It’s been a while since I made any papercrafts, and I started jonesing for something while I was in the middle of a short break in a contract translation clean-up project. So, I went to the Canon papercrafts site and downloaded the PDF for the Subaru telescope.  The 300:1 scale model is based on the Subaru on Mauna Kea, in Hawai’i. I had some sheets of 0.15 mm and 0.22 mm paper left over from previous projects, and I debated over which to use for the telescope. 0.15 is a little stiffer than regular printer paper, but has the benefit of being easier to fold. 0.22 is getting closer to card stock and stands up better to stress, but is more likely to leave white lines all over given the thickness of the paper compared to the sizes of some of the model details. In the end, I settled on 0.15 mm, but it might have been smarter if I’d printed the PDF twice, once on each set of sheets, and then cherry-picked the correct stiffness depending on the section of the model I was working on.

I didn’t time myself, but I think I spent at least 15 hours over the course of 3 days building this thing. The most time-consuming activities were for cutting out all the pieces, and then gluing the circular walls to the round flats. The instructions suggested gluing one tab from the walls to the disks at a time, then letting it dry before going to the next tab. This is important for keeping the walls vertically straight, and for ensuring that the end of the wall strips will meet up evenly at the end. But, that was boring, and actually it’s kind of difficult to get the paper to curve right to follow the edges of the disks that way. So, I’d glue 3-5 tabs at a time, depending on the sizes of the tabs, and that worked out ok. The toughest sections to work on were the tiny buildings, and the mounting legs for the telescope stage. My hands kept sweating, making the paper soggy, and causing it to not hold the crease lines. Even so, I think the model turned out better than it had any right to.

The finished model is about 6″ tall, and a little under 7″ at its longest. The big shutters do slide open and closed, but the paper for the guide fingers is thin enough that it can bind up, so I just want to leave the shutters open all the time. The top half of the building comes off to reveal the telescope inside, and the scope and main dome can rotate 360 degrees. The telescope can also rotate up and down, but again, the paper is thin enough that I don’t want to handle the scope so much that something tears. If I were to build this papercraft again, I’d want to go at least one grade thicker on the paper.

But, overall, I’m glad I got this out of my system. I’ll show it off to some of my students, and then decide what to do with it long-term. Recommended to astronomy buffs with a lot of patience and dry hands.

(From the Google Doodles)

This is a nice twist on the theme, from Rhymes With Orange.

And now we reach the last chapter in the book – Six Sensational Discoveries. This is another April Fool’s article that ran in the April 1975 issue of Scientific American, ostensibly as a recap of the top scientific discoveries that “escaped” the attention of the scientific community and the public at large. Martin comments in the addendum that although he’d thought he’d made it pretty obvious that these are all jokes, the editors received over 1,000 letters from people that took them seriously.

(All rights belong to their owners. Images used here for review purposes only. Exploding the four-color-map theorem.)

The first “discovery” by William McGregor is of a map that cannot be colored with fewer than 5 colors. (Some of Gardner’s readers succeeded in coloring the above map with 4 colors.)

The second discovery was in number theory, where e raised to the power of pi times sqrt(163) is an integer. Ramanujan had speculated that it would be, but most calculations by hand were too tedious and no one made it past twelve 9’s after the decimal. John Brillo supposedly applied Euler’s constant to prove the number is exactly 262,537,412,640,768,744. Actually, Ramanujan knew that this number is transcendental, and the next digit after 262,537,412,640,768,743.999999999999 is 2. The name Brillo is a play on John Brillhart.

In the computer sciences, Richard Pinkleaf designed a computer with the help of ex-world-chess champion Mikhail Botvinnik, designated MacHic because it plays like it’s intoxicated. MacHic is a learning machine, and is able to play 1 game against itself every 1.5 seconds. After 7 months of self-play, MacHic declared that pawn to king’s rook 4 is a win for White. Kissinger and Brezhnev were to meet to discuss the impact on world chess, and Bobby Fischer offered to play MacHic as long as it is silent during the game, and he’s guaranteed $25 million, win-or-lose. Gardner writes in the addendum that MacHic is a play on Richard Greenblatt’s MacHack.

(Special relativity – busted!)

