February 2012
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0^0 still wishes it were equal to 1 (but it still...
Here’s a little supplement to the last post. I mentioned before that there are several reasons somebody might try to define 0^0 := 1. The big reason before (flawed though it turned out to be) involved calculus and limits. But there’s an entirely different way to interpret the expression y^x which puts 0^0 in an intriguing different light.
If X and Y are sets, with |X|= x and |Y| =...
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0^0 wishes it were equal to 1 (but it's not)
Hey folks. It’s been a long time. I’ve had lots of things on the brain, along with lots of new responsibilities, and I just kind of fell out of the posting spirit around here (though I’ve still been reading others’ blogs regularly).
Anyway, maybe a short and easy math post will get me back into the swing of things (no promises on that). Here’s something irksome. ...
October 2011
1 post
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September 2011
4 posts
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A solution
When I first heard the submarine puzzle in the post below, I wasn’t able to solve it until I got some hints. The trick, as with many puzzles and even serious math problems, is to replace the problem at hand with one easy enough you can solve it, but which still captures the essential difficulty of the original problem. This tends to be easier said than done, unless you have a great deal of...
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A puzzle
It’s the Cold War, and you command a missile station tasked with bombing an enemy submarine. Intelligence has provided you with these facts:
1) The submarine travels with some constant integer velocity, due either east or west (i.e., travels along the standard number line, either to the right or left).
2) At any integral time, the submarine’s position on the number line is an...
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“In the case of my music, there is no meaning in it if it does not have concrete...”
– Toru Takemitsu, Yume to Kazu, p.21; translation from Taniyama, ‘The Development of Toru Takemitsu’s Musical Philosophy’, p.90.
(via Peter Burt’s ‘The Music of Toru Takemitsu’, p.251.)
August 2011
5 posts
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Geometrically Finding the Area of a Trapezoid
I hope the picture says it all. The last step uses the fact that the area of a parallelogram of base b and height h is simply bh, which can easily be proved geometrically by slicing off a triangle and sliding it to the other side to form a rectangle with the same base and height.
Sorry for the lack of posts recently, but they should perk up as the school year begins and I have lots of...
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My good friend and partner-in-crime Daniel Walsh made this wickedly cool 3-manifold, a deformed hexagonal toroidal-ladder of twist-degree 1 (my own fake terminology). Learn how he did it and follow Dan’s wild geometric projects at http://danielwalsh.tumblr.com/.
July 2011
11 posts
mequeme asked: Awgh! Why did you stop writting in LaTeX? Everything is way easier to follow when written in LaTeX :/
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Hopf stabilization of a fibered knot, illustrated by Ken Baker (Sketches of Topology). I have no idea what I’m seeing but it’s beautiful.
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Works of Walter Russell, via but does it float.
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i²=j²=k²=ijk=-1: An exercise, if you're... →
leafdude:
spinor:
leafdude:
Consider a sphere which has 5 points placed randomly on its surface. Demonstrate that at least 4 points lie in the same hemisphere (this is a past Putnam problem).
Select a great circle going through two of the points. As there are at most three points not on this circle, one…
Yep, this is the answer! I kind of thought you would get it ;) This is one of my...
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archery: Calabi-Yau manifold →
fewpeoplematter:
A spinning 3D model of the 6-dimensional Calabi-Yau manifold. This manifold has been shown by string theory to be the topological shape of extra, curled-up dimensions on the Planck-scale at every point in the universe.
I hope to understand the definition of Calabi-Yau manifolds…one day.
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Triangle Theorem Study
My first failed attempt at proving the equilateral triangle theorem in the post below. Notice how I mistakenly thought an equilateral triangle’s centroid is located one-third up the side length of the triangle rather than the height distance, so I’m missing the needed factors of Sqrt[3]/2. Also, I had labelled the three vertices of the starting triangle as z1, z2 & z3 rather...
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A Theorem Regarding Triangles
Start with any triangle at all. You can construct an equilateral triangle on each of its three sides. These three equilateral triangles have centroids, which you can connect to form another triangle. Surprisingly, this last triangle is always equilateral, regardless of the starting triangle!
Here’s a video my friend Dan over at http://danielwalsh.tumblr.com/ made with Mathematica - it...
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Avant Garde Project - 20th Century Classical,... →
Many rare and precious LP recordings of 20th century music ripped to lossless .flac format can be found on this website, including beautiful works by Toru Takemitsu, Iannis Xenakis, Igor Stravinsky, Morton Subotnick, Harry Partch, Alban Berg and many others. Loads of more obscure treasures as well.
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June 2011
30 posts
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2^32
proofmathisbeautiful:
In a recent job interview, a friend of mine was asked to estimate 2^32 (without a calculator)! He got it right!
After he told me this…I tried…and got it wrong! :O
LOL! FAIL!
The trick is to note that 2^10 = 1,024 is very nearly 1,000.
So 2^32 = 2^30*2^2 = (2^10)^3*4 is approximately 1,000^3*4 = 4,000,000,000.
The real answer will be bigger than 4 billion. Let’s...
