Colossal Gardner, ch. 17

The next chapter on Symmetry focuses on Maurits Cornelis Escher (1898-1972). Gardner starts out by saying that there’s very little (as of that time) in the way of artwork based on mathematical concepts, ignoring the simple stuff using geometric patterns like cubes or triangles. The only real exception is with the works of M.C. Escher, who had said that he more closely associated with mathematicians than with his fellow artists. The rest of the chapter gives examples of the tie-ins to various branches of math with specific Escher prints (crystallography and the flying birds, symmetry and Day and Night, impossible objects and Belvedere). And, I mentioned Penrose’s “infinite staircase,” which was the inspiration for Ascending and Descending.

(All rights belong to their owners. Images used here for review purposes only. Escher’s Knots.)

Gardner comments that Escher’s fascination with topology can be seen in Knots. The top two knots are mirror-image trefoils, with the top left being “made up of two long flat strips that intersect at right angles.” It was given a twist so it can either be a one-sided band that runs twice around the knot, or two distinct, intersecting Mobius bands. The large knot at the bottom is a quarter-turn Mobius band.

I’ve long been fascinated with Escher’s works, and I once tried learning to draw by copying a few of them. More recently, I did kiri-e (the Japanese art of using cut paper) versions of two of his woodcuts.

Puzzle for the week:
What’s going on with Three Spheres, below?



When I first started going through the Gardner Colossal book, I made notes of things that I wanted to follow up on, or include in the blog entries. One of which was David Cope’s EMI (Experiments in Musical Intelligence). I’m not sure I’ll be able to do much with this program in the short-term, so I wanted to put the link in the blog now, in case anyone wants to check it out.

Kim Answers for Wednesday

Answer for this week:
For the queen problem, the maximum number of queens to attack each other for n=4 is 20. Can you place the 20 queens on a regular chessboard for n=4?

Colossal Gardner, ch. 16

The Amazing Creations of Scott Kim is dedicated to Scott Kim, a puzzle designer and computer game designer. He studied under computer guru Donald Knuth at Stanford University. I mentioned before that there was a shopping complex in a revamped factory building in St. Paul, called Saint Anthony Main, that had a very eclectic bookstore that I’d visit while waiting for a table to open up at the Guadalaharry’s Tex-Mex restaurant. One of the books that I got there one day was Scott’s “Inversions,” (1981) which was my first introduction to his work on making text fonts (referred to as “ambigrams”) that could be read in more than one direction.

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

The chapters in Gardner’s book generally concentrate on one specific math concept, such as spherical ball packing, or infinity, but there are a few articles that focus on people, including Kim, Escher, Penrose and Conway. In this chapter, Gardner starts out talking about left-right and up-down mirroring of various text strings, and how a so-called “symmetrical calligraphy” can be achieved through a 180 degree rotation by stretching the letters as needed. The person he claims has taken the art to new heights is Scott Kim. Additionally, a few of Kim’s logic puzzles had been printed in Gardner’s SA column before, including the problem of placing chess knights on the corners of a hypercube.

(Scott’s invertible alphabet.)

Several pages discuss Kim’s ambigrams, and the fact that Omni devoted a page of their Sept. 1979 issue on them. Readers submitted their own ambigrams and Kim acted as the judge for the contest, with the prizewinners appearing in the April 1980 issue.

(More of Scott’s artwork.)

Eventually, Gardner switches to two different math problems Kim worked on. The first is the question of how many chess queen pieces you can put on a chessboard so that no queen attacks another, known as the Queens Problem (it’s a specific case of the more general mathematical chess problems). Kim switched this around to “what’s the maximum number of queens you can have on a board so that each one attacks exactly “n” other queens. If n=0, you have the original problem. For n=1, the max is 10.

(The Queen’s problem, solutions for n=1, n=2, and n=3.)

The second problem is one of how to fill space with snakes. A “snake” is a single connected chain of identical unit cubes that are joined at their faces such that each cube is attached face-to-face with exactly 2 other cubes, except at the cube at the end of the chain. A snake can twist in any direction possible as long as no two cubes abut the face of any other cube in the chain except with its immediate neighbors. A polycube snake can be finite, with two ends; finite and closed with no ends; infinite with one end; or, infinite and endless in both directions. What’s the smallest number of snakes to fill all space? For 2D space, the answer is “2 snakes.” As of the publication of this book, no one had yet found an answer for 3D space.

(Filling a space with 2 snakes.)

With the above queen problem, the maximum number of queens to attack each other for n=4 is 20. Can you place the 20 queens on a regular chessboard for n=4?


