You’ve reached the midway! This edition of the CoM is a doozy, with loads o’ math fun.

(notes to contributers: 1) Thank you. 2) If you contributed more than one item, I might have only used the most recent. It’s kind of big… 3) If you didn’t mean to contribute, but something of yours ended up here, I’m the person to yell at.)

(Notes to everyone: 1) Scroll, 2) Click, 3) Enjoy, 4) Repeat. If you like, go ahead, submit to the next. It’s appearing at Good Math, Bad Math (MarkCC’s blog) 13 days from now, and you can use the submission tool or maybe Mark will post alternate directions)

A Neighborhood of Infinity
Arboreal Isomorphisms from Nuclear Pennies
Game; Data Structures
But who cares about nuclear coin reactions? … Believe it or not, each coin shuffle above corresponds to an isomorphism between certain types and solving the puzzle above actually demonstrates a really neat isomorphism between tuples of data structures.”

MathTrek
Beating the Bush for Patterns
Scientific Modeling
In Africa’s Kalahari Desert as well as some areas around the Mediterranean, trees and bushes grow in clumps scattered in seemingly random locations across an otherwise barren landscape. Two new studies have discovered a fractal pattern in this seeming randomness, and they offer a novel explanation of how it comes about.”

Pissed Off
Calculators
Secondary Mathematics Education
Kids don’t learn arithmetic anymore. Adults are forgetting everything they ever knew about it because calculators will do the computations for them

johnkemeny.com
Car Car = Fuelish Hyperbole
Modeling
…a practical problem in automotive gas economy which involves a pricing anomoly, a Greek mathematician who may have tutored Alexander the Great, and an 18th century Scottish math professor who almost loses his job by taking an unauthorized 2-year sabbatical…”

Pencils Down
Choking Down Technology
Secondary Mathematics Education; Texas Instruments
…recent happenings… my attendance at a T3 conference for pre-service math teachers. I’m not sure “conference” is the right word; in truth, it was a two day sales pitch

Halshop
Class of 64
College Math Instruction
A class with 64 students? That sounds like an institutional problem. How can you possibly be pedagogically effective with a class that large?

The Jose Vilson
The Common Factor
Elementary Mathematics
I’ve grown more excited about the possibilities I have to nurture and inspire the kids I have….I was teaching the Fundamental Theorem of Arithmetic with the 6th graders…

Michi’s Blog
Coq and simple group theory
(see title)
Trying to make the time until my flight leaves tomorrow go by, I played around a bit with the proof assistant Coq. And after wrestling a LOT with the assistant, I ended up being able to prove some pretty basic group theory results.”

TerryTao
The Crossing Number Inequality
Graph Theory
Today I’d like to discuss a beautiful inequality in graph theory, namely the crossing number inequality

Secret Blogging Seminar (Ben Webster)
Does anybody actually read cover letters?
Applying for postdocs
But there are things that should be on the page that aren’t. Particularly missing is info about preparing application materials. Any guesses as to why that is?

Ars Mathematica
Eliminate Cut Elimination
Theorems
Is there any major theorem in mathematics drier than Gentzen’s cut elimination theorem?

Britannica Blog
Failing Our Geniuses
Mathematics Education
Aside from continuing to portray the gifted as oddities, the author appeares to think that such students don’t need special attention, using the peculiar argument that if Einstein didn’t get it, no genius should

Secret Blogging Seminar (Joel Kamnitzer)
Frobenius Splitting
Algebraic Geometry
I’d heard of this Frobenius splitting years ago from my advisor, but had no idea how it worked before

Ramblings of a Math Mom
Gifted Math Education: acceleration, enrichment, and the Calculus Trap
Mathematics Education
Quite often the question arises as to what is better for gifted kids — acceleration or enrichment

The Math Less Traveled
Golden Powers
Recreational
So, we know from a previous challenge that . That’s a pretty interesting property, which is shared only by its cousin,

Continuities
Half Full?
Secondary Mathematics Education
When talking with coworkers about these classes today, this was the advice I was given: If half of them understand half of what you’re teaching, it’s time to move on.

dy/dan
The Homework I Gave
Secondary Mathematics Education
I assigned homework to four of five classes this week, which is something like a personal record

The Hunt for a New Solution Concept
Game Theory
This is a call for theory building: As computer scientists, we have to find a solution concept that is as appealing as Nash’s, but that is also feasible in a world with computational limitations.

