# Xah Math Blog

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

xtodo to watch

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)

- lots updates and reorg Xah Math

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.

- xtodo 2021-08-09 read [ Inversive geometry ] [ 2021-08-09 https://en.wikipedia.org/wiki/Inversive_geometry ]
- xtodo work on Math Graphics Gallery

- Coordinate Transformation
- Plane Curves Books
- Math Font ℤ
- Unicode Math Symbols ∑ ∫ π² ∞
- Unicode Symbol for “e.g.” (exempli gratia)
- HTML Entity List
- Schmidt Arrangement, Algebra Integer, Gaussian Integer, Eisenstein Integer, Modular Group

- Complex Inversion
- Mobius Transformations
- Geometric Transformation on the Plane
- Helicoid-Catenoid
- Mathematical Simulation Ideas

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

### 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 Leon Post ] [ 2021-02-21 https://en.wikipedia.org/wiki/Emil_Leon_Post ]

then watch

[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)

- [ Post's lattice ] [ 2021-03-09 https://en.wikipedia.org/wiki/Post's_lattice ]
- [ Clone (algebra) ] [ 2021-03-09 https://en.wikipedia.org/wiki/Clone_(algebra) ]

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.

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

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

[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

repost

[2020-08-09 https://twitter.com/EyePoppingFact/status/1292467149705904139 ]

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.

random article

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.

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)`

Visualizing Quaternions

### 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 Alexander Grothendieck = Obi-Wan

### Notes on the Riemann Hypothesis by Ricardo Pérez-Marco

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

### Octonions

[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.

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

### impredicative

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 ] [ 2019-08-09 https://en.wikipedia.org/wiki/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: [ twitter KangarooPhysics ] [ 2019-08-05 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

### Mathematician Adrien-Marie Legendre

truly despise the jargon injection surjection, created by Bourbaki

- [ Ettore Majorana ] [ 2019-08-02 https://en.wikipedia.org/wiki/Ettore_Majorana ]
- [ Edward Witten ] [ 2019-08-02 https://en.wikipedia.org/wiki/Edward_Witten ]
- [ Terence Tao ] [ 2019-08-02 https://en.wikipedia.org/wiki/Terence_Tao ]
- [ Barry Simon ] [ 2019-08-02 https://en.wikipedia.org/wiki/Barry_Simon ]

### shape of space

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

### applied group theory. 🤩

### math. Field GF(4)

one of the smallest non-trivial field

### 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 ] [ 2019-07-23 https://en.wikipedia.org/wiki/Loop_braid_group ]

See also:

[ Braid theory ] [ 2019-07-23 https://en.wikipedia.org/wiki/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.

[ Euler-Mascheroni constant ] [ 2019-06-21 https://en.wikipedia.org/wiki/Euler-Mascheroni_constant ]

[ Continued fraction ] [ 2019-06-21 https://en.wikipedia.org/wiki/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.

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 ] [ 2019-04-29 https://twitter.com/johncarlosbaez/status/1122976661132021760 ]

[ Peano axioms ] [ 2019-04-29 https://en.wikipedia.org/wiki/Peano_axioms ]

[ Tennenbaum's theorem ] [ 2019-04-29 https://en.wikipedia.org/wiki/Tennenbaum%27s_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 ] [ 2019-04-29 https://en.wikipedia.org/wiki/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 ] [ 2019-04-29 https://en.wikipedia.org/wiki/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) ] [ 2019-04-29 https://en.wikipedia.org/wiki/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 ] [ 2019-04-29 https://en.wikipedia.org/wiki/Lindstr%C3%B6m%27s_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 ] [ 2019-04-29 https://en.wikipedia.org/wiki/Modal_logic ]

these shapes form hex stars. #geometry [ twitter CGTNOfficial ] [ 2019-04-08 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 ] [ 2019-04-09 https://en.wikipedia.org/wiki/Zernike_polynomials ]

todo

#### Math and Illustrations

- Algorithmic Mathematical Art
- Art of M C Escher
- Escher Chameleon Polyhedron
- What is Technical Drawing, Descriptive Geometry, Projective Geometry, Linear Algebra
- Outline Rendering of 3D Models
- A Fantastic Auto-stereogram
- Visual Illusions
- Projective Illusion
- What Is Perspective Drawing
- Illustrating Geometry with POV-Ray

### 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 ] [ 2019-02-22 https://en.wikipedia.org/wiki/Geometric_algebra ]

i no unstand.

help:

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

https://enkimute.github.io/ganja.js/examples/coffeeshop.html

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

some old articles.

