Burning calories

There was a discussion at lunch today about how eating a big chocolate bar can require 2 hours on an exercise bike to burn it off. I've always found these comparisons a bit misleading. It makes it sound like that's the only way to keep from getting fat.

Fact is, a 100 g of milk chocolate has about 520 calories, which will indeed take 1 to 1.5 hours to burn off on an exercise bike. Or you could just sleep for 7 or 8 hours and it would also be burned off.

So go ahead and have that candy bar! Just be ready to either cycle for 2 hours or sleep for 8 hours. OR... sleep for 10 hours and lose weight. Mmmm... chocolate + naps = weight loss. I think I could start a very popular weight loss program.


Shadow knitting pi

Here's a 3 meter long shadow-knitted pi scarf made by Anne Bruvold in Tromsø. Shadow knitting creates patterns that are only visible from an angle. Seen straight on, as in the middle picture, the pattern is invisible.

Black on white digits viewed from an angle
The digits are invisible viewed straight on
A portion of the back of the scarf
Lovely! Anne used 3 plus 14 extra digits, which is my favorite length of pi: 3.14159265358979. Try saying those digits a couple of times, they nicely roll off your tongue like a poem.


Natural rhombic dodecahedra

My kids and I like to go up near Trolla and find garnet crystals in the rocks near the water or cliffs. The stone cracks open easily with hammer blows, and dark red garnet crystals spill out. The crystals are in the shape of rhombic dodcedahedra – polyhedra with 12 faces which are rhombuses. The shape is very special, it is the shape you'd get by putting a pyramid on each of the faces of a cube so that the faces of the pyramids are in the same plane as the neighboring pyramids. The resulting rhombuses have diagonals in the proportion 1:√2. A bunch of these crystals the same size would pack together tightly to fill space, which is pretty cool also. Not many shapes will do that. There's lots of other properties of this shape and connections to cubes and octahedra.

Here's some pics during and after one of our hunts:


Erno Rubik

At the Bridges conference in Pécs, Hungary in 2010, we had a rare presentation from Erno Rubik, inventor of the Rubik's cube. Earlier that day I'd gone shopping and tracked down a couple of cubes and a permanent marker, and hit him up for a couple of autographs afterwards. It was a high point of geek culture!


Almost a Reuleaux, real Reuleaux and 3D Reuleaux

I saw this Nokia trinket on a colleague's cellphone. It's close to a Reuleaux triangle, but not quite. Darn! A missed opportunit from the folks at Nokia.

A Reuleaux triangle is a shape with constant width. If you roll it, you can set a board on top of it and the board will not bob up and down, it will stay at the same height just as if the roller were circular. You can make a triangle like this easily, just draw three circles that pass through the centers of each other, and the region in the middle is a Reuleaux triangle as shown here:

You can make these with 5, 7, 9 or any odd number of segments by constructing arcs on regular polygons.

George Hart made these solids of uniform width. "Reuleaux acorns" he calls them. They're a 3d generalization of Reuleaux triangles - if you place a flat board on top of them they will roll around smoothly and the board won't wobble up and down. Weird...


Shower decoration code

Last week I wrote about embedding a meaning in artwork. Here's an example from my house back in the United States. We tiled the bathroom, and I embedded some of my wife's glass pieces in the cement. There were long squiggly pieces and small round pieces, perfect for dots and dashes to embed words in Morse code. At the head of the shower:   -.  .-  -.--  .-..  ---  .-.  ...

At the end of the tub the beautifully symmetric:  .-  -.  -.  .-

And along the side of the tub:  .--.  .-  --   and  --  ..  -.-  .   and  .--.  .  -  .  .-.   and --  .-  --.  --.  ..  .

It was fun making patterns with color and shapes and noticing patterns in the dots and dashes. And when it was done, it looked simply like a random design, but closer inspection hints at a structure that just might invite someone to look deeper. None of our visitors ever thought about, but I knew it was there and it make me happy every time.


Apple Pages regular polygon?

I use the drawing tools in Apple Pages quite a bit. But every time I need a regular polygon, I struggle to find it on the shape menu. See why? The icon is the second from the bottom on the drop-down menu:

I simply cannot get myself to see this icon as a regular pentagon! I often select the 'diamond' (canted square) icon by mistake.
This is an interesting shape, though. It looks like it's the bottom half of a regular hexagon with half a square stuck on top of it. Does this shape have any interesting properties? What is the ratio of long edge:short edge? If the short edge is length 1, what are the lengths of the diagonals? These lengths and areas may not play well together; the hexagon is related to √3 while the square is related to √2. It might be fun to investigate anyways. The angles of this shape are 120°, 120°, 90°, 105° and 105°. And the average of these angle is... exactly 108°! The same as the angle measure of a regular pentagon! (Is this a surprise? Should it be a surprise?)

All right, all fun aside, the icon confuses me when I want a regular polygon (which happens more often that you'd think). I submitted feedback to Apple today asking them to change this icon... I'm just doing my job to promote good geometry. Do you use Apple Pages to make drawings? Has this bothered you? Here, tell Apple: http://www.apple.com/feedback/pages.html.


Sculpture at Statoil HQ

Between Statoil HQ and the fjord in Trondheim stands this impressive fountain scultpure. It's a kind of twist quadrilateral that looks different from every angle. There's some very nice shapes in this industrial frame.