Martin Gardner | Mathemagician | Persi Diaconis | John H. Conway | Scott Kim | Bill Gosper | David_Singmaster (1939-) | Sol Golomb (1932-) | James Randi | Donald Coxeter (1907-2003) | M.C. Escher (1898-1972) | Ray Smullyan | Jerry Andrus (1918-2007) | Jay Marshall (1919-2005)
Video
(46:04,
Vimeo)
Mystery and Magic of Mathematics: Martin Gardner and Friends
Alternate title: Martin Gardner, Mathemagician.
An episode of
The Nature of
Things
with David Suzuki (CBC, 1996)
[画像: Who is this at 1:07 ? (Please, tell me!) ]
Featuring, in order of appearance:
David Suzuki 00:48,
Meir Yedid 00:55
(finger-hiding trick
"sleight of hand")
Herb Zarrow
&
Max Maven 01:05,
Bill Gosper 01:09,
Martin Gardner 01:15,
Sol Golomb 01:18,
Ron Graham 01:27,
James Randi 01:42,
Jay Marshall 02:09,
John H. Conway 02:53,
Michael Weber 03:08,
Scott Kim 03:54,
Persi Diaconis 12:20,
Donald Coxeter 29:03,
Doris Schattschneider 31:58 and
Marjorie Rice 32:52.
After a brief illness, Martin Gardner died unexpectedly at Norman Regional Hospital at the age of 95. The precise cause of death is unknown. His passing was quick and painless. Martin Gardner is survived by two sons: James (of Norman, Oklahoma) and Tom (of Asheville, NC). He is mourned by many friends and countless professional or amateur mathematicians.
The passing of Martin Gardner has urged a few people who had crossed his path to recollect those precious moments:
In 1939, Arthur Harold Stone (1916-2000) was a British doctoral student who had just arrived at Princeton University to study general topology. Since American sheets of paper were wider than European ones, he was trimming letter-size American sheets to fit British binders. (The European size was not yet standardized as "A4".) Stone was left with lots of strips of paper to fold and play with. One day, he stumbled upon a flexagon and showed it to some of his fellow students, including Bryant Tuckerman (1915-2002), Richard P. Feynman (1918-1988) and John W. Tukey (1915-2000). They formed the Princeton Flexagon Committee. Soon, it seems everyone on campus was making and flexing hexaflexagons.
[画像: Come back later, we're still working on this one... ]
Wikipedia : Flexagon
Polyominoes were devised in 1954 by Solomon W. Golomb when he was a 22-year old graduate student at Harvard. A polyomino consists of N unit squares in the plane, each sharing at least one of its sides with another square.
According to the wording of that simple definition, there is one zeromino (consisting of an empty set of unit squares) but there are no monominoes (N=1). However, many people consider a lone square to be a monomino...
Two polyominoes are considered distinct only if they cannot be obtained from each other by rotating or flipping. There is only one domino (N=2) but there are 2 triominoes, 5 tetrominoes, 12 pentominoes, 35 hexominoes, etc.
[画像: 1 domino, 2 triominoes, 5 tetrominoes ]
1, 0, 1, 2, 5, 12, 35, 108, 369, 1285, 4655, 17073, 63600, 238591, 901971, 3426576, 13079255, 50107909, 192622052, 742624232, 2870671950, 11123060678, 43191857688, 168047007728, 654999700403, 2557227044764, 9999088822075, 39153010938487, ... (A000105)
The set of 12 pentominoes proved to be most endearing... Those twelve pieces are used in a two-player game (proposed by Golomb himself in December 1994) which is played on an 8 by 8 chessboard: The players alternate placing a piece until one of them is unable to do so (and is declared the loser).
Hilarie K. Orman proved the game to be a first-player win.
Pentominoes have been marketed whose thickness is the same as the side of the constituent squares. Twelve such pieces have a combined volume of 60 cubic units, which can be assembled into 3 types of cuboids (2 by 3 by 10, 2 by 5 by 6, 3 by 4 by 5).
The 3D equivalents of polyominoes are solids consisting of unit cubes which share at least one face with another cube.
Two such shapes are distinct only if they're not congruent by rotation.
7 distinct nonconvex shapes can be obtained in this way with 4 cubes or less (only one consists of just 3 cubes). This includes two chiral pieces which are mirror images of each other. Those are the so-called soma pieces whose combined volume is 27 units. They can be assembled into a cube 3 units on a side.
Legend has it that the Soma Puzzle was devised by Piet Hein (1905-1996) during a lecture on quantum mechanics by Werner Heisenberg (1901-1976).
Wikipedia : Soma Cube
[画像: Come back later, we're still working on this one... ]
Intriguing Tessellations
by Marjorie Rice (San Diego)
|
Links
Perplexing Pentagons
by Doris Schattschneider (Moravian College, Bethlehem, PA).
The 14 Types of
Convex Pentagons that Tile the Plane
by Ed Pegg, Jr. & Branko Grunbaum
Recipe
for finding
pentagons that tile the plane
by Bob Jenkins
The two Penrose tiles are quadrilaterals consisting of two pairs of equal sides whose lengths are in a golden ratio :
f = 1.6180339887498948482...