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Cantor-Schroeder-Bernstein made easy
Earlier this month I made a post about one of Felix Bernstein’s theorems on infinite cardinals, and briefly mentioned something much more famous with his name on it: the Cantor-Schroeder-Bernstein theorem. Though the result of this very important theorem is intuitively obvious, its proof gave me a lot of trouble as an undergrad. Recently I tried going through it again, and was surprised to...
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Shannon May: Illustration
I love the works of Maryland Institute College of Arts graduate Shannon May. Many of them take ideas from science & math and vividly breathe life into them.
Go look at more here.
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A solution
In the last puzzle I asked if anybody could find the pattern governing the sequence {1, 11, 21, 1211, 111221, 312211, …}. Leafdude got the answer first, and mequemeyahorahueloapollo rightly added that this is called the “Look and Say” sequence. You read off the digits present in the nth term to get the next one. You...
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41 Followers & A Remarkable Polynomial
I was surprised to get a modest burst of new followers overnight (thanks, rebloggers!) so here’s another little celebration post.
The polynomial $$P(n):= n^2 + n + 41$$ is special. Here’s a little table of values:
$$n \;|\; P(n)$$
$$0 \;|\; 41$$
$$1 \;|\; 43$$
$$2 \;|\; 47$$
$$3 \;|\; 53$$
$$4 \;|\; 61$$
Notice anything interesting? These values of P(n) are all prime! Does...
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A puzzle
I have another idea for a longer proof-post with nice pictures, but it’s going to take a bit of time to prepare. Meanwhile, here’s a quick and dirty puzzle.
A sequence begins as follows:
1
11
21
1211
111221
What’s the pattern here? Finding it takes cleverness, but absolutely no sophisticated math.
Below this heart-melting video of Bill Evans’ trio in 1970,...
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A solution
Last time I said:
You have a magic box with two buttons on it labeled C and D. The box is filled with an unknown number of particles flying around at random. When you push button C, a single new particle magically appears inside the box; when you push D, a single particle magically vanishes from the box.
I say there’s a subtle breakage of symmetry in how these two buttons operate. Can you...
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A puzzle
You have a magic box with two buttons on it labeled C and D. The box is filled with an unknown number of particles flying around at random. When you push button C, a single new particle magically appears inside the box; when you push D, a single particle magically vanishes from the box.
I say there’s a subtle breakage of symmetry in how these two buttons operate. Can you spot it?
Note:...
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Like, woah, dude.
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Will we ever run out of new songs?
Be advised: this will be a lengthy and somewhat speculative (if not downright fanciful) post.
There’s a question related to my two biggest interests, math and music, that has bugged me for several years now. My curiosity was piqued further by the excellent short story Melancholy Elephants by Spider Robinson, which I heartily recommend to anybody interested in the future, the arts, and the...
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Twenty Followers & The Handshaking Theorem
Well that didn’t take too long. Howdy, everyone! Feeling pretty welcome around these parts.
If we were all to shake hands, how many handshakes would occur in all? Pictured below is the complete graph on 20 vertices; every vertex is joined to every other one exactly once. The number of edges here is how many handshakes would occur among all of us.
Quite a few! There’s a...
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A solution
leafdude:
hearseethink:
A cylindrical hole 6 inches long has been drilled straight through the center of a solid sphere. What is the volume remaining in the sphere?
(Yes, there’s really enough information to solve this!)
$$36\pi \hspace{2mm} \text{in}^3.$$
(Admittedly, I knew about the Napkin ring problem before answering…)
Correct! Although you may as well have said “I knew...
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A puzzle
A cylindrical hole 6 inches long has been drilled straight through the center of a solid sphere. What is the volume remaining in the sphere?
(Yes, there’s really enough information to solve this!)
Source: Hexaflexagons, Probability Paradoxes, and the Tower of Hanoi, by Martin Gardner.
A very fun book!
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Infinite Whitehead Tangle
This never-ending knot, animated by Kenneth Baker, was made by concatenating a sequence of smaller tangles that look like this:
A quarter-turn is applied when passing from one copy of this tangle to the next in the infinite sequence. Here’s another view:
Sketches of Topology is an amazing blog for visuals like these, and I will probably crib from it frequently.
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Klein's Quartic Surface
Klein’s Quartic Surface turning itself inside out. This surface can be constructed from the equation u^3*v + v^3*w + w^3*u = 0.
Animated by Greg Egan.
edit: I’m glad this is getting some reblogs, but I cringe every time I see the error “Quartet Surface” replicated! I wasn’t able to fix that typo before it started spreading. If you care about this kind of thing...
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Everting the punctured torus and unpunctured...
idran:
intothecontinuum:
This animation shows a punctured torus being turned inside-out. So in principle, if you connected the cuffs of your pants together you could still turn them inside-out. Try it!
The other interesting bit is you’ll notice this reverses the directions of the stripes, from being around the hole to being “parallel” to it. (I don’t know enough topology to know if...
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Leonard Bernstein & Bernstein's Theorem on...
Leonard Bernstein ranks among the brightest musical talents of all time. Here’s a two-part video of him conducting Gershwin’s masterpiece “Rhapsody in Blue” …while also tearing through the piano part at the same time!!
Now that I’m inspired from that great music, I’d like to think a little bit about this theorem that came up a couple posts...