Transcript: Paul Steinhardt on Impossible Crystals | Nov 18, 2006

[Theme music plays]

The opening sequence rolls. The logo of "Big Ideas" featuring a lit lamp bulb appears against an animated green slate.
Then, Andrew Moodie appears in the studio. The walls are decorated with screens featuring lit lamp bulbs, and two signs read "Big ideas."
Andrew is in his early forties, clean-shaven, with short curly black hair. He’s wearing a brown jacket over a light orange shirt.

Andrew says WELCOME TO
BIG IDEAS.
MY NAME IS ANDREW MOODIE AND
THIS
IS THE HAND SOLO
LASER PISTOL.

He shows a Starwars toy pistol package.

He continues MY PARENTS NEVER LET ME HAVE TOY
GUNS WHEN I WAS A KID, BUT
NOW...HO!
THAT I'M AN ADULT, WELL, THANK
YOU, eBAY.
NOW THE THING I HATE ABOUT THIS
THOUGH IS ALL THE PACKAGING.
I MEAN, ALL THIS CARDBOARD, IT'S
JUST A WASTE.
BUT A PLASTIC LASER PISTOL?
I MEAN, THERE WOULD BE NO WAY TO
PACK TOGETHER SOMETHING THAT HAS
SUCH AN IRREGULAR SHAPE.
OR IS THERE?
AND COULD THIS METHOD OF PACKING
LEAD TO THE CREATION OF NEW
FORMS OF MATTER NEVER SEEN
BEFORE IN THE NATURAL UNIVERSE?
OH...
LET'S ASK TODAY'S LECTURER,
PAUL J. STEINHARDT.
HE'S THE ALBERT EINSTEIN
PROFESSOR IN SCIENCE AT
PRINCETON UNIVERSITY.

Against a blue screen, a caption reads "Prologue."

Paul Steinhardt stands behind a table next to a wooden lectern on a dark stage. He’s in his late forties, with receding dark brown hair and clean-shaven. He’s wearing glasses, a dark suit, white shirt and red tie.

Paul says I'M GONNA
BEGIN WITH A PROLOGUE TO SORT OF
SET UP A DISCUSSION THAT'S GOING
TO FOLLOW.
SO I'D LIKE TO IMAGINE THAT
YOU'VE BEEN GIVEN THE JOB OF,
LET'S SAY, TILINGS AND LARGE
FLOOR OR SOME LARGE WALL, AND
YOU'VE BEEN GIVEN ALL IDENTICAL
TILES, THE TILES ARE IN THE
SHAPE OF A PARALLELOGRAM.

He turns to look at a large screen that shows a green parallelogram on the top left corner.

He continues THE QUESTION IS: CAN YOU TILE
THE SURFACE - LET'S IGNORE THE
EDGES - CAN YOU TILE THE SURFACE
SMOOTHLY BY CLOSE-PACKING THOSE
TILES TOGETHER FULL-EDGE ON
FULL-EDGE WITHOUT ADMITTING ANY
GAPS OR CRACKS OR HOLES?
WELL, I THINK IT'S OBVIOUS THAT
THE ANSWER IS YES, YOU CAN
PROBABLY ENVISAGE THE SOLUTION.
YOU CAN IMAGING PACKING THEM IN
AN ARRAY LIKE THIS, FOR EXAMPLE.

The screen shows a large parallelogram made up of 16 smaller parallelograms.

