curiosamathematica
curiosamathematica:

The Wallace-Bolyai-Gerwien theorem states that any two polygons are equidecomposable: it is possible to cut any polygon into finitely many polygonal pieces and then rearrange them to obtain any other polygon.
The theorem doesn’t rely on the axiom of choice (unlike, for example, the more famous Banach-Tarski decomposition). Moreover, the decomposition and rearrangement (which consists of rotations and translations only) can by carried out “physically”: the pieces can, in theory, be cut with scissors from paper and reassembled by hand.
The problem about whether a hinged dissection exists, such as the ones in the animation, remained open until 2007. The paper Hinged Dissections Exist presents a method which always works to find a hinged dissection.

curiosamathematica:

The Wallace-Bolyai-Gerwien theorem states that any two polygons are equidecomposable: it is possible to cut any polygon into finitely many polygonal pieces and then rearrange them to obtain any other polygon.

The theorem doesn’t rely on the axiom of choice (unlike, for example, the more famous Banach-Tarski decomposition). Moreover, the decomposition and rearrangement (which consists of rotations and translations only) can by carried out “physically”: the pieces can, in theory, be cut with scissors from paper and reassembled by hand.

The problem about whether a hinged dissection exists, such as the ones in the animation, remained open until 2007. The paper Hinged Dissections Exist presents a method which always works to find a hinged dissection.

On Homework

I’m trying to think about ways where I might not collect homework at all next year, but still make it important and required. 

Ideas

  • Drawing a distinction between homework (just for practice, answers provided) and problem sets (more involved problems, more time to work on them, connect lots of topics)
  • Making reassessment on a standard contingent on having perfect homework from that topic (it should be perfect because you have the answers for homework, and you have time to ask questions about problem sets)
  • Students’ job to ask questions about problems they don’t understand and keep it all in a notebook with full work - this makes it a good study resource
  • This lets me assign less homework
  • I think I would still want to allow collection of homework if kids *want* feedback on it, which happened this year sometimes
  • Maybe I would collect problem sets for ungraded feedback - formative assessment that doesn’t touch the gradebook and lets them see where they are in anticipation of quizzes and tests
  • I like that a lot, actually
  • Maybe make quizzes a lot like problem sets
  • I could occasionally collect homework - almost as a pop quiz, to make sure the incentive is still there

Anyone have thoughts?

crazedmicrobiologist
crazedmicrobiologist:

visualizingmath:

allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

I’ve never heard of this before!

My roommate tried to extend this bitch into the fourth dimenson….

You totally can! And beyond!

crazedmicrobiologist:

visualizingmath:

allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

I’ve never heard of this before!

My roommate tried to extend this bitch into the fourth dimenson….

You totally can! And beyond!

michaelblume
michaelblume:

allofthemath:

visualizingmath:

allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

I’ve never heard of this before!

I never had either! I was explaining the conceptual basis of Pascal’s triangle to a student and she got very excited about it, so we started talking about the three dimensional version (and the coefficients of (x+y+z)^n, and we tried to make it!
A really amazing property of this tetrahedron is how it geometrically gives us what coefficients are of what terms. Anything that’s on any of the three outer edges is pure x, pure y or pure z. If it’s on the angle bisector emanating from, for instance the x angle, then it has equal amounts y and z. If it’s right in the middle, it has equal amounts of all of them. 
Awesome, right?

Did you come up with a closed form solution for the numbers in the middle?

Yeah, the coefficient of x^n*y^m*z^o is (n+m+o)choose(n)*(m+o)choose(m)

michaelblume:

allofthemath:

visualizingmath:

allofthemath:

This, ladies and gentlemen and genderqueer folks, is Pascal’s tetrahedron, a three dimensional analogue of Pascal’s triangle, and it’s pretty freaking great.

I’ve never heard of this before!

I never had either! I was explaining the conceptual basis of Pascal’s triangle to a student and she got very excited about it, so we started talking about the three dimensional version (and the coefficients of (x+y+z)^n, and we tried to make it!

A really amazing property of this tetrahedron is how it geometrically gives us what coefficients are of what terms. Anything that’s on any of the three outer edges is pure x, pure y or pure z. If it’s on the angle bisector emanating from, for instance the x angle, then it has equal amounts y and z. If it’s right in the middle, it has equal amounts of all of them. 

Awesome, right?

Did you come up with a closed form solution for the numbers in the middle?

Yeah, the coefficient of x^n*y^m*z^o is (n+m+o)choose(n)*(m+o)choose(m)