〔The reason why Involute gears turn smoothly By Hyprodium. @ http://hyrodium.tumblr.com/post/123270340099/the-reason-why-involute-gears-turn-smoothly-fig〕

see also Involute

〔John Horton Conway: the world's most charismatic mathematician By Siobhan Roberts. @ http://www.theguardian.com/science/2015/jul/23/john-horton-conway-the-most-charismatic-mathematician-in-the-world〕

〔Genius At Play: The Curious Mind of John Horton Conway By Siobhan Roberts. @ amazon〕

Chen's theorem: every sufficiently large even number can be written as the sum of either two primes, or a prime and a semiprime (the product of two primes). Chen's theorem

〔A Guide to Plane Algebraic Curves by Keith Kending. @ amazon〕

The Stanford Encyclopedia of Philosophy is a great thing. It is much better than Wikipedia.

the Rubik cube of today have advanced. Now, there's stickerless versions, and better mechanical design.

see updated Magic Polyhedrons

ℚ = be rational, ℝ = get real.

ℭ = Cardinality of the continuum.

Math Font, Unicode, Gothic Letters, Double Struck, ℤ ℚ ℝ ℂ ℍ ℜ ℑ ⅇ ⅈ

just discovered, this classic, definitive, book on tilings ＆ patterns is in print again, now reprinted by Dover.

see also: Reading Notes on Tilings and Patterns

John Baez, gave a intro on the problem of tiling the plane with 5-fold symmetry tiles @ https://plus.google.com/117663015413546257905/posts/5CHdeLRckRm

see also 〔The trouble with five By Craig Kaplan. @ https://plus.maths.org/content/trouble-five〕

sample of tiling ＆ patterns, see:

See also: Discontinuous Groups of Rotation and Translation in the Plane

it supposed to be fair dice. Am thinking there's a major difference from the cube. This one, has 2 different sides, like a coin. In a sense, when you throw this dice, it decides which of the side will fall first, then, decide which of the 3 faces will land.

So, one can actually make a 3-sided pyramid, and have 2 of them and glue the bottom together. So, we could make a infinite shape of 6 faced dice that's still fair. Then, if we wiggle the cutting plane like a saw tooth, we could create quite a lot strange looking and still fair 6-faced dice.

comment at https://plus.google.com/+XahLee/posts/L9FP178N6eq

Cycloid (animation update)

apparently there's a Unicode char for log. ㏒. The Unicode name is “U+33D2: SQUARE LOG”. It's Japanese origin.

There's also a ㏑ “U+33D1: SQUARE LN”

for many more, see http://www.unicode.org/charts/PDF/U3300.pdf

See also: Unicode: Math Symbols ∑ ∞ ∫ π ∈ ℝ² (use the search box there to find Unicode)

(thanks to https://twitter.com/Ryuutei)

English/Chinese Math Terminology 中/英 数学术语 (minor update)

LimaconOfPascal (updated gif animation)

Trochoid (updated gif animation)

Tractrix (updated gif animation)

Conchoid (minor update)

Notes on A New Kind of Science (photo of the book added)

in early 1990s i spend 2 years playing go. At the time i bought the best go program called Nemesis for the Mac. Was able to beat it with 5 stone handicap for the computer. (the software is rated 13 kyu, which means, i'm i think 9kyu at the time)

some said that playing with computers gets you bad habits…

sometimes in 2009 or so, i suddenly find go to be not interesting at all, after i read Stephen Wolfram's “A New Kind of Science”.

the essence is that it's a kinda a cellular automata, and as such, there's no intrinsic math in it, and there are infinite cellular automata. And go players are simply those with great memory and specialized talent and seen lots of cellular automata of the go kind…

see Go Board Game as Cellular Automata

the other thing that has been interesting for me to explore is go on triangular grid or other tiling. Though, it's disappointing that i haven't seen much literature about it at all.

how to lose all interest in go. Go Board Game as Cellular Automata

Unicode: Math Symbols ∑ ∞ ∫ π ∈ ℝ² (minor update)

Here's what mathematician John Baez says:

You build it by taking the real line in the complex plane and applying all possible transformations

`f[z] ↦ (a*z + b)/(c*z + d)`

where a,b,c,d are Eisenstein integers, meaning numbers of the form

`m + n Sqrt[-3]`

The Eisenstein integers lie on a hexagonal grid!

for more math details, see 〔Schmidt Arrangement By John Baez. @ http://blogs.ams.org/visualinsight/2015/03/01/schmidt-arrangement/〕

Stereographic projection is a special case of sphere inversion.

consider a sphere of diameter d, and plane at bottom. The inversion circle is centered on North pole, with diameter 2*d.

also:

Stereographic Projection (minor update Mathematica notebook. If you have a problem running it, let me know, i'll fix.)

also:

Stereographic Projection (minor update Mathematica notebook. If you have a problem running it, let me know, i'll fix.)

