PAPER POLYGONS, BUILDINGS, CODE-BREAKING...

PAPER POLYGONS, BUILDINGS, CODE-BREAKING...

...IT'S GONNA GET A LITTLE WEIRD

People ask me what I enjoy. There are many answers but they boil down to books, optics, science and patterns. Before I could read or write I was a graphomaniac: I would scribble and scribble and scribble on paper and then fold and refold and crinkle and smooth pieces of paper. A little piece of paper can become an entire world. Whether you write a story or fold it into something dimensional.

What follows zings around quite a lot. As far as a post it's like a lot of posts mashed together. If any of the math, etc. bores you scroll down a little--it changes up.

This really should have been my flagship post for Science & Optics, but was too chaotic.

A stella octangula!

 Also called a stellated tetrahedron. Father Magnus Wenninger (cute pic of him in his office/cell filled with paper models: http://www.saintjohnsabbey.org/wenninger/) would be proud-actually since he literally wrote the book(s) on paper polyhedrons (I’ll be getting his book free with all the Chase/Amazon points I earned buying suits for work: http://www.amazon.com/Polyhedron-Models-Magnus-J-Wenninger/dp/0521098599) he’d just politely smile (he is a monk, priest and mathematician. 

Stella octangula is compound (the shape repeated-then turned (roll your own: http://mathworld.wolfram.com/StellaOctangula.html). When I need to take a break or mull an idea over I’ve decided to make a polyhedron (or other paper shape). Wenninger came up with 75 main paper polyhedrons for his first book—but the publisher demanded he make and photograph them all before they sign a contract. He did it.

I think I’d like to repeat that feat: like the author of ‘Baking With Julia’. That could be my ‘hook’—although resources on the web reveal people who have done all 75 plus tons more shapes (this is one of my favorites, click on a shape to get the folding instructions, some are easy-some I’d only attempt if I was laid up in bed with two broken legs: http://www.korthalsaltes.com/).

I thought of making a bunch of these and leaving them around work as a mystery-but my boss (who’s also a friend) saw one tiny one and instantly knew it was me; I mean *who* else would leave something so odd behind in the library. I did see a mixed up Rubik's Cube on a professor's desk and quickly solved it at about 9:30pm. I never told anyone that. He must have just walked in and wondered who solved it for him. 






By the way, I made my first paper polygon at work. My job left me alone in our library until 10pm. I also worked mornings on Saturdays. Saturdays were very mellow (usually) and I had time for experimenting with secret little paper foldings. It was very peaceful.






The 5 Platonic Solids and a few other, slightly more exotic paper polyhedrons that I made for a math display table for library help/tutoring. If you do something you like-bring it into your work if you can.









Twisted pyramid. The shape to the left is a great representation of a crystal mineral shape. I collect minerals and I've grown crystals in the past. This site provides paper diagrams for folding complex (and many times non-symmetrical) crystals out of paper : http://webmineral.com/help/Forms.shtml which is just awesome. There's a similar one called 'The Great Icosahedron' that I may make next, Charlie Brown!

This bad boy has a removable cube hidden inside! I made this about 20 years after I made my art boxes for a three dimensional design course at Wayne State University.

I got interested in how the big ball bearing (convex) looked like a concave dish when looked at through the pipe hole, but like the (real) sphere from the side. Little worlds with weird optics.

Shapes moving and transforming also interest me.

The above video (11 secs) of me inflating a rounded-cube-polyhedron super-ellipsoid origami thing. Basically a cube with bulged out sides. Colorized it in Windows Live Movie Maker so it would be less boring. These are, in a weird way, kind of like the ever-changing flexagons (little paper math machines, kind of like flat paper Rubik's Cubes) which I'll cover in another post. Folding carefully--and then blowing into a corner inflates the shape with no visible holes.

I've always enjoyed patterns. During my Fine Arts undergraduate stay at Wayne State University I made a couple 8mm films and lots and lots of photographs of patterns in nature (flowers, rain, snow falling, boiling water close-up, wrought iron, things through a home-made macro-bellows high magnification system, etc.).

Years ago I used some spare lenses out of a broken CD player to make a magnifying diopter for the front of my camera. I make close-up photos of security paper with it. Security paper has patterns like the above printed inside envelopes (so you can't hold it up to a bright light and see the check inside) or printed on checks or other documents to make reproducing them (counterfeiting) much more difficult. 

