This is an image of a Montessori Binomial Cube, one of their slew of “sensorial materials.” It illustrates how (a+b)^3 = a^3 + 3•ab^2 + 3•ba^2 + b^3. A friend of mine introduced me to this as he worked in a Montessori school for a short time.
These kinds of toys are really fascinating to me because I’ve been thinking a lot about how to represent mathematical ideas in ways other than the weird, abstruse language we use nowadays (abstract symbols whose shapes have almost no relation to their use). Instead, how can we convey these ideas in other visual ways (or, in this case, tactile ways too!).
Would love to see these kinds of things expanded upon, outside of just blocks.
9:44 • 2013511 • 3 notes
To roughly quote my applied math professor today: “Mathematics is a game of twisting problems we don’t know how to solve until they look like easy ones we can solve. Then we prove or pretend they’re the same thing.”
13:41 • 2013227 • 4 notes
Also, Eric Harshbarger makes his own dice! I’m curious as to what the process is. (also, he does professional lego sculpting, though I don’t believe he’s lego certified)
I like the math constants one…
11:25 • 201316 • 11 notes
Also, I’ve been a bit torn between studying applied or pure mathematics; I continue to hope my degree will be in applied but I enjoy studying pure quite a bit too. This is because I believe theoretical math to be the more noble, beautiful science…. but this quote, again from A Mathematician’s Apology, while ragging on applied math, is the reason it excites me so much.
It is plain that [a physicist or applied mathematician] is trying to correlate the incoherent body of crude fact [i.e. the real world] confronting him with some definite and orderly scheme of abstract relations, the kind of scheme he can borrow only from mathematics.
This kind of approximation and shooting in the dark, is exciting to me. Almost everything in applied math is new and warrants exploration; everything is dark.
22:28 • 20121231 • 1 note
“We do not choose our friends because they embody all the pleasant qualities of humanity, but because they are the people that they are. And so in mathematics; a property common to too many objects can hardly be very exciting, and mathematical ideas also become dim unless they have plenty of individuality.”
— A Mathematician’s Apology — download here if you’d like.
22:25 • 20121231 • 2 notes
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”
— A Mathematician’s Apology — download here if you’d like.
22:24 • 20121231 • 1 note
“A man who is always asking ‘Is what I do worth while?’ and ‘Am I the right person to do it?’ will always be ineffective himself and a discouragement to others. He must shut his eyes a little and think a little more of his subject and himself than they deserve. This is not too difficult: it is harder not to make his subject and himself ridiculous by shutting his eyes too tightly.”
A Mathematician’s Apology is a beautiful essay (you can download a PDF from me here if you’d like). Tons of quotables, and I’ll share my favorites here.
It’s (generalizing here) about the beauty and purpose of pure mathematics, but my favorite bits are when the author extends his idea on math into his general philosophy on life (and vice-versa). Mathematicians have a specific way of viewing the world that I really love.
22:22 • 20121231 • 2 notes
Clopen Sets (Wikipedia)
I was reading this page and was linked to this wikipedia article.
I’ve been doing a lot of thinking about mathematics education and mathematics visualization and interpretation. I had a lot of thoughts on the subject going on subconsciously, but a conversation with Ronen in July has me actively investigating these ideas. He made me realize that people who are knowledgeable in certain subjects start speaking about them with their hands, in abstract, visual terms (rather than just facts and dry descriptions). Ronen recently linked to an article largely related to this idea here
This is a relatively trivial example, but I definitely felt this when reading about Clopen sets. In my mind clopen sets have fuzzy edges, but in a way that differentiates them from open sets (that also have “fuzzy edges”… but in my mind’s eye they’re distinct differences). I can’t quite put them into words right now but if I was in front of you explaining this I would be fidgeting with my hands.
20:54 • 2012926
Here’s a little (somewhat half-assed) video about Pareto Ranking—a technique used to clump data into “ranks” of incomparable value, relative to each other.
I realize I made a mistake in my hasty ranking of the second small piece of paper. See this image for the “fixed” one. Also, see here for another drawn example of a BL>TR versus TR>BL ranking.
Some other thoughts, post-video:
- I thought maybe going one direction versus the other would produce either convex or concave curves, but this is definitely not true after a few test runs (it just depends on the data). The curves in any one rank can have inflection points.
- When two data share one coordinate value, but not the other, they must be in different ranks (my personal rule, kinda)
- I always get the same number of ranks, regardless of which direction I go. (again, I’m only saying this because I’ve yet to be proven wrong)
- There doesn’t seem to be a whole lot of information on the subject, nor any ideas of whether or not there’s an algorithm that performs the ranking without having to “choose a direction,” if you will.
A series of videos to be continued, perhaps?
21:29 • 2012819 • 5 notes
I never understood logarithmic normalization until seeing this little visualization… further proof that pictures > words for just about everything.
(Taken from this technical note about the development of the Human Development Index)
15:01 • 201286
LaTeX is basically industry standard for mathematics notation. I’m just starting to get familiar with it and found an excellent and easy-to-read guide hosted by a professor at Trinity College Dublin here. I think knowledge of it could help me hit the ground running if I end up doing research this summer, but I’m still deciding on that.
Above is Simpson’s rule, a 2nd-degree accurate integral approximation method.
12:38 • 2012126