He continues THIS IS AN EXAMPLE OF WHAT WE
CALL A PERIODIC ARRAY.
PERIODIC MEANS REGULARLY
REPEATING OR REPEATING WITH
EQUAL DISTANCES.
AND WHAT WE SEE IS THAT ALONG
EACH DIRECTION THE PARALLELOGRAM
REPEATS, THE TILE REPEATS, WITH
EQUAL SEPARATIONS BETWEEN
PARALLELOGRAMS.
ONE KIND OF SEPARATION WHEN YOU
MOVE ALONG ONE DIRECTION, A
DIFFERENT SEPARATION WHEN YOU
MOVE ALONG THE OTHER DIRECTION.
WELL, IF INSTEAD OF GIVING YOU A
PARALLELOGRAM I HAD GIVEN YOU A
RECTANGLE?
WHAT ABOUT THEN?
COULD YOU HAVE TILED THE FLOOR
WITH THAT, WITH A PILE OF TILES
LIKE THAT IN THE SAME WAY
WITHOUT ANY CRACKS OR GAPS?
WELL, SOME OF YOU MUST HAVE
ALREADY REALIZED THE ANSWER MUST
BE YES BECAUSE A RECTANGLE IS
JUST A SPECIAL EXAMPLE OF A
PARALLELOGRAM, WHICH HAPPENS TO
HAVE RIGHT ANGLES RATHER THAN
ARBITRARY ANGLES.
BUT THERE IS SOMETHING-- AND SO,
IN FACT, YOU CAN DO IT.
BUT THERE IS SOMETHING NOW A
LITTLE BIT DIFFERENT ABOUT THIS
EXAMPLE BECAUSE A RECTANGLE HAS
THE PROPERTY THAT IF I TAKE A
RECTANGLE AND ROTATE IT BY 180
DEGREES I'D GET BACK THE SAME
FORM AS I HAVE INITIALLY.
AND SIMILARLY, IF I TAKE THIS
LATTICE OF TILES AND I FLIP IT
AROUND 180 DEGREES IT, TOO, GETS
BACK TO WHERE IT HAS THE SAME
FORM THAT IT HAD INITIALLY.
SO, BY USING A RECTANGLE RATHER
THAN A PARALLELOGRAM, I NOT ONLY
MAKE SOMETHING PERIODIC, BUT I
MAKE SOMETHING WITH MORE
SYMMETRY.
IN THIS CASE, THE SYMMETRY OF A
RECTANGLE OR WHAT WE CALL A
TWO-FOLD SYMMETRY 'CAUSE I CAN
ROTATE IT BY 180 DEGREES AND 180
DEGREES AGAIN TO GET BACK TO
WHERE I STARTED.
HOW 'BOUT A SQUARE?
WELL, AGAIN, THAT'S JUST A
SPECIAL EXAMPLE OF A
PARALLELOGRAM, SO OF COURSE WE
CAN DO IT.
ANY OF YOU THAT'S TILED A
BATHROOM WITH SQUARE TILES KNOWS
THIS.
BUT NOW WE'VE INTRODUCED MORE
SYMMETRY BECAUSE A SQUARE IS NOT
ONLY REMAINING INVARIANT WHEN I
ROTATE IT BY 180 DEGREES, BUT
IT'S INVARIANT WHEN I ROTATE IT
BY ANY MULTIPLE OF 90 DEGREES.
SO I ENDED UP WITH A PERIODIC
TILING, BUT THIS ONE, WHICH NOW
HAS AN EVEN HIGHER SYMMETRY -
THE SYMMETRY OF A SQUARE.
HOW 'BOUT A TRIANGLE?
HOW MANY PEOPLE THINK - RAISE
YOUR HAND IF YOU THINK YOU COULD
TILE THE FLOOR WITHOUT ANY
CRACKS OR GAPS WITH A TRIANGLE?
OKAY, ALMOST EVERYBODY THINKS
YES, I THINK.
HOW MANY PEOPLE THINK NO?
OKAY, NOBODY.
OH, I'VE GOT SOME NO's.
WELL, YOU CAN.

[Chuckles]
[Audience laughter]

Paul says continues YOU
CAN.
OKAY, BUT IT'S A LITTLE MORE
SUBTLE AS YOU MIGHT NOTICE
BECAUSE NOW I'VE HAD TO
INTRODUCE NOT JUST TRIANGLES
WITH ONE ORIENTATION, BUT ONE
WITH A FLIPPED ORIENTATION.
NEVERTHELESS, I CAN DO IT.
I CAN FILL THE PLANE WITHOUT ANY
GAPS.
AND NOW IF I LOOK AT, UH, OF
COURSE, THE TRIANGLE AS A
SYMMETRY OF ROTATIONS BY 120
DEGREES, SO NOW IF I TAKE THIS
TILING AND I ROTATE IT BY 120
DEGREES BY THE CENTER-- AROUND
THE CENTER OF ANY TRIANGLE, I
AGAIN GET BACK TO WHERE I
STARTED.
SO AGAIN, I GET A TILING WITH
HIGH-- IN THIS CASE THE SYMMETRY
OF A TRIANGLE.
A HEXAGON?
HOW MANY PEOPLE DON'T THINK YOU
CAN DO A HEXAGON?

[Audience laughter]

Paul continues YEAH, BEES
KNOW YOU CAN DO HEXAGONS.
[Chuckles]
THEY FIGURED OUT TO DO IT, SO
CERTAINLY YOU KNOW IT COULD BE
DONE.
SO AGAIN, ONE CAN CLOSE-PACK
THINGS.
SO, IN THIS CASE, OF COURSE, IT
HAS A SYMMETRY OF A HEXAGON,
ROTATIONS BY 60 DEGREES WILL
TAKE YOU BACK TO WHERE YOU
STARTED AGAIN.
UM, A PENTAGON.
WELL, TURNS OUT THAT THIS IS THE
FIRST INTERESTING CASE BECAUSE
YOU
CAN'T
TILE SPACE WITH
PENTAGONS.
YOU CAN'T TILE A PLANE WITH
PENTAGONS.
YOU CAN TILE THE SURFACE OF A
SPHERE WITH PENTAGONS, BUT YOU
CAN'T TILE A FLAT PLANE WITH
PENTAGONS WITHOUT INTRODUCING
CRACKS OR HOLES IN IT.
SO OUR SUCCESS RATE UP TO THIS
POINT HAS BEEN 100 PERCENT BUT NOW
SUDDENLY WE'VE COME TO A SHAPE
WHERE WE CAN'T TILE SPACE WITH
IT.