I have a file (and photographs) of hundreds-upon-hundreds of security paper motifs like the one above.

Much like Iron Man gets his magic powers from, uh…actually I can't remember what his deal is…anyway, the rain gives me complex barometric migraines *and* Master-Level Chess playing ability! Also I have a chess dictionary on my desk. 9 moves and I smashed the computer at level 2000 with my ‘Wall of Pawns’ strategy! Basically through simple patterning I've solved the problem of chess: no one ever need play it again. Actually this strategy worked a few more times then I think the game ‘learned’ and I went back to being terrible at chess. I get caught up in the patterns of the moves.

I even see patterns in my coffee!

Patterns in the clouds over my house! Patterns tend to signify things. Science explains those things. 

From an early age I was exposed to Psanky (wooden Ukrainian Easter eggs) and highly patterned textiles in the form of kilim rugs. These are just a few--I have almost 100 psanky.

My Typewriter Art: Fourth order Magic Squares increasing by sums by 4 (except the second one which is just the first re-ordered). Sums of each horizontal, diagonal and vertical row or column are the same within each square. Essay Onion Skin paper. I started doing these in the 1970s on an early electric typewriter. This is a much later one from 2010, inspired in part by an old Manuel Morschopoulis style I typed up years before I started dating things:

I just discovered this trial magic square tucked into a box of Eaton's Essay Onion Skin typewriter paper (my favorite). I must have found it in 2010, used it to make the previous math square art and then tucked it away and forgot about it.

I also make scribble drawings...but with one big rule: It's a single line that cannot touch! I've done hundreds of these in boring lectures, waiting rooms, etc.

My paper buildings span from ancient obelisk, to a farmhouse, German guesthaus, 19th century bridge, 20th century skyscrapers and ultra-modern clock tower from Dubai. Why? Why not! Extreme right: Grand Rapids Amway Grand Plaza Hotel and the Renaissance Center Building. My favorite is still the Seagram Building (center/black). At center is the famous Dubai Clock Tower (thing that looks like a white spider). They're a simpler (usually) version of the more advanced paper polygons...plus they're cute!

It's the Zur Glashütte Guesthouse in Crottendorf Germany. So tiny. So cute!

Skip down to the kitty if the following math blurb bores you.

When Grigory Perelman gave his lecture solving the Poincaré Conjecture he visualized a drop of water rolling downhill: it was a variously curved surface, but in places as it rolled it could break off or close completely. He got rid of these pinches by using mathematical tricks—but what if the universe wasn’t a repeating non-infinite sphere, or a torus (donut shape), but a blob-with another blob that was separated by ‘non-existence’ that was unreachable from the blob we’re currently inside (torus, 3-sphere or other)? If you flew in a space ship off the edge of one blob, you’d show up at the opposite side of the same blob-never entering the other blob(s). Kind of like the game Asteroid where you fly off one side of the screen and show up on the other side (you’d never make it onto your neighbor’s TV screen). 

If any loop can be shrunk to a point on a manifold sphere-then it is ‘simple’; a 3-sphere, like the Earth or a tying a string around a greasy watermelon: the tighter you pull the knot, the more it slips down and off—making a single point (the knot off the watermelon); meanwhile if you try it with a torus (think donut) and you tie it around the center you can make a donut necklace, and no matter how tight you make the chain, the donut will not slip off, unlock when you try to tie a chain around a baseball or watermelon or other spheroid-thus the torus is not simple. There was ONE historian who LIED and said that people thought the Earth was flat; untrue, for over 3000 years people have known the earth was roughly spheroid. 

The argument Columbus had about going around the world was NOT about falling off the edge-it was about who big a circumference he’d have to navigate: Queen Isabella’s people thought anywhere from 18,000 miles to 26,000 miles around (it’s a smidge under 25,000 at the bloated equator). Now, if you took every available map of the Earth at the time you could assemble them into a sphere (paste the flat paper maps onto a globe), but you could have also pasted them onto a Torus (donut shape) or a globe that looked a bit more like a pear-in fact our Earth bloats out at the equator due to centrifugal/rotational force, much like how a dragster’s rear wheels get all elongated and huge vertically when the drover whomps on the gas. Columbus worried that at the top of the pair the circumference was less than the bottom of the pear—and thus run out of supplies on his voyage. Columbus worried about a smart thing--not the dumb thing your teachers thought.

