Gardner now gets into Solid Geometry and Higher Dimensions, with The Helix. He begins by asking if there can be an alternative to a straight, or a curved sword. A straight sword can be slid into a straight scabbard, and a curved sword can be put into a scabbard of the same curvature. Are there any other shapes with the same properties? The answer is, “yes, the helix”. (Think of a corkscrew with constant radius.) From here, he goes on to mention the geometrical properties of the helix: in the limiting cases you get the straight line or a circle (if A is the constant angle of the curve crossing a line parallel to a cylinder’s axis, then if A = 0 you have the circle, and if A = 90 degrees you get a straight line). The shadow cast by a helix onto a flat wall can either be a circle or a sine wave.

The helix is “handed,” in that you can have right- and left-handed corkscrews that are distinct from one another. This brings Gardner to a discussion of handedness in both man-made and natural objects. Screws, bolts and nuts by convention are right-handed, but candy canes, barber poles, cable strands, and staircases can be both. We can have conical helices, such as the inverted conical ramp in Frank Lloyd Wright’s Guggenheim Museum in New York, which is cited as an example of a curve that spirals around a cone. Then we get DNA strands, narwal whale teeth and the cochlea of the human ear.

The Devil’s Corkscrew is a 6-foot tall fossil that turned out to be the remains of burrows of prehistoric beavers. If you count the turns made along the helical path of a plant from one leaf to the one directly above it, you often get the Fibonacci series (1, 2, 3, 5, 8, 13…). This is covered in the field of phyllotaxy, or leaf arrangements. Climbing plants are usually right-handed, but twining plants often twist the other way, and when the two types intertwine, the results are kind of romantic, as remarked on in Shakespeare’s A Midnight Summer’s Dream – “Sleep thou, and I will wind thee in my arms./…So doth the woodbine the sweet honeysuckle/Gently entwist.”

Ref. Flanders and Swann, Misalliance

There are many other examples in the article, but I’ll end here with the puzzle of the week.
You have a rotating barber’s pole, with painted red, white and blue helices. The cylinder is 4 feet tall. The red stripe cuts the vertical lines at a constant angle of 60 degrees. How long is the red stripe?

Gakken has teamed up with Capcom to produce two kits in their Metal Series for Monster Hunter, one for Liolaeus and the other for Furfur. These are brand new kits, which hit the shelves on or after June 15 (depending on where in Japan you live). They’re pricey, 2,600 yen each ($25 USD) without tax, putting them on par with the Metal Legend dragon I made last year.

The box is kind of small, which is misleading. There are least 40 pieces to this kit, not including the nuts, bolts and hinge plugs. It also includes a small piece of sandpaper (useless), a sheet of decals for the eyes and teeth, and two small tools. You really can’t complete this thing with the tools provided. You also need a small pair of diagonal cutters to separate the pieces on the connected sheets and to trim lose tabs, and a small needle nose pliers for making precise folds.

The kit consists of pieces punched out of two types of metal. The softer one that’s easier to cut and mold is probably tin. The stiffer one might be aluminum. That one is much harder to smooth off the protruding tabs, and to bend the smaller tabs.

The instruction book has a few pages describing the Hunters from one of the games, and a features section showing past super beasts.

The back pages advertise the other Gakken metal kits, and suggests poses for your finished Monster.

Starting out. It’s probably unnecessary to verify that you have all the pieces in advance, but it’s still good practice. That took at least 15 minutes, just checking everything off the parts list (I didn’t bother counting the nuts and bolts). As it was, the only leftover surplus was in the form of the backup set of eyes and teeth on the decal sheet (the sheet had two sets of eyes and teeth, in case one got damaged during assembly and application). Note the two tools to the upper left. Since the small 8mm nuts are easy to lose, it would have been nice to have 2-3 spares, as well.

Bottom of photo – the spine and internal frame structure.

From top to bottom – both legs, the finished wings, and the main body and tail built around the internal frame.

The assembled body, legs and wings, plus the finished head assembly. From start to finish, this probably took at least six hours. I had to take breaks for dinner, and sleep at night, so I don’t have an accurate accounting for how long everything took. I can say that my fingers stung from having to make so many bends, and I’ve got little slice marks on both index finger tips from handling the scissors in cutting the pieces apart and trimming off protruding tabs.

Liolaeus has taken over my laptop PC. The finished beastie has a 36 cm wingspan, which is 14.5″. It’s a pretty big kit, and is about the same size as the Metal Legend dragon.

The wings fit into soft plastic hinge pieces, so you can pull the wings out and reposition them as you like. The neck and spine are built on a soft metal strip, and allow limited bending. The legs and jaw are bolted together, You can reposition them, but doing so too much will cause the nuts to loosen and fall off if you’re not careful. I left one of the smaller insect kits at the English school for the kids to play with, and they pretty much demolished it. I’m not going to make that mistake again with Liolaeus.

Please note that while the box says that these kits are for children aged 10 on up, they do include a lot of small parts that can present a choking or health hazard. In the U.S., the metal kits would probably be labelled “16 and up.”

Roar. Simply, roar.

Answers to this week’s game questions can be found at the end of the article reprint.

We close out the Plane Geometry section with the article The Wonders of a Planiverse, and the studies of 2D physics as developed by A. K. Dewdney. Dewdney was a computer scientist at the University of Ontario, and he wrote a report on his examinations of a flatworld in 1978 in response to Gardner’s article on the earlier Flatlands book by Edwin Abbott (1884). At the time Gardner wrote this article, Dewdney was still developing his physics and mechanics for a planiversal universe. His book, Planiverse wasn’t published until 1984. The article here discusses the operations of the some of the conceivable 2D machines, and the addendum adds comments by other physicists on how light, sound waves and gravity would, or wouldn’t work, in this environment. (One physicist claimed that sharp sound spikes wouldn’t decay properly, so no one would be able to complete full sentences.)