All the shapes mentioned appear on the screen.

He continues IN FACT, WE'VE RUN-- ACTUALLY,
WE'VE COMPLETELY THE FULL LIST
OF TILE SHAPES WITH WHICH YOU
COULD TILE THE PLANE - IDENTICAL
TILE SHAPES WITH WHICH YOU COULD
TILE A PLANE.
THERE WERE ONLY-- THE ONLY KINDS
OF TILINGS YOU CAN CONSTRUCT
WHICH ARE PERIODIC, WHICH HAVE
EXACTLY REPEATING, EXACTLY EQUAL
INTERVALS BETWEEN TILES IN WHICH
YOU HAVE NO CRACKS AND HOLES,
ARE TILINGS WITH A SYMMETRY OF A
PARALLELOGRAM, A RECTANGLE, A
SQUARE, A TRIANGLE, AND A
HEXAGON, AND NOTHING ELSE.
SO NOT A PENTAGON, NOT A
HEPTAGON, NOT AN OCTAGON, NOT A
43-FOLD SYMMETRY OBJECT OR WHAT
HAVE YOU.
THAT'S THE COMPLETE LIST.

A caption appears on screen. It reads "Paul Steinhardt. Princeton University."

Paul says NOW THIS
SIMPLE OBSERVATION, THIS SIMPLE
MATHEMATICAL OBSERVATION SAID IN
A SOMEWHAT MORE ESOTERIC WAY 200
YEARS AGO, LED TO THE FOLLOWING
SCIENTIFIC BREAKTHROUGH.

The caption changes to "Perimeter Institute, Waterloo. September 6, 2006."

Paul continues IT WAS
REALIZED WHEN YOU LOOK AT THIS
LIST OF SYMMETRIES THAT WERE
ALLOWED, THAT IT CORRESPONDS
EXACTLY WITH THE SYMMETRIES YOU
FIND WHEN YOU LOOK AT THE FACETS
OF A CRYSTAL.

Pictures of crystals of different shapes and colours appear.