- Mandelbrot Set Explained (no complex number needed)
- How Computing Science created a new mathematical style
- The TeX Pestilence: Why TeX/LaTeX Sucks
- English/Chinese Math Terminology 中/英 数学术语
- What is the Difference of Russell's Logicism, Hilbert's Formalism, Axiomatic System?
- Math Notation, Computer Language Syntax, and the “Form” in Formalism
- Math Notation, Proof System, Computer Algebra, in One Language
- The Codification of Mathematics
- Math Terminology and Naming of Things
- Mathematical Notation: Past and Future
- Pattern Matching vs Grammar Specification
- A Notation for Plane Geometry
- State of Theorem Proving Systems 2008
- The Problems of Traditional Math Notation
- Notes On Plane Curves and Proofs
- Math Insight: Multiplication and Multiplicative Identity

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

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

### Mathematician Gaston Julia

this guy, is the first to study julia set. #math #geometry

### Mathematician Pierre Wantzel

[ Pierre Wantzel ] [ 2019-02-02 https://en.wikipedia.org/wiki/Pierre_Wantzel ]

Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.[1]

In a paper from 1837,[2] Wantzel proved that the problems of

- doubling the cube, and
- trisecting the angle
are impossible to solve if one uses only compass and straightedge. In the same paper he also solved the problem of determining which regular polygons are constructible:

a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e. that the sufficient conditions given by Carl Friedrich Gauss are also necessary) The solution to these problems had been sought for thousands of years, particularly by the ancient Greeks. However, Wantzel's work was neglected by his contemporaries and essentially forgotten. Indeed, it was only 50 years after its publication that Wantzel's article was mentioned either in a journal article[3] or in a textbook.[4] Before that, it seems to have been mentioned only once, by Julius Petersen, in his doctoral thesis of 1871. It was probably due to an article published about Wantzel by Florian Cajori more than 80 years after the publication of Wantzel's article[1] that his name started to be well-known among mathematicians.[5]

Wantzel was also the first person who proved, in 1843,[6] that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals. This theorem would be rediscovered decades later by (and sometimes attributed to) Vincenzo Mollame and Otto Hölder.

“Ordinarily he worked evenings, not lying down until late; then he read, and took only a few hours of troubled sleep, making alternately wrong use of coffee and opium, and taking his meals at irregular hours until he was married. He put unlimited trust in his constitution, very strong by nature, which he taunted at pleasure by all sorts of abuse. He brought sadness to those who mourn his premature death.” — Adhémar Jean Claude Barré de Saint-Venant on the occasion of Wantzel's death.[1]

### Mathematician Pierre Fatou and Julia Set

this guy, is the first to study Julia set.

the math work of Pierre Fatou. Analysis, analytical functions, dynamical systems, chaos theory.

### Mathematician Adrien Douadly, and Mandelbrot Set

this guy started research on Mandelbrot set. one major result is that the Mandelbrot set is connected. #math #geometry

the Mandelbrot set, is truly one of the most incredible thing in math. Incredible is the word, and amazing, cosmetic, and in a visual way too. It shows the deep mystery of math.

See also: Mandelbrot Set Explained (no complex number needed)

here's the deepest zoom. Zoom to: 3.4 * 10^1091, and the video plays for 70 minutes.

here's the deepest zoom of Mandelbrot set, Zoomed to 3.4 * 10^1091. If you zoom this much on a atom, it'd be 10 followed by 109 zeros times larger than the screen. it's 70 minutes video.

and i always thought, if you zoom to a particular point deep enough, the face of god would suddenly appear, and the universe would blow up. It's, like, playing lottery, you wouldn't know there is no jackpot untill you played all the possible numbers. #math

### annulus, math

### differential geometry site

been helping mat professors build differential geometry site.

latest are Soliton Surface and others, see

- http://VirtualMathMuseum.org/Surface/hyperbolic_k1_sor/hyperbolic_k1_sor.html
- http://VirtualMathMuseum.org/Surface/two-soliton/two-soliton.html
- http://VirtualMathMuseum.org/Surface/three-soliton/three-soliton.html
- http://VirtualMathMuseum.org/Surface/four-soliton/four-soliton.html

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

we've been working on it in past year.

See also: Wikipedia

- [ Sine-Gordon equation ] [ 2019-01-23 https://en.wikipedia.org/wiki/Sine-Gordon_equation ]
- [ Gauss–Codazzi equations ] [ 2019-01-23 https://en.wikipedia.org/wiki/Gauss%E2%80%93Codazzi_equations ]
- [ Soliton ] [ 2019-01-23 https://en.wikipedia.org/wiki/Soliton ]
- [ Bäcklund transform ] [ 2019-01-23 https://en.wikipedia.org/wiki/B%C3%A4cklund_transform ]

### graduate level differential geometry

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

### Calculus, Gradient

[ Gradient ] [ 2019-01-06 https://en.wikipedia.org/wiki/Gradient ]

### Derivative and Jacobian Matrix

[ Jacobian matrix and determinant ] [ 2019-01-04 https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant ]

#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 ] [ 2019-01-02 https://en.wikipedia.org/wiki/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. [2019-01-01 https://twitter.com/icm7216/status/1080209582041907200 ]