Okay, wake back up: the fancy-smanshy maths are over!!!!!!!! 

BTW, The cat is called Harbooz which is Ukrainian for melon, watermelon, cantalope...she is very jealous if anyone carries a melon into the house and must be calmed by cradling her just like she was a melon being brought in with the rest of the groceries.

Patterns and little private worlds in paper, math, writing, coding, etc.

I've written about patterns and also creating little worlds. 

This is my train set, it's T-Gauge. T = 3mm between the left and right wheels (axle length). It is 1:450 scale. This is not to be confused with TT-Gauge which is way bigger! 






There are little people.

There are even smaller people.

These are (huge by comparison) Z-Scale train people next to a penny.

So tiny.

I used them to create a paracosm: a very detailed imaginary world in your mind. Only I made mine with digital photographs and little train people and household objects. 

Above is my favorite in my Paracosm Series: the huge head/idol is about the size of a tennis ball. It was a failed/practice mold in Plaster of Paris I made from some architectural salvage I was reproducing. Kind of an Indiana Jones sort of feel. I get the same feeling from playing with patterns and math and science, etc.

TRULY SECRET WORLDS: CRYPTOGRAPHY

 

People are encrypting the dumbest things these days-but it's always fun to hit the button and feel like James Bond, if only for a second. Cryptography (and decoding) follow algorithms and patterns. I've kept notebooks since getting a secret code book (which I still have) as a very young child.

In 1401 the Duke of Mantua Leon Battista Alberti invented a frequency blurring cryptographic algorithm. In 1999 two books were published with an error in the plaintext portion of that code. Tonight I found *both* of them. Basically an 's' was were a '5' should have been in the ciphertext, so that the word "Deepfreeze" was decoded (by me) as "Neepfreeze". 

Frequency blurring is where a letter can be coded as more than one symbol:

t = x, 

q or 3; 

h = y, 

i or 9; 

e = z, 

m or 22

This way you can just count the number of times a symbol was used and match that to a simple chart of English Letter usage frequencies (where 'e' is the most used letter, etc.). I've already broken 3 other codes in these books using simple frequency counting. You count the usage of the symbols and list them from most used to least and then line it up against the English (or whatever) frequency chart--in single substitution it's usually only 1 or 2 letters off and very easy to decode.

ASINTOER (a sin to err) are the most frequent letters, along with 'the'. If you only substitute each letter of your plaintext with one single ciphertext (t = x) then the code may be easily broken by looking for groups of three letters (xyz) and assuming they are the word "the". Also, the most frequent letter 'e' is probably what the most frequent ciphertext letter (or symbol) is. x=t, y=h, z=e. You count the frequency of every letter/symbol in the code and then work asintoer/the/of/a/I/etc-the frequency of letter/word occurence in English (and other languages) is known and available in simple charts.



Not surprisingly, when I was an undergrad at Wayne State University and spending lots of time in their photographic darkrooms I got interested in spy cameras, properly called subminiature  cameras. I also always wanted one since seeing (what I think was) a Minox subminiature spy camera being used in an episode of Laverne & Shirley. I think they were trying to steal the secret recipe from a rival brewery or something. LOL!

Anyway, as you can see above-I got a Minox spy camera. It's from the Cold War Era. The bumps on the chain correspond to settings on the focus dial. You put a document on a desk; hold the camera above it; count the bumps and set the focus dial to the corresponding settings: easy copying of top secret documents! This is the famous camera that advances the film by holding it like a pair of binoculars and pulling/pushing on the left and right sides: click-click. It is a fine piece of machinery, like a Rolex watch. I even obtained a film slitter for it to cut 35mm film down to 8.5mm size.

This is a teeny-tiny subminature camera from the Cold War era. It came with a dozen rolls of 8.5mm film. Penny shown for size. Nice faux-alligator leatherette covering. I like how the (original!) box seems to imply that, 'yes, you too can create something as colorfully saturated and blurry as the Zapruder Film!!!" Actually it seems to show an Asian car stopping at a security gate. It i very light and cheap feeling but cool. It's like a neat digital watch versus a Rolex.


At some point I'll post about: my crystal radio sets made with old razor blades and pencil leads (I even made my own pencil after reading a book about the history of pencils); shortwave radio, CBs and my Rubiks Cube and Knock-off Rubik-like puzzle collection.

 

Well, that's the tail end of the snake--good night!