Impossible sculpture

In the lobby of a museum of 3d art and optical illusions in Germany is this beautiful "impossible figure" sculpture. Three right angles in three directions, somehow joining into a triangle.

Walking to side reveals that there are indeed three right angles in three directions, but they're not physically joined. Only in your eye and brain!

 

This was a fun museum. I'll post more on it later.

Cutting the cake

Bjørn Petter Jelle gave a talk at a school symposium last year with a selection of interesting math problems. I thought his presentation of the cake cutting problem was brilliant.

The classic problem is this: divide a round cake into 8 equal pieces with 3 cuts. He posed the problem and after we discussed in groups, one group offered the classic solution. Quarter the cake with two cuts and then make a third cut horizontally to split the four quarters in half.

Bjørn Petter then produced a cake with lovely sprinkles and big frosting flowers. He offered a knife and asked the group to perform to operation:

 

The result: a very unsatisfying division of the cake. The bottom pieces have almost no frosting:

Back to the drawing board. Another group proposed stacking the four pieces after the second cut and making the third cut vertically. Bjørn Petter produced a second cake and had them try. It worked better, but the beautiful frosting flowers were crushed. Other ideas?

A third group proposed setting the four pieces in a line and cutting with a really big knife. Bjørn Petter produced a third cake... and a sword!

 

Afterwards, we all ate cake of course. I got one of the bad pieces. It didn't matter, his presentation and the series of surprises made for one of the best presentations of a classic problem I'd ever seen. Bravo!

Knight Maze

From the Infinity conference a few years back in Ann Arbor, Michigan. I gave a talk on knight mazes and began the talk with a puzzle. You are a chess knight. Start anywhere you like on the left edge of this board and reach the star by moving only legal knight moves on the white squares. You may not leave the board or land on a blue square.

It's a maze I designed especially to be difficult. Give it a try! The talk focused on interesting maze elements and concluded with an analysis of this maze and the key to why it is so challenging.

Fractal Jugglers

"Nothing but Jugglers"

Making a picture containing "nothing but jugglers" presents a special challenge. In order to properly represent jugglers, the people in the scene must be juggling something, but what can those objects be if the only objects allowed are jugglers? We must conclude that the scene is be recursive with the number of jugglers growing exponentially to infinity as their size decreases to nothing.

Here's the scene as an animated gif. Try to follow one red figure, if you can!

4d juggling visualizer

Click on the window to use arrow keys to rotate this model. Shift-arrow keys scale the model. Flash required.

This is a 3-ball cascade juggling pattern. Imagine filming the juggling pattern then stacking the frames of the film to make a 3d model where depth represents time. That's what this shows. Be sure to click in the window to be able to use the arrow keys. Here is a bigger version.

Tetrahedron teabags

I love these teabags! A nice question: if taken apart what is its shape? Next question, if you want to make a large tetrahedron from the same pattern with a piece of A4 paper (or letter-size paper or whatever), what should the dimensions of your template be? Will you account for the overlap? I haven't tried it yet, but I think there will a nice use of √3 in here somewhere.

Silver ratio windows

I was recently at Jyväskylä University in Finland and saw these windows on the Agora building. They looked like silver rectangles to me, so I snapped a picture to measure. Sure enough, nearly exactly by my measurements on the image.

The silver ratio, by the way, is 1+√2 : 1. It has many connections to the golden ratio. It's equal to 2+1/(2+1/(2+1/(2+1/(2+...


 (Replace all of those 2s in the fraction with 1s and you get the golden ratio).

If you add a unit square to a 1:√2 rectangle (A4 paper, for example) you get a silver rectangle. If you subtract a unit square from a 1:√2 rectangle you also get a silver rectangle.

A4 rectangle plus a unit square is silver:

 A4 rectangle minus a unit square is also silver:

And just like you can remove a series of squares infinitely from a golden rectangle, you can remove a series of 2 squares infinitely from a silver rectangle...

There's lots of other cool properties... I've been fascinated with these "precious metal ratios" lately (gold, silver and yes bronze and others!) so it was exciting to spot these silver ratio windows. I hope this was an instance of intentional mathematics!