Mathematical Zoo is a mix of real and fictional creatures that hold at least some interest for recreational mathematicians. Gardner imagines an actual zoo split up into the two sections, with a newsletter named ZOONOOZ (with permission from the Zoological Society of San Diego, which already uses this name) for announcing new arrivals.

(All rights belong to their owners. Images used here for review purposes only. Radiolaria skeletons from Ernst Haekel’s Monograph of the Challenger Radiolaria.)

One of the first entries would be visible only through a microscope – the radiolaria that live in the sea and were documented by Ernst Haekel from the Challenger expedition of 1872-76. Not only are these creatures spherical, but their claw protrusions mark the corners of regular octahedrons (8 faces), icosahedrons (20 faces) and dodecahedrons (12 faces). Martin spends some time talking about the five Platonic solids, two of which have faces that are equilateral triangles, adding that there is an infinite number of semi-regular solids that are also composed of equilateral triangles, called deltahedra since their faces look like the Greek letter delta. Eight of the deltahedra are convex, and have 4, 6, 8, 10, 12, 16 and 20 faces. Note that radiolaria #4 above is a deltahedra.

The Aulonia hexagona is accompanied by a question: Can you cover a sphere with a regular map of hexagons, three edges at each vertex? Euler proved that the answer is “no,” but many observers thought that Aulonia showed he was wrong until they looked closer and discovered that some of the cells on the thing’s surface have fewer than 6 sides.

Other real, small, creatures would include viruses that crystallize into macromolecules shaped like regular icosahedra (eg. – the measles virus), while the mumps virus is helical. For other helical structures, we’d have the cochlea of the human ear, the DNA molecule, the horns of various sheep and goats, and the devil’s corkscrews. Fish are good examples of bilateral symmetry, but animals that violate this symmetry would include the crossbill, a small red bird with a crossed beak for prying open evergreen cones; and the wrybill plover (which has a beak that curves to the right for turning over stones to find food).

For knots, we’d have the hagfish, which can tie itself into a knot, the filament-like leucothrix mucor that reproduces by tying itself into a knot, and humans that can cross their arms. Other specimens would include the four-eyed stargazer and the spiral-toothed narwhal. There are many other creatures as well, but it would be easier to scan in the pages than to type them all up.

For imaginary animals, we’d have Frank L. Baum’s Duo, Woozy, Wheeler and Ork, the curl-ups by M. C. Escher, and the Ta-Ta of Sidney Sime (Sidney is best known for illustrating the books of Lord Dunsany, but he did write one book of his own, titled Bogey Beasts). The Ta-Ta can turn itself inside-out.

“The Ta-Ta
There is a cozy kitchen inside of his roomy head
Also a tiny bedroom in which he goes to bed.

So when his walk is ended and he no more would roam
Inside out he turns himself to find himself at Home.

He cleared away his brain stuff, got pots and pans galore!
Sofa, chairs and tables, and carpets for the floor.

He found his brains were useless, as many others would,
If they but tried to use them a great unlikelihood.

He pays no rent, no taxes, no use has he for pelf
Infested not with servants, he plays with work himself.

And when his chores are ended and he would walk about,
Outside in he turns himself to get himself turned out.”

In the addendum, Martin says that readers informed him about the wheel spider, which escapes predators by extending its legs, flipping on its side and rolling away. And, of the pangolin, mammals that have keratin scales on their skin. Pangolins will curl themselves up into balls to roll down hills to avoid enemies.

Challenge: Find a pangolin backpack and give it to Alice Otterloop.

This was the second CD I received last Christmas. Here, my wording was “anything by Ryuichi Sakamoto.” My expectation was that I’d get one of his solo albums, or something from YMO, where he’d been the keyboardist/synth player. Instead, we have Soundtracks (1995), which is a sampler of some of the music he composed for the movies The Last Emperor, Merry Christmas, Mr. Lawrence, and The Revenant. It’s all orchestral, and none of it has synth work.

