One of the talks I usually give is on geometric dissections, which I begin with the following puzzle from (Lindgren 1964):
I thought it might have been more fun to give the puzzle to the prospective audience one week before I actually presented the solution, so I did that for an upcoming presentation on geometric dissections for students from Raffles Girls' School. I hope they got very frustrated by the puzzle. I also promised to give a hint to the puzzle, but I forgot to post it! Highlight the hint if you want to read it...
The cut lies on the grid lines.
Lindgren, Harry (1964). Geometric Dissections. D. Van Nostrand.
There was this thing going on Facebook:
The idea is to fill Facebook with math, surprising friends with its little-known beauty, which augments art, poetry, etc. Whoever likes this post will receive a mathematician's name, and will have to write a short introduction about them, and explain in layman terms a mathematical work (theorem, idea, book, field of math, anything important or interesting) by them. Include this text, but you may modify it for the sake of public math awareness.
Stefan Banach was one of the 20th century's greatest mathematicians. His self-taught prowess impressed the renowned Prof. Hugo Steinhaus, who supported Banach's career. His doctoral thesis sparked and grew into the field of "Functional Analysis", which has proved fundamental to Quantum Mechanics and Differential Equations (used widely in engineering). During World War II he and fellow academics were sheltered from Nazi persecution by employment as "lice feeders" at a Typhus Research Institute. (from Wikipedia)
He proved that:
This is one of many "fixed-point theorems", all of which say that if you take a "world", transform it so-and-so, then some point in the world must have remained immobile. The more famous Brouwer fixed-point theorem has been applied in Economics, to show that markets (the "world") have "fixed points" (e.g. balances between supply and demand).
I must thank a friend for telling me a basic form of this layman explanation many years ago; it has stuck with me all these years.
Next week I'm conducting an origami math workshop for Grade 8 students, and one of the activities will be to fold a skeletal octahedron (really simple modular origami). The students will be asked to choose between five octahedra I already folded in various colors, to see what kind of color schemes and arrangements appeal to them. We can then discuss about symmetry, graph coloring and more.
It would also be interesting to see which color schemes people like, even before the workshop, so here goes!
Color Schemes 1—5 (from left)
If you're studying undergrad math, physics or chemistry, chances are you've heard of this thing called a "Group" that is studied in "Group Theory". What is it and why is it so important? I'll explain in a simple way.
The most familiar group is our number system; we have a bunch of numbers, and we have this thing called "$+$" which can take two numbers and churn out another number. A group is just a set of objects, and you have one "operation" (usually called "$\ast$" or "$+$", but those are just names) that tells you how to combine objects to produce other objects in that set, subject to a few rules. Why devise such a "strange chasing game" in a set of objects? That's because such systems are omnipresent in a stupendous array of phenomena:
(The first person to uncover the hidden meaning wins my gratitude. My hopes aren't high though...)
The ABCs of Creation
Look behind Creation!
Exemplified in just a few given equations.
As part of my effort to expose myself to more art, today I attended Art Stage Singapore, which showcased more than 130 galleries' worth of art, mostly from the Asia-Pacific region. The six hours that I had weren't nearly enough for me to experience all of the art pieces there. Here's some of the pieces that I liked:
Unfortunately I could only attribute some of the pieces, but all rights belong to the respective artists.
I also liked Malaysian artist Haslin Ismail's "Book Land", which cuts and assembles parts of books and other materials to form castles of fantasy and dioramas that bring the books' contents to life.
These days I hardly fold any origami, but when I do, it's usually from either of the books Origami Art or Advanced Origami, both by Michael G. Lafosse & Richard L. Alexander from their Origamido Studio. Their designs have this elegance and beauty that attract me, and leaves me satisfied when I am able to reproduce part of that beauty with my own hands.
So when I offered to fold a present for a friend, I asked him to choose from those books. He picked their Humpback Whale.
Elephanthide paper, wet-folded
I designed this for a friend I met at an exchange programme to Qingdao because of his interest in math.
This model is mainly a tube with sealed ends. The following series of folding instructions will make this apparent:
I improvised this design in class using an envelope that was lying around.
I liked being able to use the flap of the envelope as part of the nun's gown - so I was using the envelope "fully" and not simply as a rectangular sheet of paper!
Thinking back, I was probably inspired by Vietnamese origami master Giang Dinh's evocative abstract human figures:
You must check out his achingly beautiful origami faces, animals and figures. I saw his gorgeous work in a book of photographs from an origami exhibition, titled Masters of Origami: At Hanger-7.
One of my earliest designs, born from a brainwave on creating hexagonal cells from a square grid:
One uncut square, designed in 2007.
Besides sharing my own musings and insights on various topics, I recommend some books, events or other material.
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