He continues AS TO SAY, IF YOU LOOK AT
CRYSTAL FACETS THEY COME IN
VARIOUS FORMS, BUT THEY ALWAYS
HAVE EITHER THE SYMMETRY OF A
PARALLELOGRAM, RECTANGLE,
TRIANGLE, HEXAGON, OR A SQUARE,
AND NOTHING ELSE.
AND SO WHAT THAT SUGGESTED TO
PEOPLE 200 YEARS AGO IS THAT THE
INTERNAL STRUCTURE OF A CRYSTAL
MUST BE THAT THERE'S A
FUNDAMENTAL BUILDING BLOCK THAT
SIMPLE REPEATS OVER AND OVER
AGAIN WITH EQUAL SPACING - WITH
PERIODIC SPACING - IN ORDER TO
MAKE THE STRUCTURE.
NOW TODAY WE WOULD KNOW THOSE
THAT BUILDING BLOCK AS BEING AN
ATOM OR A MOLECULE, BUT THIS IS
200 YEARS AGO.
THERE WAS NO THEORY, THERE WAS
NO AGREEMENT THAT MATTER WAS
MADE OF ATOMS.
IN FACT, THAT WAS JUST ONE OF
MANY POSSIBLE HYPOTHESES.
BUT THIS WAS THE FIRST
OBSERVATIONAL EVIDENCE, INDIRECT
THOUGH IT MAY BE, THAT MATTER -
IN PARTICULAR, CRYSTALS; BUT IN
GENERAL, MATTER - IS MADE OF
ATOMS OR MOLECULES, AS MADE UP
OF FUNDAMENTAL INDIVISIBLE
BUILDING BLOCKS.
AND NOT ONLY THAT, BUT IN THE
CASE OF CRYSTALS, THOSE BUILDING
BLOCKS ARE ARRANGED SO THEY
EXACTLY REPEAT OVER AND OVER
AGAIN WITH EQUAL SPACING IN THIS
KIND OF PERIODIC ORDER.
NOW, THIS DISCOVERY WHICH CAME
ABOUT FROM THIS SIMPLE
MATHEMATICS, SIMPLE OBSERVATIONS
OF CRYSTALS KNOWN IN NATURE, AND
SIMPLE PHYSICAL REASONING, FOUND
IT NOT ONLY THE IDEA THAT
MATTERS MADE OF ATOMS, BUT ALSO
FOUND IT A NEW SCIENCE - THE
SCIENCE OF CRYSTALLOGRAPHY, A
SCIENCE THAT GIVES US THE POWER
TO UNDERSTAND WHY MATERIALS HAVE
THE PROPERTIES THAT THEY HAVE.
IN ESSENCE, IT MEANS THAT TO
UNDERSTAND ANY PARTICULAR
CRYSTAL, ESSENTIALLY ALL I HAVE
TO DO IS FIGURE OUT WHAT THE
BUILDING BLOCK CONSISTS OF.
WE WOULD SAY NOWADAYS, WHAT
ARRANGEMENT OF ATOMS AND
MOLECULES IT CONSISTS OF.
AND ONCE I'VE UNDERSTOOD THAT I
UNDERSTAND THE ENTIRE STRUCTURE
SINCE THE ENTIRE STRUCTURE IS
SIMPLE A REPETITION OF THAT.
AND FURTHERMORE, BY KNOWING THAT
BUILDING BLOCK AND KNOWING
ENOUGH ABOUT THE PHYSICS AND
CHEMISTRY OF THAT PARTICULAR
BUILDING BLOCK, I CAN INFER
EVERYTHING ABOUT THE ELECTRONIC,
PHYSICAL, ELASTIC AND OTHER
PROPERTIES OF THAT MATERIAL.
IF YOU OPEN UP ALMOST ANY BOOK,
ANY BASIC BOOK, TEXTBOOK, ON THE
SCIENCE OF MATERIALS, OR A SOLID
STATE PHYSICS, YOU'LL FIND THE
FIRST ISSUE THEY ADDRESS IS THIS
ISSUE OF WHAT SYMMETRIES ARE
ALLOWED BASED ON THIS SORT OF
TILING ARGUMENT I GAVE YOU, AND
WHAT STRUCTURES-- AND WHAT KIND
OF FACETS AND STRUCTURES ARE
ALLOWED FOR CRYSTALS.
WELL THAT'S THE PROLOGUE FOR MY
TALK BECAUSE WHAT I'M GOING TO
EXPLAIN TO YOU IN THE REST OF
THE TALK IS THOSE THINGS THAT
YOU LEARNED, THOSE ELEMENTARY
TEXTBOOKS ARE ACTUALLY WRONG, OR
AT LEAST INCOMPLETE.
BECAUSE WHAT WE'RE GONNA BE
TALKING ABOUT FOR THE REST OF
THIS TALK IS A SET OF OBJECTS
WHICH WOULD BE IMPOSSIBLE
ACCORDING TO THOSE LAWS - THOSE
200 YEAR OLD LAWS - AND THIS IS
THE IDEA WE CALL IMPOSSIBLE
CRYSTALS, OR WHAT ARE KNOWN IN
THE FIELD AS QUASICRYSTALS.
THIS SUBJECT BEGAN ABOUT TWO
DECADES AGO WITH THE ACCIDENTAL
DISCOVERY OF THIS PARTICULAR
MATERIAL - AN ALLOY MIXTURE OF
SIX PARTS ALUMINUM AND ONE PART
MANGANESE.

A black and white picture shows abstract shapes.

He continues NOW, WHAT YOU'RE SEEING MOSTLY
IN THIS PICTURE IS ACTUALLY PURE
ALUMINUM.
WHAT I WANT YOU TO FOCUS YOUR
EYES ON ARE THESE SORT OF
FEATHERY LOOKING THINGS THAT ARE
SITTING EMBEDDED IN THAT
ALUMINUM.
THAT'S THE MATERIAL WE'RE
TALKING ABOUT, THESE GRAINS OF
ALUMINUM MANGANESE.
AND THEY HAVE A KIND OF
SNOWFLAKE APPEARANCE, BUT WHAT'S
STRIKING ABOUT THEM - WELL AT
LEAST IT'S STRIKING WHEN YOU
SAY, FOR EXAMPLE, ABOUT THIS
ONE - IS THAT IT SEEMS, AT LEAST
TO THE EYE, SUPERFICIALLY TO
HAVE A FIVE-FOLD SYMMETRY.

A blue circle highlights one of the shapes.

