Had a friend from college visiting for Memorial Day and thought it would be fun to do a video explaining the 4th dimension to all of the kids in the house this weekend. This project didn’t go quite as well as I was hoping, but I think the idea here is fun. Will probably try it again in a few months.
In the first video we walk through the concept of a zero dimensional object sliding in time. Our model for a zero dimensional object is a snap cube. We talk through how a zero dimensional object sliding in time can create a one dimensional object. The concept may seem a little strange when you talk (or read) about it, but seeing the trail of the snap cube as it moves helps the idea make sense (I hope!).
One other thing that we’ll be keeping track of in each of the videos is the number of cubes we have at every stage of the sliding. With a single sliding snap cube, counting the cubes is easy – we just get 1,2,3,4,5, . . .
Next we try to make a two dimensional object by paying careful attention the sliding zero dimensional object from the previous video. We build a two dimensional object – sort of a triangle – out of the pieces that the sliding snap cube created in the last video. In this section the number of cubes we need to build our object at each stage is 1,3,6,10,15, and etc:
Now we take the idea from the last video and apply again to make a three dimensional object. This time we have to keep track of the shapes at every stage of the “sliding” in the last film and combine those shapes together as they slide in time. The object we create this time around is a 3-dimensional pyramid. The number of blocks at each stage is 1,4,10,20,35, and etc . . .
Now for the 4-D challenge. We want to apply the same idea as in the previous two videos, but there’s a little snag. We don’t have any dimensions left in our kitchen, so how are we going to put the 3D object together? Unfortunately the 4D shape we are creating here is pretty hard to visualize, but we can at least understand what the slices look like – they are exactly the shapes from the prior video! One neat thing is that even though it is difficult to understand the picture of the full shape, we actually can count the number of cubes at each stage – 1, 5,15,35, and etc.
Finally, having build and sort of understood a 4 dimensional object, I wanted to show a neat connection this project has to Pascal’s triangle. In every video we found an interesting sequence of numbers by counting the number of blocks needed to build our object. Each of those sequences comes from a diagonal in Pascal’s Triangle! Pretty amazing that Pascal’s triangle tells us how to count blocks in 4 dimensional pyramids. The kids even speculated that other diagonals count blocks in higher dimensions. Pretty fun:
So, although this one didn’t go as well as I’d hoped, it was still really fun. At least it was nice to end on a really cool note with the connection to Pascal’s triangle. Will definitely try to improve on this one later.