The Fitful Flog

This is a section of Giovanni Maria Cassini’s globe, a very beautiful object, which you can view in great detail at David Rumsey’s map site. You can see that Cassini knew something I was just guessing at, how to find the radius of the curve of map gores. Dr. Math tried to explain it to me, but he’s a mathematician and I’m a dilletante and we have trouble communicating. Every time I read the term “radians,” a wee pixie comes in through the cat door and blows out my pilot light. The eyes glaze over and the frontal lobes grow cold and I wake up eight hours later, surrounded by empty 40 ounce malt liquor bottles and professional wrestling, blaring on the television.

This is not the royal road to geometry. This is not even the on-ramp.

I kept thinking it was a spherical triangle thing. It probably is, but since I can’t manage to get the concept into my head, I tried doing it with good old Pythagoras and a piece of paper. (Well, actually, I was sketching madly on my tablet pc, amusing the other bus passengers no end.) It is not beautiful, but I think it works. Feel free to disagree; I won’t argue.

Gore Math

(For our foreign readers, I will mention that a pixie is a small, semi-mythical being, very like a leprechaun, but with a post-punk sensibility.)

Hey, speaking of Pythagoras, are any of our South Asian readers familiar with the medieval mathematician, Bhaskara II? Came across a reference to his having done a proof of the Pythagorean theorem by folding paper, but I can’t find any details on this.

I know that Eric has already blogged this, but this will not stop me from noting that lilzabubba (who is also known as Bekah) has made a rather more rational crease pattern for the lunoid model. And made a nice double lock on those troublesome corners — very clever, never occurred to me. lilzabubba’s Crease Pattern

Hmm, lots of diagramming going on in the Land o’ Lakes — do we see a book in Bekah’s future?

This was one of my first models, I think – I was about 13. It’s corrugated for extra goodness.

I had thought it one of my simpler models, but making a CP has taught me otherwise. It’s still fun.

Here’s the crease pattern.

And a Happy New Year’s to all!

Gore Square (deprecated – feh, uck, ptui)

Better Gore Square

For experimental use – it’s not quite done yet.

(Oh, and if any of those fancy New York design firms are curious, we may be open origami evangelists, but we can be bought licensed.)

Don’t mind the cat – Basil’s just been into the catnip and he’s a little stoned.

I decided it would be fun to color the gores of the curved surface waterbomb, to enhance the general eggyness of it.

“Easter egg” is a term programmers use to describe little secret treats they leave in their code, like a little video game hidden in a spreadsheet program. I reckon that by hiding a an origami crease pattern for an Easter egg in my blog during the Advent season, it’s sort of similar. When folders go googling for egg diagrams in late Lent, they’ll find this one, maybe.

So, some file to play with:

The crease pattern

The two-sided crease pattern with colorful fractal images on the back

The two-sided crease pattern with just lines on the back (you draw the pattern, yes?)

With the two-sided ones, you can print them back to back, depending on your printer and your level of comfort with your printer driver. The crease pattern part is oriented to fold up inside the egg. The flaps on top fold over and tuck into the curvature of the buttress thingee there — I left that in white, again, to call attention to the essential eggyness.

Yes, you can hide jellybeans inside. But not Jelly Belly® brand, for that kind is inextricably associated with the memory of that father of lies, Ronald Reagan.

A Chad Stewart from some place Southern writes in to say, “Hey, y’all! where’s the phreakin’ diagrams?”

Em. Haven’t made them yet. We will, but we’re still waiting for understanding to take place. (Ours, that is; yours will come later.)

Of course, if the world stopped every time we didn’t understand something, we’d still be negotiating with Khrushchev. Here’s a slightly improved Crease Pattern for your experimentation pleasure.

After I drew this, I realized this model has up and down to deal with. Okay, this is the underside of the rose you’re looking at, and the spiral will come towards you as you twist it, to form the top of the stem. The other side of the paper will be surface of the petals — no lines showing.

I’m not positive this CP is 100% accurate, but we’ll see. It’s the side locking that’s the real tricky part.

Here are some other shots:

Yes, getting a little swirly in here. Curved surface folding adds a new axis to the fold: mountain/valley, left/right, up/down, curved/flat….I like it. Gritty, textural, doesn’t work 3 times out of 5. Oh, yeah.

This is a Coriolis Bowl, named for that swirling effect the Earth’s rotation adds to storms and whirlpools. Astute folders will note that this one has a counterclockwise rotation, suggesting that it was created somewhere in the Northern Hemisphere.

However, I urge our folding public to exercise caution. Coriolis storms have been known to move people out of (and into) Kansas. Which, we hear, is quite the cultural mecca these days, though we’re reserving judgment on this head — heard the same thing about Elk City, Oklahoma.

The crease pattern.

Update: Better crease pattern and some nice hints can be found here.

Say what? Well, we must call it something and that name may be unlovely, but it is not unapt.

It’s in two pieces, the top wrapping over the bottom. Or the other way around, doesn’t matter much.

If you’ve spent anytime at all with circles, you’ve noticed that it’s very easy to fold a hex grid. Circles were built for this. But where, you ask, do the curves come from? Easy — the edge. Circle’s only got the one. Look at the CP and note where the curves begin and end. Now, fold the edge over, so it connects the two points. See? Just bend the paper a little and start pinching in the curve. As you practice this, you’ll see how much more accurate your lines will become.

The Crease Pattern

This is the same as the Triskelion Dish, but with the corners folded under – I didn’t quite like the white showing. A trifle more complicated, but logical and more aesthetically satisfying. Like a nice bowl of Spicy Beef Soup.

We offer two files here, one the crease pattern: Sam-Taeguk Dish Crease Pattern

And the other a colored printout that suggests the Korean national symbol: Sam-Taeguk Dish in Color

And we say howdy to our readers in Korea.

I think I mentioned at some point that it was easy to make an elegant pentagon from a circle of paper. It is, but still, it’s not intuitive. Making polygons from circles has a lot in common with compass-and-straightedge work. Except paper is easier to work with and the straightedge can be marked up like crazy.

Pentagon from a Circle Sequenced Crease Pattern

Based on a method described by H.W. Richards in 1893 – not that I’ve seen this. I read about it at MathWorld.