He continues AND WE JUST PROVED TO OURSELVES
THAT FIVE-FOLD SYMMETRY IS
IMPOSSIBLE FOR CRYSTALS, IT'S
IMPOSSIBLE FOR ANY KIND OF
MATTER WHICH IS MADE FROM
REPEATING BUILDING BLOCKS.
NOW, JUST SEEING A PICTURE LIKE
THAT IS UNCONVINCING TO REALLY
CONVINCE YOURSELF WHAT THE
SYMMETRY IS.
WHAT YOU DO IS YOU PERFORM A
KIND OF DIFFRA-- WHAT WE CALL AN
ELECTRON DIFFRACTION EXPERIMENT.
WE SCATTER ELECTRONS THROUGH A
GRAIN OF THIS MATERIAL AND LOOK
AT THE SCATTERING PATTERN IT
PRODUCES, BECAUSE THAT
SCATTERING PATTERN REVEALS THE
ORDER AND SYMMETRY OF THE SOLID
THAT THE ELECTRONS PASS THROUGH.
SO, FOR EXAMPLE, IF YOU PASS
ELECTRONS THROUGH A CUBIC
CRYSTAL IT WILL PRODUCE A
LATTICE OF SHARP SPOTS WHICH
INDICATES-- THE SHARPNESS
INDICATES THAT THE STRUCTURE IS
MADE FROM EXACTLY REPEATING
ELEMENTS OVER AND OVER AGAIN,
WITH EQUAL SPACING.
AND FURTHERMORE, THE PATTERN OF
SPOTS IF I SHOOT THE ELECTRONS
ALONG AN AXIS OF, SAY, FOUR
SQUARE SYMMETRY, THE PATTERNS OF
SPOTS WILL HAVE A SQUARE PATTERN
OF SPOTS.
WHAT HAPPENED IN THIS CASE
THOUGH IS THEY GOT A PATTERN
LIKE THIS.

A picture of white spots on a black background appears.

He continues THEY GOT A PATTERN WHICH HAD
SHARP SPOTS.
THESE ARE SHARP; THE REASON WHY
THEY LOOK KIND OF BROAD IS
SIMPLE THAT THEY OVEREXPOSED THE
FILM, BUT THEY'RE ACTUALLY QUITE
SHARP.
BUT WHEN YOU LOOK AT THE
SYMMETRY OF THIS PATTERN, IT
REVEALS SOMETHING WHICH WAS
SHOCKING, AT LEAST FOR THE TIME.