This yields angles that are multiples of p/5 and allow various types of pentagonal patterns around any vertex where several tiles meet (without voids).
[画像: Penrose's Kite and Dart ]
The convex tile is called a kite, the other one is dubbed dart. They bear a specific color pattern like the one pictured above. The colors must match along any side where two such tiles touch.
It's nice to have circular arcs centered on vertices for the inner boundaries between colors but more creative designs can be used, as long as the colors on equal sides enforce the same matching as what's illustrated here. Another (colorless) alternative would be to enforce the side-matching with jigsaw-type notches.
Mathworld : Penrose Tiles Numericana : Quasicrystal
Scott Kim is a friend of Martin Gardner who practices an elaborate type of calligraphy where the spellings of words changes when they are rotated or viewed in a mirror. The example at right has mirror symmetry whereas the title of this page is symmetric with respect to its central point.
If you learn to be good at a game, you find
what it is you should have been thinking about.
John
Horton Conway (1937-2020)
The Game of Life (GOL) invented by John H. Conway (in 1970) is a zero-player game. Once a board configuration is set up, it just evolves according to fixed rules, like life would unfold in a completely deterministic universe. The point is to discover life forms with an interesting evolution... A very rich catalog was eventually compiled which provided a few components that allow the simulation of any imaginable deterministic computer!
In Conway's game, the board is just an infinite grid of square cells. Each cell is either dead (empty) or alive (occupied by a black dot). The neighbors of a cell are the 8 cells which share a side or a corner with it. There are just two rules which govern the evolution of a configuration from one generation to the next:
[画像: Block ] Block The so-called block is the simplest stable life form. It consists of four live cells in a square configuration. Each of them survives because of its 3 live neighbors and no cell is born because no other cell has 3 live neighbors.
The blinker consists of a row of 3 live cells which oscillates between a vertical and a horizontal configuration. The center cell survives, both extremities die and two cells are born which replace them at a right angle... Again and again.
[画像: Vertical Blinker ] [画像: Horizontal Blinker ] [画像: Vertical Blinker ] [画像: Horizontal Blinker ] [画像: Blinker ]
The most interesting of the small life forms is the glider, which consists of 5 live cells and moves diagonally one unit in 4 steps:
[画像: Glider ] [画像: Glider ] [画像: Glider ] [画像: Glider ] [画像: Glider ]
Dart Spaceship Dart Spaceship Spaceships : The life patterns which move p cells in q generations are called spaceships and are said to be moving at p/q times the speed of light (c). The aforementioned glider moves at c/4 diagonally.
At right, is the dart spaceship which moves at c/3. It was discovered by David Bell in May 1992.
Three small spaceships were discovered by Conway (in 1970) which move at speed c/2 (namely: 2 cells in 4 generations) either horizontally or vertically: The small fish (or float) the medium fish and the big fish. They are also respectively known as the lightweight, middleweight and heavyweight spaceships (abbreviated LWSS, MWSS, HWSS).
Small Fish. Medium Fish. Big Fish.
One early question about the Game of Life was the existence of board configurations which cannot result from the evolution of a previous population. Such a configuration is known as a Garden of Eden. The existence of Gardens of Eden can be demonstrated by the following numerical argument:
A population contained in a square which is 5n-2 cells on a side has either no parent or at least one parent fully contained in a 5n by 5n square.
Such parent configurations can be partitioned into 5 by 5 squares. The key remark is that two parents clearly have the same children if one of those small squares is either empty or has only its central cell occupied. So, the number of distinct children of parents contained in a 5n by 5n square is no greater than :
( 225 - 1 ) n2
If that number is less than the number of 5n-2 by 5n-2 configurations, some of those must have no parent ! Let's simplify the relevant inequality:
( 2 25 - 1 ) n2 < 2 (5n-2)2
With k = lg ( 2 25 - 1 ) = 24.999999957004336643612528... this inequality becomes (taking the binary logarithm of both sides):
k n2 < 25 n2 - 20 n + 4
The leading term of the polynomial (25-k) n2 - 20 n + 4 being positive, it is itself positive for sufficiently large values of n. Numerically, the inequality holds when n is beyond 465163191.59... So, there must be Gardens of Eden among the populations contained in a square 2325815956 cells on a side! QED
The above can be used to show that a (very) large configuration is most likely to be a Garden of Eden (the probability that it isn't vanishes exponentially as a function of its size). It's still a challenge to find small Gardens of Eden, though.
The first explicit Garden of Eden to be discovered was the following pattern, inscribed in a 9 by 33 rectangle. It was found by Roger Banks, Mike Beeler, Rich Schroeppel et al. at MIT in 1971. Curiously, Achim Flammenkamp noticed many years later (on June 16, 2004) that the 5 rightmost columns of this historical example are essentially not needed (yielding a 9 by 28 Orphan ).