What’s funny is that the jacket notes are in Japanese, so I guess this is an import (and here I am, living in Japan…) Overall, the music is beautiful and atmospheric. If you like movie soundtrack music that is classically-influenced, then you’ll enjoy Soundtracks.

I think the below idea would be more interesting if you put a piece of paper with a vertical slit in it in front of the lens, and then have the slit go from the left edge of the projector lens to the right edge, and then back, producing a beam of light that travels faster than light speed across the screen. Unfortunately, for the light to be bright enough to see on the screen from Earth, the projector would cause the Earth’s atmosphere to ignite before you could finish the joke.

Movie Screen

We have finally come to the final section in the book, Miscellaneous, and the first chapter, Melody-Making Machines. This is an extension of the earlier Fractal Music (ch. 27). Gardner starts out by saying that there’s a trivial way that any work of art can be treated as a combination of a finite number of discrete elements. In essence, he’s describing digitizing those works on a computer. A poem is a sequence of digits that represents the letters in the alphabet (ASCII codes), and paintings are just RGB values. Now, say that there’s a big museum that has every combination of matrices of a maximum limiting size. Martin claims that somewhere in that museum will be every picture that has been painted, or can be painted. Could we develop an algorithm to “search on a code number for a great painting not yet painted?” (Note: This question is a call-back to Turing’s approach to mechanically generating math proofs with a Turing machine.) He then moves to music again.

He mentions Athanasius Kircher, a German Jesuit, who in 1650 published Musurgia universalis sive ars magna consoni et dissone, which was based on Ramon Lull’s Ars Magna. Lull thought “that significant new knowledge could be obtained in almost every field simply by exploring all combinations of a small number of basic elements.” Kircher viewed music as a combinatorial problem, and he described a Lull-type machine for creating polyphony by sliding columns alongside one another and reading the numbers on the rows to get a variety of combinations and permutations. Samuel Pepys owned a copy of Kircher’s book, and he had the machine built. The wiki article calls it the arca musarithmica, while Gardner says it was the Musarithmica mirifica, and that it’s now in the Pepys Museum at Magdalene College, Cambridge.

(All rights belong to their owners. Images used here for review purposes only. Cover of Kircher’s book, by J. Paul Schor. It’s hard to tell from the scan, but the angels are singing, Musica holds Apollo’s lyre and the panpipes of Marsyas, and Pythagoras is to the left with his hand pointing to his theorem.)

There’s a waltz pamphlet attributed to Mozart (but discredited), entitled Musikalisches Wurfelspiel, published in 1792 (one year after Mozart’s death) that claims anyone can write any number of waltzes they like using just a pair of dice. You can buy copies of the text or run app versions by searching on Mozart’s musical dice game.

Italian composer Antonio Calegari used two dice to compose pieces for pianoforte and harp, and the book for his system was published in 1801. The book Melographicon was published anonymously in 1805, with 4 different forms for providing music for poetry, without needing dice. The Kaleidacousticon was a card-based system that could compose 214 million waltzes. The Componium was a pipe organ invented in the early 1820’s in Amsterdam, by M. Winkel, and it could play its own compositions. The Quadrille Melodist was another card-based system that could compose up to 480 million quadrilles.

(New York City skyline set to music.)

Joseph Schillinger, a Columbia University teacher, published his system in 1940, as the booklet Kaleidophone. George Gershwin reportedly used this for Porgy and Bess. Then, we get Heitor Villa-Lobos again, who used this system with the New York City skyline for a piano composition. In the 50’s, J. R. Pierce and others tried to apply information theory to musical composition, and chemist Richard C. Pickerton released an article, “Information Theory and Melody,” which included a graph for his “banal tune-maker.” (You flip a coin to determine paths along a network.) Finally, we get into the 60’s and 70’s, and the use of synths and computer-generated music. Martin talks about Markoff chain melodies based on an analysis of Chopin’s works, but adds that as of the date of this article, that all computer-generated music is “mediocre, frigid and forgettable.”