The Godelian Puzzle Book

(All rights belong to their owners. Images used here for review purposes only.)

The Godelian Puzzle Book: Puzzles, Paradoxes and Proofs, by Raymond Smullyan.
I first encountered the works of Raymond Smullyan in the book What is the Name of this Book? He started as a magician, then turned to mathematics with a degree from the University of Chicago in 1955, and a Ph.D. from Princeton in 1959. Several of his books are designed to teach the math principles of logic, and as such are great primers for this subject. The problem, as I see it, is that many of the readers that did get his puzzle books never actually learned anything from them, and were turned off by the apparent repetition of the puzzle set-ups. I can see that, having read two of his books back to back in the mid-80’s, and then not continuing on with his other titles. On the other hand, I liked his sense of humor, and I think he was a funny writer. Unfortunately, he passed away in February at age 97, and now I’ll never have a chance to meet him in person.

The current title was released in 2013, and was written as an introduction to Kurt Godel’s incompleteness theorems. The first few chapters set up the concepts of logic, infinity and paradoxes, many of which mirror the related chapters in the Martin Gardner Colossal Math Book. In fact, Gardner mentions Raymond in relation to Newcomb’s Paradox, and Raymond specifically talks about Martin’s interest in Newcomb’s paradox in this book. Again, everything is connected, and I love it

There’s one particular joke here that I want to repeat. “A certain man was in quest of immortality. He read many occult books on the subject, but none of them gave him any practical advice on how to become immortal. Then he heard of a certain great sage of the East who knew the true secret of immortality. It took him twelve years to find the sage, and when he did, he asked, “Is it really possible to become immortal?” The sage replied, “It is really quite easy, if you just do two things.” “And what are they?” the man asked quite eagerly. “First of all,” replied the sage, “from now on, you must always tell the truth. You must never make a false statement. That’s a small price to pay for immortality, isn’t it?” “Of course!” was the reply. “Secondly,” continued the sage, “just say ‘I will repeat this sentence tomorrow.’ If you do these two things, I guarantee you will live forever.” After thinking for a few minutes, the man said, “Of course if today I truthfully say that I will repeat this sentence tomorrow, then I will indeed say it again tomorrow, hence again the next day, and the next and the next and so on, but your solution is not very practical. How can I be sure of truthfully saying that I will repeat this sentence tomorrow if I don’t know for sure that I will be alive tomorrow? Your advice, though interesting, is simply not practical.”
“Oh,” said the sage, “You wanted a practical solution! No, I deal only in theory.”

The second half of the book is more textbook oriented, discussing the actual principles of mathematical logic theory and introducing symbolic logic. And, this is where my eyes started glazing over. If you actually want to sincerely learn the proofs, and you sit down with a pencil and paper and write out the symbols consistently to learn what they represent, coupled with the nomenclature of the proofs, then yes, this is a good, useful book. But there isn’t much in the way of real world examples to link the theory to practical application, so I got lost. I mean, I understood the basic concepts, and I could read some of the expressions in the proofs, but I’d never be able to explain the proofs to someone else at this point.

(Case in point.)

But, the main point is that in mathematics, mathematicians want to create specific types of math with clear rules that either hold, or don’t hold, and can be proven one way or another. Examples would be, in the world of natural numbers (positive integers starting with 1): that even numbers are divisible by 2; odd numbers leave a remainder of 1 when divided by 2; when adding two numbers the order doesn’t matter (2 + 3 = 3 + 2 = 5); if you add two numbers to get a third value, you can use either set when adding a third number (5 + 7 = 2 + 3 + 7 = 2 + 3 + 2 + 5 = 2 + 3 + 2 + 2 + 3); etc. So, we have these math guys trying to prove that the true statements are true, and that the false statements are false, and along comes Kurt Godel with a paper in 1931 proving that in any math system, there will be statements that you can not prove are either true OR false using the rules of that system. You CAN prove the statements one way or another if you analyze the one system using the next system up, but that becomes a meta-analysis, and that’s where logic theory steps in, with a vocabulary that groups statements, formulas, variables and sets together to generalize the system you want to look at. There is something of an example showing how this analysis works involving the Peano axioms as applied to the natural numbers, but that’s at the end of the book, is kind of short, and isn’t actually explained in simple words.

If you look at Smullyan’s puzzles, they often take the form of Knights and Knaves. The knights always tell the truth, the knaves always lie. In some cases, he introduces “certified,” where the status of the knight or knave has been established (or “uncertified” if it’s not established). And, we also get “magicians,” who can be either knights or knaves, but they don’t actually do anything different. Examples of these puzzles are:

1) You’re walking along a road and you get to a fork. One tine of the fork goes to the town of Knights, and the other goes to the town of Knaves. At the fork are two people, A and B, one is a Knight, the other is a Knave. What question can you ask one of them to determine which road goes where?