Gooseania
I can has a mathzburger
Thesis (non)-writing
To prevent me from being booted out of the club, I had better write about maths occasionally

Rational Mathematics Educations
Looking Further at Multiplication
Elementary Mathematics Education
I wish I could understand what appears to be an irrational rejection of a perfectly sensible approach that is just as mathematically sound as the “traditional” way most Americans were supposed to learn to do multi-digit multiplications.

meeyauw
NCLB Assessment Quality
Elementary Mathematics (Assessment)
Can you find the problem that I have with it? What are the standards for testing validity for high stakes tests?

Halfway There
Ned Ludd does technology (And he’s wearing sabots!)
Campus Politics; Technology
We labored diligently, sometimes against unexpected obstacles (like the faculty member who volunteered for the committee for the specific purpose of trying to sabotage it) …

Aside by your carnival host: zenoferox (host of Halfway There) weaves classroom stories – he teaches undergrads – together with campus politics, and politics in general. Amusing, diverting, fun. Go, poke around. – jd2718

Jan’s Diary
Online tutoring
Hmm.
1. Start a project. 2. Find no real big success. 3. Repeat cycle with another project.

Rigorous Trivialities
Parallel Parking
Lie Groups
Well, as it happens, if your car has length $L$, then for any $\epsilon>0$, it is possible to parallel park, assuming some things like that the driver can make arbitrarily small movements.

mommy bytes
Photo Hunters – Paper
Origami
These paper folded polyhedra are assembled using identical interlocking pieces of paper with no tape or glue

Learning Games
Physics for games
Secondary Mathematics Education
Author prepares for “Physics Modelling” class (hybrid math, physics, computer programming) by assembling a collection of first rate links.

Let’s play math
Elementary Mathematics Education
This time I will demonstrate these problem-solving tools in action with a series of 3rd-grade problems based on the Singapore Primary Math series, level 3A

MathNotations
Products of Digits: Challenges for Everyone…
Puzzles
Some number puzzles that can be used to introduce middle school students to the special methods of solving Diophantine Equations (and proof).

Polymathematics
Proof of the Coolest Math Fact Ever
Secondary Mathematics
Put n equally spaced marks around the circumference of a unit circle (radius of 1). Then from any one of those marks, draw the chords that connect it to all the other (n-1) marks. The lengths of these chords are then multiplied together, and amazingly, that product is always n.

jd2718 (that’s me, your carnival host)
Puzzle – a conic twist
Puzzles
I took a number stumper and turned it into a locus problem. Try it.

TerryTao
Pythagoras’ Theorem
Secondary Mathematics
My colleague Ricardo Pérez-Marco showed me a very cute proof of Pythagoras’ theorem, which I thought I would share here; it’s not particularly earth-shattering, but it is perhaps the most intuitive proof of the theorem that I have seen yet.

Halshop
Humor ?

Mathematics Under the Microscope
A questionnaire about languages in mathematics
This is an appeal, to those who speak languages other than Russian or English. If you can help out, it would be nice.
In my University, I teach a preparatory course in mathematics for Foundation Year students, many of whom came from overseas. My experience suggests that, in communication with foreign students, lecturers too frequently ignore difficulties arising from variation in logical structures of human languages.”

Killing Mind
The Results are in
Puzzle
…at pub trivia … was a fairly benign looking magic star puzzle… I figured a bit of sensible number distribution ought to allow me to work towards a solution. A good number of failed attempts later, the puzzle’s elusive solution beckoned some further exploration.

Science and Reason
Rings of algebraic integers
Algebraic Number Theory
The time has come,” the Walrus said/ To talk of many things: /of shoes–and ships–and sealing-wax– / of cabbages–and rings

Gli studenti di oggi
Scomposizioni
Secondary Mathematics (Global)
In tutte le scuole italiane, in prima o in seconda superiore, si insegnano le scomposizioni dei polinomi.