For the physical sciences, Humbert Pringle discovered a fatal logical flaw in the special theory of relativity. In his thought experiment, a meter stick is flying through space at a speed so that it is Lorentz-contracted by a factor of 10. Simultaneously, we have a plate with a one-meter diameter hole, and the plate is traveling perpendicular to the stick, so that the two will intersect when the stick is centered in the hole. Both the stick and plate are idealized to have 0 thickness. If the plate is the inertial frame, the stick will be contracted to 10 centimeters and will easily pass through the hole. But, from the stick’s inertial reference frame, the plate’s hole will have a width of 10 cm, and the stick won’t be able to pass through. “The two situations aren’t equivalent, and thus a fundamental assumption in special relativity is violated.” For this one, Pringle is a play on relativity denier Herbert Dingle, and the solution is included in George Gamow’s Mr. Tompkins in Wonderland.

(da Vinci taking a break.)

Discovery five is of two of Leonardo da Vinci’s “lost” notebooks, which include a series of ball bearings surrounding a conical pivot similar to the Sperry Gyroscope invented in the 1920’s; a worm screw; a bicycle with a chain drive, and the first valve flush toilet. (The English patent for the valve flush toilet was granted to Alexander Cummings in 1775, but da Vinci had the idea first.) Really, the artwork here was produced by graphic artist Anthony Ravielli (Ravielli died in 1997 at age 86).

(Making a psi motor.)

Last but best is the psi energy motor constructed in 1973 by Robert Ripoff (if that isn’t a dead-giveaway, nothing is). To construct the motor, cut a rectangle as shown at the top of the figure out of heavy bond paper, 3″x7″. Make two slits as shown, and roll the rectangle into a cylinder and glue the ends together. Using a file card or pasteboard, cut a strip 3/8″x 3″. Insert a needle through the center of the strip, and push the ends of the strip through the slots in the cylinder. Place the assembly on top of a bottle 4″ tall. The bottle must have a glass or hard plastic top.

(Testing the psi motor.)

Set the psi motor on a copy of the Bible or the I Ching, aligning the book’s spine due north-south. Cup your left hand (right if you’re in the southern hemisphere) as close to the motor as possible without touching it. “Make your mind blanker than usual” and concentrate your psi energy on the motor. It can take upwards of one minute for the motor to start turning. If you’re having trouble getting the motor to spin, it helps to lean in closer and breathe shallowly.

(McCullogh and the motor.)

Gardner says that the ripoff motor is a variation of a psychic motor detailed in Hugo Gernsback’s Science and Invention magazine (Nov. 1923). Pictured above is Warren McCulloch as he demonstrates the motor. Interestingly, Gernsback himself had been described as a crook and ripoff artist who often refused to pay his writers, while collecting a $100,000/year salary, so maybe we could have called the above invention the Gernsback motor.

——-

And there we have it. Personally, I’m loving The Colossal Book of Mathematics. There’s a lot of things in here that I hadn’t known before, and when I went through the book several more times for writing up these blog entries, I kept finding new things that had escaped me the first time. Additionally, I ended up reading quite a few wiki articles, and math and science web articles on these topics to see which ones had updates above and beyond what Martin wrote, or as part of the process of adding support links to my blog entries. But, I do have to put this book away now and move on to the next one (I got 6 new books for my birthday (I’m writing this in August, 2 days after my birthday) and I need to finish those before I get any more new books for Christmas. Plus, there’s a very good chance that I’m going to start up writing apps in three different languages (two of the books are for programming in Prolog and for the Android tablet, plus I want to make a decryption program in Free Pascal). I have too many things I want to do…

Final challenge of the week: Following Ripoff’s procedure, construct two psi motors, and turn them into a reciprocating motor for harnessing the free energy that surrounds all of us.

Ok, you now have 50 entries for a Gardner-a-week calendar. Please use them for good; and I definitely recommend buying your own copy of the book – you’ll be glad you did.