Thomas O’Beirne of Glasgow University noted that some musical systems resemble the Honeywell Simplified Integrated Modular Prose (SIMP) buzz-phrase generator. The earliest form of this technique dates back to W. Hooper’s Rational Recreations (London, 1794). Martin then ends by saying that no one knows when, or if, computers will be able to create great works of art.

My notes: I was really hoping to find one of the rules for the above music generating systems on-line, but all I got were links to books you have to pay for, and a few simple software apps. I’ve reviewed the serialized DVD-magazine volume set for Rana for Vocaloid, and their process for generating music isn’t quite automated, but it is pretty much a cookie-cutter approach. The Vocaloid singing voice software still sounds mechanical, but for a while 1-2 years ago some of the Vocaloid composers were actually getting serious airplay on Japanese radio stations, and the vocals are very recognizably shrill. It may not be much longer before computer-generated music rivals that of human composers, but I expect that the fear of musicians losing their jobs will keep it from really catching on (that, and so many Americans are too lazy to even use automated systems for making their own music.)

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

This last Christmas, I asked for a couple CDs. In the first case, I simply said that I wanted a punk album. What I received was Leatherface’s Mush. The band formed in 1988, and broke up in 1993. They got back together after 1998, and issued 4 albums between ’98 and 2012. According to the wiki entry, The Guardian called them “the greatest British punk band of the modern era.”

Mush, their third album, was released in 1991, and Allmusic called it “one of the most intense records of the 90s, with some of the fiercest playing and song dynamics.. considered one of the best albums of the decade.” I have to agree. The vocals are raw and angry, while the music is fast and tight. There’s not a bad track on the disk, including a cover of Police’s “Message in a Bottle,” which anchors the CD as the last track. The lyrics are still topical, especially Pandora’s Box, with the line “Don’t f*ck with Pandora’s Box, or don’t make it so obvious.”

Highly recommended if you love 90’s punk (although, if you do love punk, you probably already have Mush).

Escher Gang Sign

Going from Nothing, we end the section on Physics with Everything. Martin starts with a few jokes about having nothing more to say about Nothing, and everything to say about Everything, and quotes Pascal as saying, “What is man in nature? A nothing in comparison with the infinite, and all in comparison with nothing, a mean between nothing and everything.” He then moves to Venn diagrams as a way of showing how logic and set theory diagrams “things.”

(All rights belong to their owners. Images used here for review purposes only. Venn diagram for “No humans have feathers.”)

In showing the overlap between “a” (humans) and “b” (feathered animals), the intersection set is dark because it has no members. It is in fact the empty (Null) set. But, look at the points outside both circles. They represent things that are both not humans and not feathered animals. Martin asks “how far-ranging is this set?” Augustus De Morgan coined the phrase “universe of discourse” to discuss the range of all of the variables we care about. In set theory, it’s called the universal set, or the universe, for short. “It can be whatever we want it to be.” In the above illustration, we only have living things. Below, we extend the set to the set of all typewriters. The intersections of all three sets are empty, but the range of the null set has been extended. As Gardner writes, “The complement of a set k is the set of all elements in the universal set that are not in k.” Meaning, the universe and the empty set “are complements of each other.”

(Making the universe bigger by contributing nothing to the whole.)

If we try to keep extending the universal set, we get the question of how far we can go. We can include irrational numbers like e and pi, and universes where Sherlock Holmes actually fell off the cliff at Reichenback Falls and died. Can we finally make the universe so large that it contains the set of all possible sets? No, because this would result in a contradiction that we’ve already addressed in the chapter on Cantor’s Aleph Null and Aleph One. The set of all subsets would have the highest cardinal number, but that can’t be the highest number because the set of all those subsets must be one cardinal number higher.

(World lines for 2 particles.)

