Over the last couple of days we’ve done two projects that started from a couple of easy to state questions:
(i) Given some squares with area 1, how do you make a square with area 2?
(ii) Given some squares with area 1, how do you make a square with are 3?
Those project are here:
A neat and easy to state geometry problem
Some simple proofs of the Pythagorean Theorem
Tonight my older son is at a school event. That gave me time to do a fun little extension of these two projects with my younger son.
First I reviewed the original problems:
My son solved the 2nd problem above by making triangles with sides 1, and . For this part of the project I wanted to show him a different triangle that has a side length of – a 30-60-90 triangle:
Now – for a little extra fun – we made a Zometool cube. That cube shows that the face diagonal (of a 1x1x1) cube has length . It also shows that the internal diagonal has length
Here’s the surprise – if we extend basically the same geometry to 4 dimensions, we find that the “long” internal diagonal of a 1x1x1x1 cube has length 2, and that there’s a secret little 30-60-90 triangle hiding in the cube!
We did a similar project a few years ago:
Did you know that there is a 30-60-90 triangle in a Hypercube
It was nice to revisit this idea today 🙂