2) You meet three people (A, B, C), one of whom is a magician. They tell you:
A) B is not both a knave and a magician.
B) Either A is a knave or I am not a magician.
C) The magician is a knave.
Which types are all three people, and which one is the magician?

3) You find an island of certified and uncertified knights and knaves. You meet someone who says “I am not a certified knight.” Is he/she a knight or a knave, and certified or uncertified?

What I found ultimately useful in this book is that Raymond finally explains his puzzles’ tie-ins to logic theory.
1) Knights represent statements that are true.
2) Knaves represent statements that are false.
3) Certified means that the statement is provably true or false.
4) Uncertified means that the statement is undecidable. You can’t prove it is either true OR false.
5) “Magician” represents some property of the statement, e.g. – that natural whole numbers can be even, odd or prime.

So, you could have a statement regarding the natural numbers that is true about a specific property and still not be able to prove it. The rest of the book then goes on to develop the vocabulary and theorems for how to show Godel incompleteness for any system, including logic theory itself.

Overall, I like this book, and I now have a slightly better idea of what Smullyan was trying to demonstrate in his earlier puzzle books. I would have preferred more examples of unprovably true or false statements in number theory, because I do want a practical answer. Also, there are quite a few obvious typos in the book (misspellings, repeated or missing words, and the use of “proportional logic” instead of “propositional logic” on page 256) that were distracting but otherwise non-malicious. This is not a simple child-friendly introductory primer to Godelian theory. But, it could be good for math undergrads. Recommended if you like Smullyan’s other books.

Answer 1) If you ask A, say something like “If I asked B ‘does the left fork goes to the town of Knights?’, what would he say?”
Answer 2) A and B are both knights, C is a knave, and A is the magician.
Answer 3) Let’s call the person “A”. If A is a knave, they are lying and then HAVE to be an uncertified knight. But, knights don’t lie, so A can’t be a knave that is lying. However, if A IS a knave, certified or uncertified, then they are telling the truth in not being an uncertified knight, and knaves don’t tell the true. Alternatively, if A is a knight, who always tells the truth, then there is no option but for A to be an uncertified knight. (That is, A is a true statement that can’t be proved.)

Answer Rotations

Answers for rotation and reflection:
1) Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

Rotated 180 degrees, the plate reads “OLLIE LEE”.

2) A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

x = 9 and y = 9. x + y = 18. x * y = 81. Turn 18 upside-down and you get 81.

If the problem allowed any number of eggs, then x = 3 and y = 3 (6 and 9) would also work.

Colossal Gardner, ch. 15

We’re now entering the section on Symmetry, starting with the chapter on Rotations and Reflections. First, from a geometrical viewpoint, something is considered to be symmetrical if it looks the same (its physical characteristics don’t changed) if you perform a “symmetry operation” on it. That is, if you hold a mirror up to the capital letter “A” at the halfway vertical point, it’s unchanged. “A” is said to have vertical symmetry. If you hold the mirror horizontally halfway up the letter “B”, you’ll see it has horizontal symmetry. The letter “S” is the same if you rotate it 180 degrees (twofold symmetry). “H,” “I,” “X” and “O” have all three symmetries. If the letter “X” is written as a cross, it’s unchanged if you rotate it 90 degrees (fourfold symmetry). The letter “O” is considered to be the “richest” in symmetry because it’s unchanged whatever you do to it.

The entire chapter is then dedicated to giving other examples of rotations and reflections. Gardner states that because the Earth is a sphere, with a center of gravity that pulls everything on it downward, living things have developed strong vertical symmetry, but very little horizontal or rotational symmetry. Things humans have created reflect this bias towards vertical symmetry – just look at how chairs, tables, dishes, cars, planes and office buildings are designed. Most normal representational art is similar in this respect. The exception is completely non-representational art, and in some cases museum curators have hung abstract paintings upside-down and no one noticed for months. Can’t do that with the Mona Lisa without realizing the mistake pretty quickly.

Pictures that look one way when right-side up, and something else when rotated have been popular tricks for political cartoonists since the 1800’s. A couple more recent examples were used by Life magazine. In the Sept. 18, 1950 issue, a reproduced Italian poster had the face of Garibaldi, and when turned over showed the face of Stalin. The back cover of the Nov. 23, 1953 issue showed an Indian brave inspecting a stalk of corn, and upside-down was a man looking hungrily at an open can of corn. Gardner mentions Peter Newell (1862-1924), who published 2 children’s books of color plates of scenes that transform when rotated: Topsys and Turvys and Topsys and Turvys Number 2. And there’s !OHO!, by Rex Whistler.