He approaches the screen and says IF YOU LOOK AT, FOR EXAMPLE, AT
THESE SPOTS OVER HERE, THEY
FORM A RING OF SPOTS.
IF YOU COUNT THEM AS THEY GO
AROUND - THERE'S ONE, TWO,
THREE, FOUR, FIVE, SIX, SEVEN,
EIGHT, NINE, TEN - TEN-FOLD
SYMMETRY, ONE OF THE MANY
DISALLOWED SYMMETRIES FOR
CRYSTALS, EVEN THOUGH THE
PATTERNS ARE SHARP, WHICH IS
INDICATING THAT IT'S ORDERED
LIKE A CRYSTAL.
THERE'S SOME REGULAR ORDERING TO
THIS SAMPLE.
AND THEN IF YOU LOOK MORE
CLOSELY YOU MIGHT EVEN BE ABLE
TO PICK OUT PENTAGONS AND
DECAGONS IN THE PATTERN AS WELL.
SO THIS PATTERN WAS A REAL
PUZZLE TO THEM.
ALL THEY COULD SAY IS THAT IT
DISOBEYED THE LAWS OF
CRYSTALLOGRAPHY, BUT THEY HAD NO
EXPLANATION FOR HOW THIS WAS
POSSIBLE, WHAT THE INTERNAL
STRUCTURE WOULD BE THAT WOULD
ALLOW THIS.
WELL AS IT TURNED OUT, THE
ANSWER TO THE PUZZLE FOR
EXPLAINING THESE MATERIALS - OR
A POTENTIAL ANSWER WAS LAYING IN
THE WINGS.
AT THE SAME TIME THAT THIS GROUP
DANNY "SHEKTWAN" AND HIS GROUP
AT THE NATIONAL BUREAU OF
STANDARDS OUTSIDE WASHINGTON
WERE EXPLORING THIS AND TRYING
TO UNDERSTAND THEIR PUZZLING
GRAIN OF ALUMINUM MANGANESE.
MY STUDENT AND I AT THE
UNIVERSITY OF PENNSYLVANIA WHERE
WORKING ON THE IDEA OF A
HYPOTHETICAL NEW KIND OF SOLID
THAT WE THOUGHT COULD VIOLATE
THE LAWS OF CRYSTALLOGRAPHY, AND
WE CALL THESE THINGS
QUASICRYSTALS.
WE WANTED TO MAKE SOMETHING
WHICH WAS AS CLOSE TO A CRYSTAL
AS POSSIBLE BUT DIDN'T HAVE TO
OBEY THE STANDARD RULES OF
SYMMETRY, THAT COULD BREAK THOSE
200 YEAR OLD RULES.
'COURSE, MOST PEOPLE ASSUME THAT
THAT WAS IMPOSSIBLE BASED ON THE
PROOFS-- THE 200 YEAR OLD
PROOFS, BUT WE HAD A NOTION OF
HOW WE MIGHT GET AROUND THOSE
CONSTRAINTS.
SO WE STILL WANTED TO HAVE
SOMETHING WHICH HAD AN ORDERLY
ARRANGEMENT, AND IN PARTICULAR,
WHEN YOU DIFFRACTED ELECTRONS
THROUGH IT WOULD GIVE YOU SHARP
SPOTS.
WE STILL WANTED TO HAVE
SOMETHING THAT HAD A CLEARLY
DEFINED ROTATIONAL SYMMETRY.
IN FACT, WE WANTED TO GET
ROTATIONALLY SYMMETRIES THAT ARE
NORMALLY IMPOSSIBLE FOR
CRYSTALS.
WE ALSO WANTED THE STRUCTURE TO
BE WHAT THEY MIGHT CALL SIMPLE,
SIMPLE IN THE SENSE THAT IT'S
MADE FROM REPEATING BUILDING
BLOCKS.
WE WANTED TO HAVE ALL THOSE
PROPERTIES BUT WE WANTED TO
BREAK THOSE RULES OF-- 200 YEAR
OLD RULES.
TO DO THAT WE INTRODUCED ONE
SUBSTANTIAL CHANGE, WHICH HAD TO
DO WITH THE ORDERING OF THE
ELEMENTS.
INSTEAD OF LOOKING-- INSTEAD OF
CONSIDERING STRUCTURES OR
TILINGS OR ATOMIC ARRANGEMENTS
WHICH WERE PERIODIC, WE SWITCHES
TO THINKING ABOUT ONES WHICH ARE
WHAT ARE CALLED QUASI PERIODIC.
NOW QUASI PERIODIC MEANS THAT
THERE'S NOT A SINGLE REPEATING
DISTANCE BETWEEN ELEMENTS.
THERE IS TWO OR MAYBE THREE OR
SOME OTHER FINITE NUMBER OF
THEM.
BUT NOT ONLY THAT, BUT WHEN YOU
LOOK FOR THE DISTANCES, THE
RATIO IS NOT A RATIONAL NUMBER -
IT'S AN IRRATIONAL NUMBER.
THAT MEANS IT CAN'T BE EXPRESSED
AS A RATIO OF INTEGERS.
SO OF COURSE, TWO IS AN INTEGER
AND IT'S A RATIO OF INTEGERS -
TWO DIVIDED BY ONE.
SEVEN SIXTHS IS A RATIONAL
NUMBER BECAUSE IT'S A RATIO OF
INTEGERS - SEVEN TO SIX.
BUT NUMBERS LIKE THE SQUARE ROOT
OF TWO, SQUARE ROOT OF FIVE,
ETCETERA - THOSE ARE EXAMPLES
IRRATIONAL NUMBERS.
WHEN YOU EXPRESS THEM AS
DECIMALS, THE DECIMAL NEVER
REPEATS AND THEY CAN NEVER BE
EXPRESSED AS A RATIO OF
INTEGERS.
SO WE HAD THE IDEA THAT YOU
MIGHT HAVE, FOR EXAMPLE, A
STRUCTURE WHISH IS DEFINED BY
TWO REGULARLY REPEATING
REPETITIONS, WHOSE RATIO WAS AN
IRRATIONAL NUMBER, AND THEN WE
SHOWED YOU COULD GET AROUND THE
RULES, THE 200 YEAR OLD RULES
ABOUT ROTATIONAL SYMMETRY.
NEW SYMMETRIES ARE POSSIBLE,
INCLUDING FIVE-FOLD SYMMETRY IN
THE PLANE, SEVEN-FOLD SYMMETRY
IN THE PLANE, 43-FOLD SYMMETRY
IN THE PLANE,
AND
THE
SUPER FORBIDDEN CASE OF
ICOSAHEDRAL SYMMETRY IN THREE
DIMENSIONS.
THE OTHER ELEMENT, OR SLIGHT
CHANGE TO THE STAND OF RULES IS
THAT WE IMAGINE THAT WE DIDN'T
HAVE TO STICK WITH STRUCTURES
WHICH JUST HAD A SINGLE
REPEATING ELEMENT, THEY MIGHT
HAVE TWO OR A FINITE NUMBER OF
REPEATING ELEMENTS.
THAT WOULD BE A WAY BY HAVING
ELEMENTS OF DIFFERENT SIZES OF
GETTING SORT OF TWO LENGTHS IN
THE PROBLEM WHOSE RATIO WOULD BE
IRRATIONAL.
SO ESSENTIALLY WHAT QUASI
PERIODIC MEANS, WHERE IT WOULD
BE ANALOGOUS IN MUSIC WOULD BE
DISHARMONY - TO CHOOSE TWO
FREQUENCIES WHICH ARE NOT
HARMONICS OF ONE ANOTHER.
A QUASICRYSTAL IS THE ANALOG IN
SPACE, IT'S A DISHARMONIC
CRYSTAL, IF YOU LIKE, WITH AT
LEAST TWO REPEATING INTERVALS
WHOSE RATIO IS IRRATIONAL.
NOW OUR INSPIRATION FOR THINKING
ABOUT THIS IDEA CAME FROM A
TILING THAT WAS FIRST PRODUCED
BY A PHYSICIST BY THE NAME OF
ROGER PENROSE.

