I’ve only watched the trailer so far, but this looks extremely cool! Some beautiful, fascinating videos about math, with lots of extra accompanying material and explanations on the website.
I’ve written before about the mathematics of the traditional song “The Twelve Days of Christmas”. For a different angle, check out PNC Bank’s “Christmas Price Index”: apparently, every year they compute the total cost of all the gifts in the song as a whimsical measurement of the economy. This year they have a really cute video along with games and activities that can be used by educators—give it a look! This year, a True Love would have to shell out $87,402.81 for all 364 gifts (up only 1.8% from last year)!
(Note, when I visit the site I get a warning about a bad security certificate—but after poking around a bit nothing fishy appears to be going on, it should be OK to tell your browser to make an exception if you get a similar warning.)
If you want to play with one, here’s a simulator that someone made. If would be even more fun to play with one in real life, of course, but I hear they are rather expensive now! I saw one in real life in a display case in the Princeton computer science department—but of course I didn’t get to play with it.
I also encourage you to go read Rod’s post, which includes a number of other links to information about these fascinating machines!
Hooray for summer! Now that finals are finally over, you can expect a lot more from me over the next few months. I’ve got a lot of exciting things planned, including
a description and explanation of an algorithm for finding square roots by hand (which most people don’t know anymore);
going through a neat little paper by Ivan Niven giving a proof that pi is irrational;
a little number theory, focusing on the Fundamental Theorem of Arithmetic;
fun links to interesting mathematics around the web;
and whatever else tickles my mathematical fancy!
To get things started, here’s a link for you: Norman Wildberger’s MathFoundations, a series of videos explaining the foundations of mathematics. I don’t agree with all his opinions—particularly regarding infinite sets—but this is a subject on which I’m willing to agree to disagree; as a legitimate, published research mathematician who’s been doing mathematics longer than I’ve been alive, he’s certainly entitled to his opinions! And I certainly do agree with many of his opinions, especially regarding mathematics education. In any event, the videos are very well-done, and there’s lots of interesting stuff there!
For something on the lighter side, here’s a fun video that I’ve seen linked to from a number of places. It’s a video of Arthur Benjamin, a math professor at Harvey Mudd College, performing a fifteen-minute “mathemagics” show. He multiplies numbers in his head, figures out the day of the week on which people were born, and some other things, all while keeping up an entertaining banter. It’s quite fun, watch it if you’ve got a spare fifteen minutes!
Welcome to the 23rd Carnival of Mathematics: Haiku Edition! First, I must apologize for the delay: I usually have very little trouble with my hosting provider, but of course it went down just when the CoM was supposed to be posted. But it’s free, so I can’t complain! It’s back up now, and will hopefully stay that way.
For this edition of the CoM, I decided to write a short seventeen-syllable haiku about each of the excellent seventeen submissions I received (along with additional commentary of the more prosaic variety). I’ve arranged the posts more or less in order of required mathematical background, but don’t stop halfway through because then you’ll miss the pretty pictures at the end. Enjoy!
English pols want to
make math more interesting.
It’s not already?
Need socks in the dark?
The pigeonhole principle
comes to your rescue!
Mary Pat Campbell (aka meep) presents a cute video explaining the pigeonhole principle. Did you know that at least two people in the US have the exact same number of hairs on their body? You can’t argue with math!
A counting problem:
how many bracelets are there?
Harder than it looks…
A tricky puzzle:
rectangles and angle sums.
I solved it, can you?
JD2718 shares a gem of a puzzle involving the sum of some angles. It’s tricky—are you up to the challenge? I would especially encourage would-be solvers to come up with a nice geometric solution (I couldn’t)!
Pascal’s Triangle:
writing it out is a chore.
How fast does it grow?
Foxy, of FoxMaths! fame, presents an interesting two-partanalysis of the asymptotic growth of the rows of Pascal’s triangle—not the growth of the actual values in the rows, but of the space needed to write them!—making use of some clever algebraic gymnastics and asymptotic analysis.
In how many ways
can the Nauru graph be drawn?
The answer: a lot!
David Eppstein of 0xDE presents The many faces of the Nauru graph: a collection of diverse ways to visualize a particular graph which he dubs the “Nauru graph”, due to the similarity of one of its drawings to the flag of Nauru. Planar tesselation, hyperbolic tesselation, embedding on the surface of a torus… all that and much more, with, yes, pretty pictures for everything! Even those who don’t understand the article itself should still go take a look, solely for the sake of the pictures. =)
Thanks to everyone for the great submissions, I had a fun time reading them and putting this together. The next CoM will be hosted at Ars Mathematica. As always, email Alon Levy (including “Carnival of Mathematics” in the subject line) if you’d like to host an edition.
A fun video about , including a nice visual explanation of the formula for the area of a circle, and some interesting places that shows up where you might not expect it to. The video was made by Tom M. Apostol and Jim Blinn as part of Project MATHEMATICS!.
Also, don’t forget to post your discoveries about perfect number factorizations as comments to the previous post!
For your viewing pleasure, a fantastically beautiful video about Möbius transformations, which are functions of the form
where z, a, b, c, and d are complex numbers, and . For example, is a Möbius transformation with b=2, c=1, and a=d=0. is also a Möbius transformation. However, isn’t, because . Since any Möbius transformation sends a complex number z to another complex number , it can be thought of as a transformation on the complex plane. The question is, what sorts of transformations are possible? That’s what the video is about.
The video was made by two mathematicians at the University of Minnesota, Douglas Arnold and Jonathan Rogness. There’s more information about the video here.
Check out the following totally sweet video of zooming into a Menger sponge!
This video was made by David Makin, who has lots of other cool images and videos at his website. You can probably figure out what a Menger sponge is just from watching the video, but it’s a fractal object which is very easy to make. Here’s what you do:
Start with a solid cube.
Slice the cube into 27 equal little cubes, by making two parallel slices in each dimension (just like a Rubik’s cube).
Remove the cube in the very center, and the six cubes in the center of each face of the big cube. You’ll be left with a cube-shaped object with square holes going straight through the middle on each side.
Repeat this procedure on each of the 20 remaining little cubes, and so on recursively forever.
The Wikipedia page has some nice pictures that should make this pretty clear if it’s not already. In case the video didn’t blow your mind enough, you should note that Menger sponges have zero volume but infinite surface area! (“How is that possible?!” I hear you cry in dismay. Well, infinity plays very weird games with your intuition!)
, including a nice visual explanation of the formula for the area of a circle, and some interesting places that 
. For example, 
is a Möbius transformation with b=2, c=1, and a=d=0. 
is also a Möbius transformation. However, 
isn’t, because 
. Since any Möbius transformation sends a complex number z to another complex number 
, it can be thought of as a transformation on the complex plane. The question is, what sorts of transformations are possible? That’s what the video is about.