This then brings us to the physical universe and whether there is enough mass for it to be closed, and eventually stop expanding. The steady-state universe theory was eliminated by the universal background radiation that is reasonable proof that things started with the Big Bang, but will things go the way of the poof bird (which flies in decreasing circles until it disappears up its own butt), or is the universe just one in a series of unending big bangs/big crunches? Martin mentions Archibald Wheeler as the physicist that had gone the farthest in building models of the universe. His Superspace consists of an infinity of universes, of which ours is only one. The above illustration shows two particles, one black, one of some color. They come into existence and intertwine to the end of time. This is a one-dimensional graph of time and space, but the particles move in two dimensions, and the way to view it is to take a piece of paper, cut a slit in it, and run the paper with the slit from the bottom of the graph to the top. This particular approach uses “world lines” for the particles.  Where is/was/will be the black particle at time k? Run the paper to time k, move over to the black particle, and then look at the bottom of the graph for its relative position. Add more particles, move the paper back and forth, and you have the birth, life and death of the universe.

(Configuration space, with the position of the colored particle on the y-axis, and the black particle on the x-axis.)

The kinematics approach is to draw the changes of the system of particles as the motion of a single point in a higher space called configuration space. The space is 2D, but both coordinates are spatial. One coordinate is assigned to the black particle, and the other to the colored one. Both particles can be drawn as one “configuration point.” This is not a time graph (that would add a third axis). If the system is closed, the line forms a closed loop. It can not have branches, but it can intersect itself. If we want to graph 100 particles, we just go up to 100 dimensions. Add one more dimension for time and we have a full space-time graph. But, this graph won’t let us reconstruct the particles’ past or predict their future.

Martin then gets philosophical in speculating on a “higher dimensional being” that can view our universe in its entirety, commenting on past beliefs, then comes back to reality by quoting Edgar Allan Poe’s Eureka: A Prose Poem. Poe wrote this near the end of his life, and was convinced it was his best work. He wrote a friend that this would “revolutionize the world of Physical and Metaphysical Science.” He tried to get his publisher, George Putnam to print 50,000 copies. Putnam advanced him $14 and printed 500 copies. The reviews were mostly unfavorable, and it was taken seriously only in France. But, Gardner says that it is a theist’s version of Wheeler’s cosmology. Harold Beaver, editor of The Science Fiction Works of Edgar Allan Poe, wrote that the “I” in Poe’s Dreamland is the universe itself.

“By a route obscure and lonely,
Haunted by ill angels only,
Where an Eidolon, named NIGHT,
On a black throne reigns upright,
I have reached these lands but newly
From an ultimate dim Thule-
From a wild clime that lieth, sublime,
Out of SPACE- out of TIME.”

Gardner ends with C. S. Lewis: “‘Everything’ is a subject on which there is not much to be said.”

Challenge: Define the universe. Give two examples.
(From the Ultimate Final Exam.)

Some of my more long-term readers (one or both of you) may have figured out by now that I like stuff I can animate. In the case of numbers and math, that’s whatever I can automate in a VBScript and feed into an Excel spreadsheet, then save as individual jpegs that I can stitch together in a movie editor. So, I’m always looking for some way of doing that when I find a new formula or algorithm. Unfortunately, Collatz doesn’t immediately lend itself to any kind of animation I consider interesting, but that doesn’t stop me from at least drawing up different kinds of charts in an attempt to try.

So, what exactly is the Collatz Conjecture telling us? In effect, the algorithm creates a series of “ladders” through the application of the first rule (if the number is even, divide it by 2). The ground level of the ladder is any one of the odd integers greater than 1. The ladder itself consists of that “ground” times powers of 2 (i.e.: 3, 6, 12, 24, 48, 96…, which is just 3*(1, 2, 4, 8, 16, 32…)). The exception to this is the ladder that has 2 for the ground, and we can think of that as 1*(2, 4, 8, 16, 32…)