In mathematics, nonlinear programming (NLP) is the process of solving an optimization problem defined by a system of equalities and inequalities, collectively termed constraints, over a set of unknown real variables, along with an objective function to be maximized or minimized, where some of the constraints or the objective function are nonlinear.[1] It is the sub-field of Mathematical optimization that deals with problems that are not linear.

how many pieces can you get by cutting a donut 3 times? (you are not allowed to re-arrange the pieces after each cut)

rephrase: Maximal number of regions of dividing a torus with n planes? (n=3 here.)

answer: http://oeis.org/A003600

“group theory” should be called “symmetry theory”

In mathematics, a Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries that is equal to its own conjugate transpose—that is, the element in the i-th row and j-th column is equal to the complex conjugate of the element in the j-th row and i-th column, for all indices i and j.

Hermitian matrices can be understood as the complex extension of real symmetric matrices.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. example

{ {1, 7, 3}, {7, 4, 5}, {3, 5, 6}, }

Great Math Board Game Software (minor update)

Great Math Board Game Software (minor update)

sad to know. Alexander Grothendieck died 2 days ago. Alexander Grothendieck

geometric design on sphere. http://taffgoch.deviantart.com/

old goodie. State of Theorem Proving Systems 2008

Bezier Curve (minor update)

#math When an attractive girl flips her wet hair, the water stream forms a Fibonacci spiral. Math Mysticism: is Hurricane Shape a Fibonaci Spiral?

“Distress not yourself if you cannot at first understand the deeper mysteries of Spaceland”

FLATLAND: A Romance of Many Dimensions. The best book to understand higher dimensions.

reading math is just immense pure pleasure. Some people like video games, some movies, some drink bars… these are considered pleasure. But really, nothing beats reading just math. The purity, the beauty, the depth, and the austerity.

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

2 decades ago, i know about gamma function, which is a extention of the factorial to all positive real numbers (to say the least). i didn't understand how it is done. i just knew it's something advanced. but today, reading about it, i can understand it! Gamma function must be thanks to, my study of complex analysis sometimes in 2006!

just remembered, that i had a dream few days ago, that the plane curves website at St Andrews University (Scotland) http://www-history.mcs.st-and.ac.uk/Curves/Curves.html converted their Java curve applet to modern JavaScript. Which is something i've been planning to do for my Visual Dictionary of Special Plane Curves site.

my old rival beats me!

added a hyperbolic tiling interactive software to Great software for Tilings, Patterns, Symmetry

Found this on math overflow, a pure hogwash http://mathoverflow.net/questions/43690/whats-a-mathematician-to-do/44213#44213 YOU know? the kinda pleasing and meaningless things to say, that everyone loves to hear. Which id wrote that? Then i noticed, the name Bill Thurston. Bill Thurston? The William Thurston, geometry god of the century who died few years ago? Indeed.

〔Groups, Tilings, and Finite State Automata By William P Thurston. @ http://timo.jolivet.free.fr/docs/ThurstonLectNotes.pdf〕

the story of looking up math. so today, i want to lookup ultra-filter. “…an ultrafilter on a poset P is a maximal filter on P, …” so, i have to lookup filter: “filter is a special subset of a partially ordered set.” but don't remember what's “partially ordered set” so, lookup it is: “A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the set, one of the elements precedes the other.” ok, but i want to lookup “total order” to see if that's something i recall: “a linear order, total order, simple order, or (non-strict) ordering is a binary relation (here denoted by infix ≤) on some set X which is transitive, antisymmetric, and total.”

great, i remember having worked out this. That's total satisfaction. (actually, i discovered the properties of the Equivalence relation. When around 1997 i was writing code to decide if 2 polygons in 3d are equivalent, and discovered inconsistancies in my code, namely a==b, b==c, yet my code a≠c. And, i discovered, to great happiness, that to define equivalence is equivalent to partition of a set.)

well, that's a bit excursion. But i was writing about my typical trip to math these days. When trying to understand one thing, involves some 10 or 20 other Wikipedia articles.

by the way, also, some'd suggest math books instead. But no, i prefer dictionary style learning, esp for math. I prefer, the cold, logical, senseless, definition. The human touch of “motivation”, i want after the fact. (side note: sometimes, the human touch are often mis-leading, and there are many different takes, depending on the author. The human touch is often necessary though. But, i've been thinking, it is possible to do without entirely, because, math (defined as the essence of something), is how things are. And, to some degree (perhaps 100%), you really just need to know that gist, anything else is fluff, and possibly even harmful. But why do we have the need for the human touch? i gather possibly it's pure habit. As in, new thinking usually happens with newer generation (as is, only older generation have problems with imaginary number, whereas newer generation who are taught its definitions directly never have this problem and moves on)) (the gist of this thought is that, what happens, when people learned math ONLY by their logical definitions and never the story behind it.)

(side note: xah's edu corner: linguistics: by the way, in math lingo, often you'll encounter the phrase “Motivation” as a section title. It is a idiom among math texts. The context is that, math becomes so abstract, that just definining something seems out of the blue. So, one needs to provide a context, so that readers can see how the definition came to be. And that, is often called “Motivation”, which is kinda a math idiom. I can't help but finding it funny, when reading math papers (the “formal” type), you encounter a section titled “Motivation” pro forma)

after writing all these, i haven't understood ultra-filter yet. Now, back to procrastination…

comment at https://plus.google.com/+XahLee/posts/hxUPxeYyN59

#math #education