A picture of a turquoise diamond-pattern tiling appears.

He continues AND WHEN ROGER MADE THIS TILING,
WELL, HE ACTUALLY HAD NO
INTEREST IN CRYSTALS, HE DIDN'T
ACTUALLY KNOW THE TILING HE
MADE WAS QUASI PERIODIC - WHAT
HE WAS TRYING TO DO WAS SOLVE A
CERTAIN KIND OF PUZZLE THAT HAD
BEEN AROUND FOR, OH, DECADES.
THE GOAL OF THE PUZZLE, OR THE
IDEA OF THE PUZZLE WAS, CAN YOU
FIND A SET OF SHAPES WHICH CAN
ONLY TILE THE PLANE
NONPERIODICALLY?
SO, WE KNOW WE CAN TAKE SHAPES
WITH SOME COMBINATIONS, TILE
THEM PERIODICALLY OR
NONPERIODICALLY.
THE QUESTION IS COULD YOU FIND A
SET OF SHAPES WHICH WHEN YOU PUT
THEM TOGETHER ARE FORCED BY THE
WAY THEY JOIN TO ONE ANOTHER TO
ONLY TILE THE PLANE
NONPERIODICALLY.
AND THIS IS THE TILING THAT HE
PRODUCED.
NOW YOU MIGHT LOOK AT THIS AND
SAY THERE'S SOMETHING A LITTLE
BIT FUNNY ABOUT THIS BECAUSE,
WELL, IT'S MADE OF THESE TWO
UNITS WHICH RHOMBUSES - A SKINNY
ONE AND A FAT ONE - AND WE JUST
SAID BEFORE THAT I CAN TAKE A
SKINNY RHOMBUS, SAY, OR A FAT
RHOMBUS, AND I CAN TILE THEM
PERIODICALLY.
SO, THE TILING IS NOT QUITE AS
SIMPLE AS I'VE SHOWN IT ON THE
LEFT.
IN ORDER TO FORCE IT TO BE
NONPERIODIC YOU HAVE TO IMPOSE
SOME RULES WHICH ONLY ALLOW THE
PIECES TO MEET ALONG SOME WAYS
AND NOT OTHER WAYS, IN SUCH A
WAY AS TO PREVENT IT FROM MAKING
ANY KIND OF PERIODIC PATTERN.
SO HOW DO YOU IMPOSE THOSE
RULES?
WELL, THERE ARE VARIOUS WAYS.
LOTS OF SCHEMES WHICH ARE
VARIATIONS OF THE SAME BASIC
IDEA.
ONE WAY IS TO TAKE THE SIDES OF
THOSE TILES THERE AND CURVE THEM
A BIT SO THEY KIND OF ONLY
INTERLOCK IN SOME WAYS BUT NOT
IN OTHER WAYS, SO THEY CAN'T
MEET UP IN ALL POSSIBLE
DIRECTIONS.
SO THAT WAS KIND OF THE WAY
PENROSE THOUGHT ABOUT IT.
IN FACT, HE MADE A PARTICULARLY
CLEVER VERSION OF IT.
THESE ARE CALLED PENROSE
CHICKENS, AND THEY HAVE
INTERLOCKS.

The screen shows his hands fitting in colourful chicken figures.

He continues THEY'RE SHAPED IN SUCH A WAY
THAT THEY HAVE KIND OF
INTERLOCKS SO THAT, FOR EXAMPLE,
I CAN TAKE TWO OF THESE FAT
CHICKENS AND I CAN TILE THEM
TOGETHER THAT WAY, AND THEN I
CAN TAKE ONE OF THESE SKINNY
CHICKENS AND PUT IT IN THERE.
HERE'S ANOTHER SKINNY CHICKEN.
AH, HERE'S A FAT CHICKEN.
OKAY, SO, UM...
AND SO THE INTERESTING THING
ABOUT THESE RULES IS, THE MORE
YOU THINK ABOUT THEM THE MORE
AMAZING IT IS.
THE INTERESTING THING ABOUT
THESE RULES IS THAT NUMBER ONE:
THEY DO
ALLOW YOU TO FILL THE ENTIRE
PLANE AS MUCH AS YOU WANT.
BUT NUMBER TWO: IT'S PURELY
NONPERIODIC.
NEVER-- EVERY TIME YOU THINK THE
PATTERN'S GONNA REGULARLY
REPEAT, YOU FOLLOW IT FAR OUT
ENOUGH, IT DOES STOP REGULARLY
REPEATING - SOMETHING CHANGES
OUT THERE THAT MAKES THE PATTERN
NOT REPEAT.
SO IT'S REALLY QUITE AN AMAZING
CONSTRUCTION.