(Image from Amazon UK, used for review purposes only.)

Probably one of the more impressive displays of this kind of art is Gustav Verbeek’s The Upside-Downs of Little Lady Lovekins and Old Man Muffaloo, a weekly strip that ran from 1903 to 1905, for a total of 64 strips. You read the first 6 panels normally, then turn the page over and read the same panels again to finish the story. GoComics carries the reprints now.

(From GoComics)

Another example of pictures changing when you rotate them 90 degrees is the rabbit duck.

(Image from the wiki article on ambiguous images.)

Plus two landscapes by German painters from the Renaissance.

Salvador Dali illustrated Maurice Sandoz’s The Maze, which supposedly has several plates where turning the image causes it to look like something else, but I can’t find any example artwork on the net showing this.

Finally, a couple puzzles.
Oliver Lee, age 44, lives at 312 Main Street. He asked the city to give him the license plate 337-31770 for his car. Why?

A basket contains more than 6 eggs, which are either white or brown. If x is the number of white eggs and y is the number of brown eggs, then what values are needed such that the sum of x and y turned upside-down is the product of x and y? That is, how many eggs are there?

Penn and Teller Magic and Mystery Tour

(All rights belong to their owners. Images used here for review purposes only.)

Penn and Teller’s Magic and Mystery Tour (2003)
I’ve seen bits and pieces of this DVD on youtube, and it really is better to watch the DVD itself. This was a 3-part TV mini-series directed and produced by the Canadian Broadcasting Corporation, along with Channel Four Film. The producers had wanted to do a piece on street magic around the world, and they asked Penn and Teller to act as hosts, as well as to actually visit other countries and interview the magicians they met. The countries selected were: China, Egypt and India. Each episode runs about 50 minutes, and consists of local color footage, footage of various street magicians performing, a couple of interviews, and Penn and Tell performing a couple tricks for the street magicians. The DVD extras section has video that didn’t make it into the shows. For China, that was Penn leading the village kids in a cheer (“Penn and Teller Rule”); an Egyptian that eats razor blades and glass; and an Indian that does a bullet catching routine.

China was kind of a special case, because the People’s government outlawed street performances, so most of the performers were working either in tea houses, failing government-run amusement parks, or small theaters. I’ve seen the one guy that does the mask switching dance here in Kagoshima, and I loved watching him again on the video. But, the rest of the Chinese performers were great, too. The Egyptian acts alternated between the cups and balls, and snake swallowing. I liked the primary cups and balls guy, but Penn’s commentary for everything happening in Egypt struck me as unnecessarily shallow. The Indian magicians, on the other hand, have absolutely mastered the geek act, cutting off bits of their children, killing them and then bringing them back to life. Disturbing, in a way. But, still a great DVD to get if you like Penn and Teller.

On a side note, I got this DVD as part of a birthday gift, shipped to me in Japan. When I received it, the package was neatly wrapped in gift paper, but it rattled when I picked it up. Removing the wrapping showed that it was indeed a DVD, but generally when you shake DVDs they don’t rattle like they’re in multiple pieces. At least, not if the idea is to put the pieces in a DVD player and actually play them. So, I was a bit hesitant to open the clamshell, but it was ok. The clamshell itself was so old that it had gotten brittle and was self-destructing. I took this to be a form of commentary on the DVD itself.

How to Play With Your Food


(All rights belong to their owners. Images used here for review purposes only.)

Penn and Teller’s How To Play With Your Food (Villard Books, 1992)
If you have read Cruel Tricks for Dear Friends, you know what you’re getting into here. How to Play With Your Food is both an instruction manual, and an introduction into the world of magic. Penn and Teller take turns describing different tricks and scams you can play on your friends, or people you don’t like, that all involve food in one way or another. From how to get free meals at restaurants to setting people up for reveals on card tricks, from the Glowing Pickle Machine to descriptions of how they pulled off tricks on the Letterman show.

There’s no point in actually reviewing this book, or describing the tricks. If you want to know what they are and how they work, buy the book. But make sure you get a new copy that comes with the goodies packet. Previous owners of used copies probably tore out the goodies packets and played with them before listing the book on the reseller sites. It’s an old book, released in ’92, but still a great read at 210+ pages. You get to learn how to make desserts bleed, and shove watches into fish. What else do you want?

