Ballooning with Vi Hart

Vi Hart ran a mathematical balloon sculpting workshop at the Bridges conference in Pécs Hungary summer 2010. Here's the effervescent Vi with an icosahedral balloon sculpture, and me inside her group project Sierpinski pyramid. Squeaky fun!

Pi acknowledgement

In Sandra Kring's novel "Thank You for All Things" (amazon.com link), one of the characters is an autistic child who memorizes pi and is close to the world record. Here's an excerpt about one of his practice sessions:

As I pass Milo’s door—eight hours after he began—he’s resetting his timer.  He sees me and calls out, “I’m just starting position 48,551,” he says.  “I’m averaging 6000 digits per hour.  Right up there with the current record holders,” He sets the timer down and starts, “three, seven, two, five, four—I really like that part!—eight, two, five….”  The sounds of Grandpa Sam’s rutted breaths and rattling bed bars are filling every corner of the house.  Milo hears them too, of course, and I can tell by his eyes that he’s grappling hard to see the digits, rather than to see Grandpa struggling to breathe.  I feel sorry for him, so I say, “Good job. Catch you later.”  

 The part he likes, 3 7 2 5 4, is my name in sound numerals. A cool 'secret reference'.

Find out more about sound numerals here: http://folk.ntnu.no/krill/home.htm

Human Hypercube

Speaking of hypercubes, here's representation with people. We can build up to a hypercube by starting with a 0-dimensional point. 'Stretching' that in one direction gives us a 1-dimensional line segment. Stretch the line segment in a direction at 90° from the line segment and we get a 2-dimensional square. Stretching that 90° from the plane of the square gives us a 3-dimensional cubes. Now, stretch the cube at 90° from its volume (yes, I know it hurts to try to think like that. It's only impossible in the real-world, not in your mind!) and you'll have a 4-dimensional hypercube.

The pictures above show the changes from 0-d to 4-d.

Puzzles:
1. Describe how the number of vertices changes. Write a formula.
2. Describe how the number of line segments changes from one figure to the next. Write a formula.
3. Describe how the number of squares changes. Yep, formula.
4. Describe how the number of cubes changes.
5. Predict the properties of a 5-d hypercube. Can you draw one?

Nidaros Cathedral Hypercube

This lamp (one of several) in front of the Nidaros Cathedral in Trondheim bears a striking resemblance to a hypercube.

Magic Static Cat

Two pieces of transparency film printed with static, taped on opposite sides of a piece of glass. From the right angle, the static aligns and the Cheshire Cat appears. This is hanging outside of my office in the elevator lobby. It's fun to point out to visitors. How does it work?

The trick lies in starting with 2 identical copies of static and inverting the static in one of the images in the area you want to appear black. That way, when the copies are aligned the regular static is still just static, 50% gray, but in the other area the black pixels are aligned with the clear pixels and vice versa, so that part is dark.

Note that inverted static holds as much information as non-inverted static, that is, zero. Neither of the sheets of static contain an image of the cat. It's only by juxtaposing the two images that the information can be generated/retrieved.

Skolelaboratoriet Logo

Skolelaboratoriet is a department that is housed on the same floor as Matematikksenteret where I work. My first day on the job I was pleased to see their logo:

 

Can you see what is special about it?

 

It show a dissection of a square into an equilateral triangle. The white piece in the middle stays in place and is common to both the square and equilateral triangle. The three upper pieces which are part of the square are moved below, each with a 180° rotation, to create the equilateral triangle on the bottom. I've never tried to work out the angles and lengths involved to accomplish such a dissection, but it looks like a nice little problem. I'll try to remember to think about this next time I have a long flight.

The dissection is even a little better than a simple cut-and-rearrange: the pieces can be hinged so that one form turns inside out to become the other.

George Hart presents a coffee table that performs this trick:

http://blog.makezine.com/2011/04/04/math-monday-mathematical-coffee-table/

Cobblestone scallops

These scallop patterns in the cobblestone in downtown Trondheim are interesting. If you were laying cobblestone, how would you determine the layout for these stones? Using your technique, what other interesting curvy designs are possible?

Lady in Glass

Peter Sutton is a glass artist in Trondheim, Norway. He and I collaborated on this project, a lady in glass:

 

I designed 23 cross-section slices which he printed on sheets of glass and assembled into an elegant block. The result is a 3d model floating in a solid chunk of glass. Here's a view of the cross-sections:

Extra second, and celebrating a gigasecond

Last night an extra second was added to world time. It happened at 23:59:60 UTC, or Universal Time which is the same as Greenwich Mean Time minus daylight savings. Here in Norway we're at GMT +1 which is UMT +2, so it happened just before 2:00 a.m. here.

I stayed up late to celebrate. It was an eerie feeling, to experience a minute go by that was 61 seconds long. It dragged on and on and on and we began to wonder when this minute would ever end. Whoa. Good thing they only happen every 4 years or so.

Reminds me of when I turned one billion seconds old. It happens sometime when you're 31 years and some months old. I remember thinking about it when I turned 31 and I calculated the date. It was 252 days away and I soon forgot about it.

One morning I got an email from a guy I hadn't from in 10 years. The email simply said "Happy Gigasecond!" I panicked! I'd forgotten! Was it today? Had I missed it? I furiously calculated. I was born in Indiana at 1:02 p.m., now I was in Washington, I had to account for time zone changes and daylight savings. I found out the exact second... and I still had 4 hours to go. I celebrated by popping open a malty beverage at the exact second while in my hot tub. It felt good to be aware of and celebrate the exact second.

But I still had a mystery – how did this guy know? I wrote him back to thank him. Turns out 10 or 15 years ago at a juggling club meeting we all figured out the exact dates we'd turn one billion seconds old. Dave L. had written down all of these dates in his calendar, and being a very organized person at the end of each year he would buy a new calendar and transfer all of the relevant information from the old calendar to the new. This he did again and again, and when it came to be my turn, he tracked down my email address.

Dave, you're my hero! If I'd missed it I would have needed to wait until 63 to celebrate the next one.