GEB answers:
Puzzle 1: What’s the smallest positive integer that can be expressed as the sum of two different squares in two different ways?
Answer: 50
5^2 + 5^2 or 1^2 + 7^2

Puzzle 2: Is “MU” a theorem of the MIU system?
Answer: No

The problem is that the only way to introduce “U” to the existing theorem is to either add it to the end of a string ending in “I,” or to reduce “III” to “U”. Because you can only introduce new characters by appending them to the end of the string, doing something like MI -> MIU -> MIUIU means that you can’t reduce the “I”s in the middle of the string. And doubling “I” with the “Mx -> “Mxx” rule means that “I” is going to be a factor of 2. Reducing “III” to “U” will always leave one or two “I”s at the end of the string.

Example:
MI
MII (can’t reduce)
MIIII (MUI -> MUIUI)
MIIIIIIII (MUUII -> MII)
MIIIIIIIIIIIIIIII (MUUUUUI -> MUI)

Godel, Escher, Bach.
I’ve been looking forward to this chapter. I read GEB, by Douglas Hofstadter, shortly after it got listed as a bestseller, and I absolutely loved it, both for all the information on the three top title people, and the insights into the state of AI development at the time. I’ve reread parts of the book when I’ve gotten the chance, and I still consider it a good read. The majority of Martin’s comments can be taken as a book review, and just reiterate the content of GEB as a whole. I do want to repeat his insights a little bit regarding the wood block on the GEB cover. Hofstadter made the block himself from a piece of redwood, and dubbed it a trip-let, for “triple letters.” He also reprints Scott Kim’s (creator of Inversions) “Figure” as an example of the background of a set containing the complementary information of its foreground.

The subtitle for Godel, Escher, Bach is “The Eternal Golden Braid,” in part for how the works and thoughts of the three main characters intertwine with each other, and in part because the initials can be derived from the GEB (EGB) trip-let.

(All rights belong to their owners. Images used here for review purposes only. Scott Kim’s “Figure“.)

Going through this chapter, I am finally able to appreciate Douglas’ pun character, the Indian mathematician Najunamar (Ramanujan), who proved three theorems. “He can color a map of India with no fewer than 1,729 colors; he knows that every even prime is a sum of two odd numbers, and he has established that there is no solution to a^n + b^n = c^n when n is zero.” (1729 was the number of the taxi G. H. Hardy rode in when he visited Ramanujan in the hospital, and is the smallest positive integer that is the sum of two different pairs of cubes. Note that color mapping also shows up in the next chapter.) Puzzle 1: What’s the smallest positive integer that can be expressed as the sum of two different squares in two different ways?

For the second puzzle, I have to lay down the logic first. Hofstadter has a “formal system,” which is made up of the letters M, I and U. They can be arranged in strings that he calls theorems, but only by following 4 very specific rules (taken from the stanford.edu site):

Rule 1: xI -> xIU (only if I is at the end)
Rule 2: Mx -> Mxx (only if M is at the beginning)
Rule 3: xIIIy -> xUy (any III anywhere)
Rule 4: xUUy -> xy (any UU anywhere)

In other words, if the string ends in “I”, we can append “U” to it.
We can copy-paste the string after “M” only if “M” is at the beginning.
If the string contains “III,” we can reduce the “III” to “U”.
If the string contains “UU,” we can eliminate the “UU”.

The only “axiom” is that we have to start with the theorem “MI”. Any string that we can make by applying the above rules to “MI” is a theorem of the system.

Example:
MI (start)
MII (rule 2)
MIIII (rule 2)
MIIIIIIII (rule 2)
MUIIU (rule 3)

Puzzle 2: Is “MU” a theorem of this system?

(The Two Mysteries, by Magritte (1966))

Gardner describes one section of GEB where Turing and Babbage are engaged in a form of Turing game, each accusing the other of not being real, when Douglas enters the scene and tells them that both of them are products of his imagination, as is fictional Hofstadter himself. Martin is reminded of Magritte’s “The Two Mysteries,” and the statement “this is not a pipe.” Martin asks, “And how real was Magritte? How real are Hofstadter, you, and I?” “We are back to a Godelian Platonism in which reality is infinitely layered. Who can say what reality really is?”

Indeed, who can say? And if they do, why do you believe them?

Raymond Smullyan never mentioned there being guards on those islands…

Tricky Question Guards

I’d be lying if I didn’t say there was a grain of truth to this one…

Well, that’s all the science/technology strips I’ve been wanting to highlight here.