Rule 2 (if the number is odd, multiply by 3 and add 1) is what allows us to change ladders. At a minimum, the new number will place us on rung 1 of the new ladder (e.g. – n0 = 3; n1 = 9 + 1, the new ladder has a ground of “5”). Looking at this a little closer, numbers in a sequence get bigger by a factor of about 1.5 when rule 2 applies, because it’s immediately followed by rule 1 at least once, assuming rule 2 lands on the first rung of the new ladder. Otherwise, hitting the ladder farther up (say with n0 = 5, n1 = 16, n2 = 8, n3 = 4, n4 = 2, n5 = 1) causes the ladder to “collapse” all the way down to the odd-numbered root by a factor of 4 or greater. It’s this collapse of the ladder that prevents the sequence from going to infinity any faster than by 1.5, if at all. The Conjecture, in fact, states that any positive integer you start with will eventually go to 1, which is due specifically to rule 2 switching to rungs relatively high up on certain ladders, and then that ladder collapsing.

Given that rule 1 results in the sequence going to odd numbers before rule 2 can take effect, it would kind of make sense to remove all even numbers from a chain and only consider the odd-numbered points. Looking at the above badly-drawn chart, that’s essentially what we have. 7->22->11->34->17->52->26->13->40->20->10->5->16…1 turns into 7->11->17->13->5->1. There’s no real pattern that I can see easily, but there is an implication.

Actually, there’s an unstated third rule that I feel needs to be put forth now. Rule 0: If n = 1, stop.

I know it’s part of the description of the algorithm, but I want it to be more blatantly obvious. If we mindlessly apply rules 1 and 2, we get into an infinite loop, where 1->4->2->1… With rule 0 in place, we can claim that 2 is no longer the exception for the ladder ground values, and that 1 is in fact the lowest-most ladder and 2 is the first rung on it. The above odd-number reduction chain (7->11->17->13->5->1) makes more sense this way.

Getting back to the Conjecture, if we treat all of the chains that branch off of the root value of 1 as a tree, then every positive integer should appear in this tree once and only once. The implication is that we will never see an odd number showing up twice in the reduction chain (i.e. – 7->11->17->13->5->7…) which would produce an infinite loop.

If we pretend that the Conjecture is wrong, it means that there is either a second (or more) tree independent of the main root=1 tree, or that there is at least one chain that forms a closed loop. For an independent tree, we’d need either some minimum number that the tree reduces to and goes no farther in the sequence (the role that 1 plays in the main tree), or that goes to infinity so there is no “root” per se. Remember, rule 2 represents a maximum growth factor of 1.5, while rule 1 is unpredictable, and gives a growth factor anywhere from 0.5 to almost 0. Since we can’t have a minimum root value that isn’t 1, the root sequence has to go to infinity, but that means that the combination of rules 1 and 2 MUST have a growth factor in this root sequence that is greater than 1 by some fraction. But, that means that every time we switch ladders via rule 2, we hit rung 1 of the next higher odd number more often than we don’t. In other words, for Nn+1 = (3Nn + 1)/2; Nn+1 must almost always be odd. For this to hold true, there must be an entire, infinite class of numbers for which the Collatz sequence never intersects any branch of the main tree. This is unlikely at best.

This leaves the second case, where there’s an infinite loop. Say we start with 9. The sequence would be 9->28->14->7->22->11->34->17->52->26->13->40->20->10->5->16->8->4->2->1. BUT, rather than collapsing down from 40 to 10 (we hit the third rung up on the 5 ladder) what if we’d collapsed down from 40 to 9? Looking only at the odd numbers in the chain, we’d get 9->7->11->17->13->9. Then, applying rule 2 again, we’d just repeat the loop. What this means is, at large enough numbers, there could be a case where Nn/2^x = N0. If this happened, it would invalidate the Conjecture, which says that, in effect, (as mentioned above) no number can appear twice in one chain. Naturally, this situation would also create a Swiss cheese effect in the main tree, where an infinite number of points branching off the closed loop are not part of the main tree (those points that are factors of N*2^x).