But for me, the greatest thing was in seeing that one of the tricks in here (Cutting Off Your Thumb (Not really)) was suggested by Martin Gardner. P&T mention Martin twice in the book, plus in the Thanks section, saying that they were fans of his. It’s all connected.

Note, if you look at the cover, you’ll see 47 of the 49 foods mentioned in the book, and there’s a diagram next to the table of contents with numbers over each of the foods to show what chapters they’re from. Are these guys helpful, or what?

P.S. – If there’s anyone reading this that lives in Minneapolis, do me a favor. See if the Palomino Euro-Metro Bistro is still in business. They used to be in LaSalle Plaza, at 825 Hennepin Ave. If they are, check with the staff and see if they still remember the Penn and Teller James Bond card trick. Then get back to me with what you find out. Thanks.

Colossal Gardner, ch. 14

The last chapter for Solid Geometry and Higher Dimensions is on Non-Euclidean Geometry. Gardner starts by talking about Euclidean Geometry, and how Euclid was smart enough to recognize that his fifth postulate was different from the first four. That is, the statement “through a point on a plane, not on a given straight line, only one line is parallel to the given line” can not be proven without relying on some other approach that is derived from the parallel postulate. That is, if you assume that the sum of the angles of every triangle equals two right angles, you can not prove this assumption without using the parallel postulate as well.

Gardner goes on to talk about Hungarian mathematician Farkas Bolyai, who, in his attempt to prove the parallel postulate in the early 1800’s, not only realized it was independent of the first 4 axioms, he went on to show that a consistent geometry could be created assuming that through a point you could have an infinite number of parallel lines. Unfortunately, Farkas waited too long to publish his work, and Nikolai Lobachevski beat him to it. Even Farkas’s friend, Gauss, claimed to have worked out the same findings several years earlier and never bothered to publish them. Then there’s Italian Jesuit Giralamo Saccheri, who worked out both forms of non-Euclidean geometry in a Latin book published in 1733, titled “Euclid Cleared of All Blemish.” The problem was that claiming that non-Euclidean geometry was as true as the Euclidean form would have been very dangerous at that time, so he published the book while simultaneously denouncing his own findings

If the parallel postulate is altered to have infinite parallel lines through a point, you get a hyperbolic space where all triangles have sums of their angles that add up to less than 180 degrees, and the sum decreases as the triangle gets bigger. The circumference of a circle is greater than pi times the diameter, and the measure of curvature of a plane is negative. The other type of geometry, called “elliptic,” was simultaneously developed by Ludwig Schlafli and Bernhard Riemann (yes, the zeta function Riemann). In this geometry, NO parallel line can be drawn through a point to the given line. Here, the sums of the angles of the triangle add up to more than 180 degrees, and the circumference of a circle is less than pi times the diameter.

One of the last mathematicians to doubt non-Euclidean geometry was Lewis Carroll. H.S.M. Coxeter is quoted as saying “It is a strange paradox that he, whose Alice in Wonderland could alter her size by eating a little cake, was unable to accept that the area of a triangle could remain finite when its sides tend to infinity.” Gardner presents M. C. Escher’s Circle Limit III woodcut as an example of what Coxeter was talking about for hyperbolic space. In fact, Escher based Circle Limit III on a 1957 paper on crystal symmetry that Coxeter wrote and sent him.

(Circle Limit III, from the wiki entry. All rights belong to their owners. Images used here for review purposes only.)

Note that in this woodcut, the lines running through the fish are perpendicular to the circle’s edge. If you walked along one of these lines, every fish would be of exactly the same size. The thing is, you’d be getting smaller as you got closer to the edge of the circle, as would any measuring stick you used to check the size of the fish. You’d never get to the edge of the circle, because there is no “edge” from your perspective. Both you, and the fish, just keep getting infinitely smaller as you go (George Gamow talked about curved space in exactly this way in one of his books).

Elliptic geometry then can be modeled on the surface of a sphere. Euclidean straight lines become great circles, and no two are parallel. Einstein adopted a generalized form of this geometry, formulated by Riemann, to show that the curvature of physical space “varies from point to point depending on the influence of matter.” According to Gardner, one of the greatest revolutions in physics that came about from the General Theory of Relativity was that physics can be simplified by assuming physical space to have an elliptic structure.

The rest of the chapter is dedicated to “cranks” that were convinced that Einstein was wrong, that you can’t have non-Euclidean space, and that they were sure they’d proved Euclid’s parallel postulate.

Challenge: Prove the parallel postulate.
It’s ok, I’m willing to wait.