(Example of a simple 2D machine.)

Dewdney eventually took over the Mathematical Games department from Gardner in 1984. And, the concept of 2D physics has expanded to explore planar phenomena, including one-molecule-thick films and two-dimensional electrostatic and electronic Hall effects. There’s also a mention in the addendum of Arthur Clark’s Childhood’s End, in which intense gravity on a gigantic planet causes the evolution of creatures only 1cm thick.

(Example of the 2D planiverse creatures living at home.)

Some months back I was writing on topology, and I wanted to mention Planiverse, but I couldn’t remember the name of the book at that point, and instead got stuck with Flatland. There used to be a shopping center in St. Anthony Falls, Minneapolis, in Minnesota, that had been a repurposed factory building. They had a tex-mex restaurant that I loved, called Guadala-Harry’s. What was great about this place was that they’d give you wireless buzzers to let you know your table was ready, and you could spend up to an hour wandering the other shops. One of which was a very eclectic bookstore with a good selection of math and science books. I’d go to St Anthony Main every Friday for dinner, and I’d almost always find a book I wanted to buy. Naturally, one of those books was Planiverse.

(Planiveral steam engine.)

My main complaint about the planiverse world that Dewdney called Astria, and the Astrian lifeforms on it, revolved around what I considered to be a minor glitch with an illustration on one page. There was a small lip on a hatch door that was otherwise flush with the ground. If Astria had weather, it’d have an equivalent of rain. And any standing water on the ground would back up from any obstacles to find its own level. That is, flooding would be inevitable, and any Astrians outside in a deluge would be washed into a massive lake or ocean. Not to mention that humidity in the air would probably suffocate everyone. But, that’s a small complaint for something that was obviously written as fiction.

(How locks work.)

One of the readers of the article wrote in to say that this lock design is very similar to that of the 1895 Mauser military pistol. And, you can model these machines with cardboard cutouts, which is how John Browning used to work when designing his firearms.

Puzzles and games:
Examples of 2D games. (a) is the start of a checkers game. Pieces only move forward, 1 square at a time. Jumps are mandatory. On an 11-square board with pieces on the first 3 lines on each side, which player wins, the first one to move or the second?

(b) is a 2D variant on chess. Knights move two cells in either direction and can jump pieces of either color. Given rational play, which side will win? Or, can anyone win?

(c) is linear Go, called Pinch. The rules are available at the end of the following reprint of this chapter. (d) and (e) are further examples of the Pinch game.

Phil Foglio, creator of the Girl Genius webcomic, likes to occasionally make papercrafts, wallpapers and other things. Here we have an other thing – a papercraft puzzle featuring the jagermonsters. The idea is to rotate the cubes so that you only get one each of the images on each row. I had so much trouble with this that I actually wrote a VBscript to find the solution to this puzzle, to no effect. Then, I went back and corrected some errors, and the script located the first of the many possible correct solutions. So, yes, it’s difficult (for me), but it can be solved. Either way, I printed the PDF on 0.15mm plain stock, which turns out to have the right stiffness without being so thick as to throw off the fold lines on the images.

I’m going to bring this puzzle to my classes and see if any of my students can do it before they crush the paper up in frustration.

The next Gardner article in the section on plane geometry is on Piet Hein’s superellipse.

Piet Hein was a Danish mathematician, inventor, designer, author, and poet who lived from 1905 to 1996. He created the games Hex, Nimbi, Tower and the Soma Cube. For poetry, he mastered something he called a grook – a short rhyming aphorism. He wrote over 7000 grooks, which have been published in 20 volumes. You can find copies on amazon. I first encountered a book of grooks in the late 1970’s, early 80’s, and I liked them a lot.

Example from the wiki entry:
It may be observed, in a general way,
that life would be better, distinctly
If more of the people with nothing to say
were able to say it succinctly
– Piet Hein

In 1959, the architectural team tasked with rebuilding a congested section of old houses and narrow streets in the heart of Stockholm, Sweden, couldn’t figure out how to do the layout. They approached Piet Hein, who eventually came up with the idea of the superellipse – a curve that fits pleasantly into a rectangle without having the sharper deformations of an ellipse.

The general form of an ellipse is:
abs(x/a)^n + abs(y/b)^n = 1

As n approaches infinity, the shape becomes a rectangle. “a” and “b” are arbitrary constants that represent the semiaxes of the curve, and n is any positive real number.

Piet Hein settled on n = 2.5.

The superellipse has been used as the basis for furniture, park landscaping, and the patterns for Danish postage stamps. When made 3D out of solid brass, referred to as a superegg, it will easily sit on one end without falling over.

Superegg, from the wiki article.

Challenge of the week: Draw your own superellipse and stick it on stuff.

This is what science does when you’re busy looking under the couch for the remote.

Direct youtube link

Solutions to this week’s puzzle can be found in the wiki entry on rep-tiles.

Rep-Tiles!
Rep-Tiles on a plane!

Sorry, wrong movie. Rep-tile was coined by mathematician Solomon Golomb to describe self-replicating tilings that can cover a plane surface. Solomon pioneered the study of these types of tilings, and developed much of the language for it.

(Image from wikipedia for a sampling of rep-tile types.)

When Gardner originally wrote this article, the link between fractals and rep-tiles wasn’t understood. Objects like the Sierpinski triangle and Sierpinski carpet are now known as self-similar fractals and are a lot of fun to make.

Hand-drawn Sierpinski triangle

Sierpinski triangle

Sierpinski carpet