Welcome to the Carnival of Mathematics! Finding that the 60th is apparently the “diamond anniversary,” I was reminded of the symmetry in the Buckyball , which has the shape of a truncated icosahedron. You can make pretty nice ones using modular origami:
Before getting to this month’s links, allow me a diversion to talk about some geometry I learned a little of this month.
There are 6!=720 ways to order the letters A, B, E, I, L, and S. If we declare that two orderings are the same if one is obtained from the other by cyclic permutation (for example, ABEILS and ILSABE are the same), there are 6!/6=5!=120 combinations. If we also declare that a word and it’s reverse are the same (ABEILS = SLIEBA), we have arrived at 6!/(6*2)=60 combinations.
Pick any 6 distinct points on a circle (or any conic section). Choose any of the points as a starting point, and draw a line to any of the other points. Then draw a line to one of the remaining 4 points. Continue until all of the points have been hit, and then draw a line back to your starting point. How many different pictures can you make in this process? 60, again, because you could label the points A, B, E, I, L, S, and then pictures correspond to words from the previous calculation.
Each picture you draw is a figure with six edges. These six edges can be put into three set of pairs, where two edges are paired if they are “opposite.” In the process of drawing the lines, above, the line opposite the very first line is the fourth line you draw. Similarly, the second and fifth form a pair, and then the third and sixth.
Now, if you extend all of the lines, each pair of opposite edges will determine a point of intersection (or infinity… maybe try another setup for your original points ). So each picture you draw determines 3 points in the plane (or infinity). When he was only 16, Pascal showed that these three points are always colinear.
So, given 6 points on a conic, the process outlined above determines 60 lines, called Pascal Lines. Mathworld has more on Pascal Lines, for the inquisitive, so it’s probably about time to direct you over there and get on to this month’s blog posts!
In honor of 60 being both a colossally abundant number and a superior highly composite number, I thought it fitting to include as many links as divisors of 60. I ended up with slightly more links than that, so here are (more on ) groups of links from the previous month:
1) At the beginning of the month, Charles Siegel, at Rigorous Trivialities decided to parallel the National Novel Writing Month (NaNoWriMo) by introducing Math Blog Writing Month, MaBloWriMo. After putting it to a vote, he wrote a series on intersection theory. Also taking up MaBloWriMo were Akhil Mathew at Delta Epsilons, Qiaochu Yuan at Annoying Precision, Harrison Brown at Portrait of the Mathematician and, well, yours truly. I found it to be a great experience, and hope next year brings many more authors. If you like your daily math in bite-size fashion, and not just in MaBloWriMo, you might check out probfact on twitter for daily probability facts.
3) Riemann’s zeta function, the lead character in his hypothesis, is connected to primes by Euler’s product formula. If you are interested in the distribution of the primes, Matt Springer at Built on Facts has a post about the function Li(x), as part of his running Sunday Function series. If natural number primes aren’t exciting enough for you, Rich Beveridge at Where The Arts Meet The Sciences has a post for you on Gaussian Primes.
4) It would hardly be a month of math posts without some puzzles:
- Pat B at Pat’sBlog: An Interesting Counting Problem
- Yan at Concrete Nonsense: M-2: Forcing Properties onto integer pairs
- JD2718: Puzzle: Who am I?
- Sam Shah at Continuous Everywhere but Differentiable Nowhere: Circles, circles everywhere
If you prefer unsolved puzzles, Bill the Lizard has recently written posts about the Collatz Conjecture and the Perfect Cuboid Problem. Alternatively, for some behind-the-scenes on the notoriously difficult Putnam exam (and yet more puzzles), head over to Izabella’s post at The Accidental Mathematician.
5) It’ll take a while to get to the 3435th Carnival of Math, so I think I’m not stepping on too many toes if I point you at Mike Croucher’s quick post at Walking Randomly and Dan MacKinnon’s slightly longer post at mathrecreation that talk about what makes 3435 interesting.
6) Brian, at bit-player, finds some interesting math in a collection of staples, as described in The birth of the giant component.
12) This month also saw some nice calculus posts. Daniel Colquitt at General Musings describes the fascinating trumpet of Torricelli. Kalid at BetterExplained asked Why Do We Need Limits and Infinitesimals? and had A Friendly Chat About Whether 0.999… = 1.
15) Kareem at The Twofold Gaze points out that asking for a Best Proximate Value has two reasonable answers.
20) Plus Magazine had an article entitled Pandora’s 3D Box, talking about a recently discovered fractal inspired by the Mandelbrot set.
30) Dave Richeson at Division By Zero reports on a case of mistaken identity in Legendre Who?
60) Finally, Samuel at ACME Science discusses the fractured state of the current mathematics community, noting that Mathematics Really is Discrete. This post was closely followed by Abstruse Goose’s Landscape.
That’s it for now. Look for the next Carnival, Math Teachers at Play, in two weeks!