge. At http://chalkdustmagazine.com/features/too-good-to-be-truchet/ ]

[Truchet By Cameron Browne. At http://cambolbro.com/games/truchet/ ]

[Truchet, Braille and Euler By Peter Rowlett. At https://aperiodical.com/2010/02/truchet-braille-and-euler/ ]

Zernike polynomials


Module (Algebra)

Linear Map, Bilinear Map, Multilinear Map

Geometric algebra

todo read

[Let's remove Quaternions from every 3D Engine (An Interactive Introduction to Rotors from Geometric Algebra) By Marc Ten Bosch. At http://marctenbosch.com/quaternions/ ]

Geometric algebra

geometric algebra 2019-02 7sspb
geometric algebra 2019-02 7sspb

i no unstand.


[Linear and Geometric Algebra By Alan Macdonald. At Buy at amazon ]


[Geometric Algebra for Computer Science By Leo Dorst , Daniel Fontijne , Stephen Mann. At Buy at amazon ]

some old articles.

  1. Mandelbrot Set Explained (no complex number needed)
  2. How Computing Science created a new mathematical style By Edsger W. Dykstra (EWD 1073)
  3. The TeX Pestilence: Why TeX/LaTeX Sucks
  4. English/Chinese Math Terminology 中/英 数学术语
  5. What is the Difference of Russell's Logicism, Hilbert's Formalism, Axiomatic System?
  6. Math Notation, Computer Language Syntax, and the “Form” in Formalism
  7. Math Notation, Proof System, Computer Algebra, in One Language
  8. The Codification of Mathematics
  9. Math Terminology and Naming of Things
  10. Mathematical Notation: Past and Future
  11. Pattern Matching vs Grammar Specification
  12. A Notation for Plane Geometry
  13. State of Theorem Proving Systems 2008
  14. The Problems of Traditional Math Notation
  15. Notes On Plane Curves and Proofs
  16. Math Insight: Multiplication and Multiplicative Identity
special relativity 2019-02-14 cpy4j
special relativity 2019-02-14

Hyperboloid of Two Sheet http://VirtualMathMuseum.org/Surface/hyperboloid2/hyperboloid2.html

Dirac Belt Trick http://VirtualMathMuseum.org/Surface/dirac-belt/DiracBelt.html

annulus, math

annulus math 2019-02-01 xh8vd
annulus math 2019-02-01

differential geometry site

been helping mat professors build differential geometry site.

latest are Soliton Surface and others, see

visit the whole gallery at http://virtualmathmuseum.org/index.html

we've been working on it in past year.

See also: Wikipedia

graduate level differential geometry

math Three-Soliton Surface http://VirtualMathMuseum.org/Surface/three-soliton/three-soliton.html

Breather Surface http://VirtualMathMuseum.org/Surface/breather/breather.html

Calculus, Gradient

gradient 2019-01-06 7fa94
gradient 2019-01-06


Derivative and Jacobian Matrix

Jacobian Matrix
Jacobian Matrix

Jacobian matrix and determinant

derivative 8f19a

#math derivative. if you are rusty with calculus, or programer who never learned it, it's good time to revisit. join my journey. read my daily snippet, and lookup Wikipedia.

Wikipedia is usually chaotic, rambling on and on, touching highschool stuff to research stuff. if calculus is new to you, read textbooks. Here's Free Math Textbooks i verified quality. Free Math Textbooks

Proof theory

#math programers, if u haven't seen Conway's Game of Life yet, look into. it's eye opening. In 1990s, i spent years “playing” it. 1st deep theory you learn: deterministic system can be unpredictable. https://twitter.com/icm7216/status/1080209582041907200