At this writing, the smallest known Garden of Eden is a pattern of 72 live cells in an 11 by 12 rectangle. It was discovered by Achim Flammenkamp on June 23, 2004.
Early on, Conway had conjectured that there were finite life forms which would grow indefinitely but he could not find one... So, he put up a 50ドル reward for an example.
Bill Gosper (1943-) claimed the prize with the following grand thing, obtained by studying the interaction of two queen bee shuttles (stabilized by blocks ).
This was the first example of what's called a glider gun. The Gosper gun emits a steady stream of gliders but its core returns to its former self after 30 steps.
Glider guns have since been devised for any arbitrary period above 14. They are a key ingredient in the so-called universalization of the game of life performed independently by Gosper and by Conway (using the same approach). As described in the next section, this establishes, essentially, that anything boils down to a question about Conway's game!
Alan M. TuringIn the last chapter of the first edition of Winning Ways (1982) John Conway proves that his automaton is just as powerful as a Turing Machine (or any other type of computer with an unbounded amount of read/write memory).
Remarkably, an engineering approach is used to show how all the components of modern computer circuitry can be simulated within the Game of Life (program and input data being encoded in the starting configuration).
Basically, Conway uses clocked streams of rarefied gliders as the basic digital signals (the presence or absence of a glider in a stream at a scheduled time is interpreted as a specific bit being 1 or 0). Such streams are produced by guns and absorbed by eaters (guns with arbitrarily low output rate exist, so that synchronized wires will not interact as they cross each other).
Conway uses a large zoo of special configurations and a bunch of clever techniques to simulate logic gates and all the circuitry of a finite computer endowed with an unbounded external memory which it can read and write...
As a beautiful final touch, he shows how such a simulation can completely self-destruct to indicate that the corresponding computer program has halted (otherwise, something remains on the board).
The engineering details are quite intricate but the guiding principles are simple and the glorious conclusion is inescapable: The Game of Life is an automaton which is just as powerful as a Turing machine. Any problem which (like most interesting logical questions) is equivalent to the ultimate halting of a computing machine (with unlimited storage capabilities) can actually be rephrased in terms of the ultimate vanishing of a specific starting configuration in Conway's Game of Life . In other words: Life is hard. Isn't it?
October 1970
(Martin Gardner)
|
Conway's Game of Life
(Wikipedia)
What is the Game of Life?
by Paul Callahan
and Alan Hensel
Golly
by Andrew Trevorrow and Tomas Rokicki
(with Dave Greene, Jason Summers & Tim Hutton)
Logicell 1.0
by Jean-Philippe Rennard, Ph.D. (implementing logic functions in Conway's game).
26-Cell
Pattern with Quadratic Growth (Bill Gosper and Nick Gotts, March 2006)
Eric Weisstein's Treasure Trove of
the Life Cellular Automaton by Eric
Weisstein (2000-2005)
Open
Directory Life Index (initially maintained by Mirek Wojtowicz)
John Conway Talks About the Game of Life
Part 1
|
Part 2
The notation which is now standard to describe sequences of moves in Rubik's cube was invented by David Singmaster in 1979.
A capital letter indicates a clockwise rotation of a quarter turn. The same letter primed denotes a counterclockwise rotation. The following six letters are used, which refer to the location of the center of rotation, irrespective of its color:
F (front), R (right), L (left), U (up), D (down) and B (back).
In practice, B is rarely used.
[画像: Come back later, we're still working on this one... ]
Rubik's Cube (Wikipedia)
|
Solving the Rubik's Cube
Systematically by Alex Fung Ho-San
Beginner Solution to
the Rubik's Cube by Jasmine Lee
A simple trick to crack all Rubik
puzzles (14:26) by Burkard Polster (Mathologer, 2016年01月15日).
You are allowed to lie a little,
but you should never mislead.
Paul R. Halmos (1916-2006)
John Conway 2 minutes and 53 seconds into the aforementioned video presented by David Suzuki (CBC, 1996) John Conway says:
This, I'm sure, was in Martin's column sometime. You know, it's impossible to tie a knot without leaving go of the ends of the strings the way I just did...
Up to this point, Conway did not lie but he did mislead... Indeed, the last thing he said could be parsed as applying only to the locution "leaving go of the ends of the strings" (which is precisely what Conway did, secretly).
This perfect example of a misleading true statement is followed by Conway's concluding remark which either turns the whole thing into a straight lie (expected of an illusionist, amateur or not) or can be forcefully reparsed into a true statement. Your pick:
... but Martin will tell you many different ways of doing it.
Conway's performance is flawless and well photographed. Even if you know what to look for and play the video frame by frame, you simply won't detect the fallacy.
Martin Gardner loved the following Indian legend. He first run across it in The People, Yes (1936) by Carl Sandburg (1878-1967):
The white man drew a small circle in the sand and told the red man, "This is what the Indian knows," and drawing a big circle around the small one, "This is what the white man knows." The Indian took the stick and swept an immense ring around both circles: "This is where the white man and the red man know nothing."