Reasonable Deviations
The Scottish Book
History
I wrote a post about the Scottish Café in Lwów (now Lviv, Ukraine), where, in the 1930s, Polish mathematicians from the Lwów School of Mathematics would gather to pose, discuss and solve problems. At first, the mathematicians would write directly on the tables’ marble tops…

Walking Randomly
Secret messages hidden inside equations
Puzzle? Mathematica?
Suitably intrigued, I issued the required Mathematica commands and got the plot below which spoke to me in a way that no equation ever has before

Elliptica
A Sequence
Puzzle (post-secondary)
If you’re an experienced mathsy type, try to prove that the sequence contains every possible positive fraction exactly once (yes, people, sequences like that are possible). I did it in four hours — working with a friend who was obsessed with Fibonacci numbers :-). Another friend of mine got it by himself in three, but our proof was nicer.

Mathematical Painting and Sculpture
Staircase knot
Topology
This strange composition of staircase reminds Möbius strip, but unlike Möbius strip this figure has two sides

OxDE
Surfaces for symmetric groups
xyz graphs
Inspired by my previous post on a truncated-cube Cayley graph for permutations on four elements, I tried looking at similar presentations for larger permutation groups.

MathNotations
Taking the ‘Unsummable’ Numbers to a higher level: An Algebraic Proof
Secondary Mathematics
The challenge for my readers and for students is to use methods from Algebra 2 and basic number theory (primes, factors) to prove a conjecture…

xkcd
Tapping
Humor

Casting Out Nines
Textbook-free Modern Algebra update
College Teaching
I decided to get away from any kind of lecture at all, whether it was given by students or by me. Instead, I ended up changing the whole structure of the course to be a sort of modified Moore method

Quomodocumque
They had not the habit of definition
Meaning
High school math teacher Polymathematics delivered a magisterial series of posts on [does 0.9999 = 1] last year, which covers with admirable thoroughness every one of the many, many strange trails this argument likes to wander down. So I’ll leave that to him, and just use the question as an excuse to copy in one of my favorite quotes from G.H. Hardy

The Universe of Discourse
Van der Waerden’s problem
Integer sequences
In particular, I wanted to calculate V(3, 3). These days you can just look it up on Wikipedia, but in those benighted times such information was hard to come by

Textsavvyblog
Volume of a Cone (Part 1Part 2Part 3Finale)
Secondary Mathematics
Vlorbik semi-challenged me to explain, sans calculus, how to find the formula for the volume of a cone. Well, after what was likely about two miles worth of pacing–and with the help of a certain Greek who got to this about 1,700 years before I did–I think I came up with a simple (though windy) way of explaining how to find the formula.

The Geomblog
The “Voronoi” Trick
Redistricting and Computing
I’m impressed at their use of power diagrams and the Voronoi trick, something that many theoryCS folks may not have encountered….can you evaluate the quality of a state’s redistricting plan ? Their answer is in the form of an approximation ratio

Andrew McKie
The Wandering Mathematician
History
Today is the anniversary of the death of the great Hungarian mathematician Paul Erdos in 1996

The Ridgewood Eclectic Educator – Release Two
We don’t all learn math the same way
Mathematics Education
Our brains are more pattern recognizers than rule recognizers so we really focus on patterns in mathematics.

Bug Girl’s Blog
We’re taking our math and going home in a huff
Secondary Mathematics (Global)
The US is bowing out of the next TIMSSA, and Bug Girl says “this is just silly”

Political Calculations
Why Isn’t the U.S.’ National Debt per Capita Higher?
Economics
how come the share of the national debt that would be shared equally among all American men, women and children only amounts to \$26,727?

That’s it for the October 5, 2007 Carnival of Mathematics #18 (published on October 6). CoM 19 will be at Good Math, Bad Math October 19. Until then!

October 6, 2007 pm31 9:12 pm 9:12 pm

oops – the link to Science and Reason – Rings of algebraic integers is not working.

Great Carnival!

October 6, 2007 pm31 9:13 pm 9:13 pm

ok, now it is. disregard the last comment.

3. October 6, 2007 pm31 10:21 pm 10:21 pm

Wow, Jonathan! I’m impressed by the length and variety of this carnival offering. If I had a hat, it would be off to you right now. Keep up the good work.

4. October 7, 2007 am31 6:50 am 6:50 am

Great post. Thanks for including me. I enjoyed the links and found some new bloggers to visit.