Intuitively, it seems clear that the Conjecture has to hold true, but no one’s ever proved it conclusively. In a way, the Collatz algorithm is a lot like the generation of prime numbers. Using a brute force Sieve of Eratosthenes approach of drawing out the chains for each new odd number not yet attached to the main tree leaves a bunch of holes (i.e. – “prime numbers”), which then represent a new branch on the main tree for some odd-numbered node that is much larger than the “prime” we started with (e.g. – starting with 7 takes us to the node point at 13, on the 5-ladder). The question then becomes, is there a way to analyze any random positive integer, without using rules 1 and 2, and determine the number of steps it would take to reach 1? If the result was infinite, then that integer would have to be in the closed loop.

For such a simple algorithm, it’s actually a lot of fun to play with.

Nothing.
You know, it’s VERY tempting to leave this entry empty. But I won’t. This time, but you have been warned…
On the other hand, Martin pretty much pulls out all the stops in listing jokes about nothing, with the book of blank pages entitled “What I know About Women,” Elbert Hubbard’s “Essay on Silence”, and John Cage’s 4’33” (the composition is 273 seconds long, and -273 Kelvin is absolute zero). He asks how much dirt is in a hole 5′ long, 2′ wide and 4′ deep, comments on laws that punish people that don’t take action in certain circumstances, and presents song lyrics including “I ain’t got nobody,” “Nobody loves me,” “I’ve got plenty of nothing,” “Nobody lied when they said I cried over you,” and “There ain’t no sweet gal that’s worth the salt of my tears.” He goes on at length about Ad Reinhardt’s black Abstract Painting, 1960-61 and the first annual award of the Elves’, Gnomes’, and Little Men’s Science Fiction Chowder and Marching Society in 1950 to Ray Bradbury (the award was an invisible little man standing on a brass plate with a polished wood pedestal; you could tell the invisible man was there because of the footprints on the brass plate).

(All rights belong to their owners. Images used here for review purposes only. “Portrait of Its Immanence the Absolute“, Mind, 1901.)

In 1901, the Special Christmas edition of Mind ran the “Portrait of Its Immanence the Absolute” on the cover, which was a blank rectangle protected by a covering of tissue. The text reads, “Turn the eye of faith fondly but firmly on the center of the page, wink the other, and gaze fixedly until you see it.” And, “This side up.” (The Absolute was a phrase used by Hegelian philosophers.)

The most solid of the math elements in this chapter revolves around John Horton Conway’s use of the Null set to create his surreal numbers (chapter 28). Conway had been at the University of Cambridge at the time, and he wrote up a 13-page typescript titled “All Numbers Great and Small.” It starts with “We wish to construct all numbers. Let us see how those who were good at constructing numbers have approached the problem in the past.” And ends with, “Is the whole structure of any use?”

The null set is the only set that has no members, and is the only set that is a subset of all other sets. It denotes, although it doesn’t denote anything. “For example, it denotes such things as the set of all square circles, the set of all even primes other than 2, and the set of all readers of this book who are chimpanzees.”

The Surreal numbers are constructed in stages, and involve pairing subsets of numbers already constructed. From the wiki page: “given subsets L and R of numbers such that all the members of L are strictly less than all the members of R, then the pair { L | R } represents a number intermediate in value between all the members of L and all the members of R.”

When you start out, you only have the empty (null) set: { | }.
{   |   } = 0
{ 0 |   } = 1
{ 1 |   } = 2
{   | 0 } = -1
{ 0 | 1 } = 1/2
{ 0 | 1/2 } = 1/4
{ 1/2 | 1 } = 3/4

Conway presented his typescript to Donald Knuth, computer scientist at Stanford University, in 1972. Knuth was so blown away that he wrote an introduction to Conway’s method in the form of a novelette in 1973, and it was published as a paperback in 1974. The novelette was titled, “How two ex-students turned on to pure mathematics and found total happiness” and can still be ordered from Addison-Wesley.

Challenge: In the above hole, add a -1 cubic feet of dirt.