The audience listens carefully.

He continues NOW, THE PROBLEM WITH THIS WAY
OF SHOWING THE TILES, ALTHOUGH
IT'S A LOT OF FUN, IT DOESN'T
ACTUALLY SHOW YOU THE HIDDEN
SYMMETRY IN THE TILINGS.
A MUCH BETTER WAY OF ENFORCING
THESE MATCHING RULES, RULES
WHICH CONSTRAIN THE WAY YOU PUT
THEM TOGETHER, IS THE ONE SHOWN
HERE.
THIS IS THE ONE WE GOT
INTERESTED IN, IS TO LABEL EACH
OF THE FAT TILES WITH A CERTAIN
ARRANGEMENT OF LINES.
EACH OF THE SKINNY TILES WITH A
DIFFERENT ARRANGEMENT OF LINES,
BUT EVERY FAT TILE THE SAME WAY,
EVERY SKINNY TILE THE SAME WAY.
AND THEN WE MAKE THE RULE THAT
WHEN YOU JOIN TWO TILES ALONG AN
EDGE YOU'RE ONLY ALLOWED TO JOIN
THEM SO THAT THE LINES CONTINUE
ACROSS THE INTERFACE.
THAT, IT TURNS OUT, PREVENTS
SOME JOININGS, LIKE THE ONES
WHICH GIVE YOU PERIODIC TILINGS,
AND ADMITS OTHERS, LIKE THE ONES
THAT GIVE YOU THE TILING ON THE
LEFT.
BUT WHAT'S MORE IMPORTANT FOR
OUR PURPOSE IS THAT WHEN YOU
JOIN THESE THINGS UP THIS WAY,
THOSE LITTLE LINE SEGMENTS END
UP REVEALING THE HIDDEN SYMMETRY
OF THE STRUCTURE.
SO WHEN I JOIN THEM UP, IF I DO
THIS FOR EACH TILING, THIS IS
WHAT HAPPENS.
THEY FORM CONTINUOUS LINES
THROUGH THE WHOLE STRUCTURE.
SO YOUR EYE MIGHT HAVE TOLD YOU
THERE WAS SOME HIDDEN ORDER TO
THIS, BUT NOW YOU
KNOW
THERE'S AN ORDER TO IT

He approaches the screen and points to the diamond-pattern tiling picture.

He continues 'CAUSE
YOU NOW KNOW, FOR EXAMPLE, WHEN
I PUT THIS TILE DOWN HERE
THERE'S A LITTLE LINE SEGMENT
ALONG HERE, IT FORCES AN
INFINITE ALIGNMENT OF TILES
GOING OUT IN THIS DIRECTION, AND
SIMILARLY IN THIS DIRECTION,
ETCETERA.
SO THERE IS DEFINITELY-- THIS IS
NOT A RANDOM STRUCTURE, THIS IS
NOT SIMPLY A NONPERIODIC
STRUCTURE; IT'S SOMETHING MUCH
MORE SPECIAL THAN THAT.
NOW WHAT IS IT?
WELL, IF YOU LOOK AT THE
PATTERNS OF LINES YOU'LL FIND
THERE ARE FIVE OF THEM, EACH ONE
OF THE PARALLEL TO ONE OF THE
EDGES OF A PENTAGON.
FURTHERMORE, THE SEQUENCE OF
LINES IN EACH OF THOSE FIVE
DIRECTIONS TURNS OUT TO BE THE
SAME.
SO NOW YOU HAVE EXPLICIT PROOF
THAT THE STRUCTURE REALLY HAS AN
OVERALL GLOBAL FIVE-FOLD
SYMMETRY, THE FAMOUS SYMMETRY
THAT'S DISALLOWED FOR CRYSTALS.
YOU MAY HAVE BELIEVED THAT
BECAUSE YOU SAW SOME PENTAGONAL
SHAPES IN THERE, BUT BELIEVE ME,
YOU CAN MAKE LOTS OF SHAPES WITH
PENTAGONAL MOTIFS WHICH ARE NOT
FIVE-FOLD SYMMETRIC.
ISLAMIC ART IS FULL OF SUCH
STRUCTURES AND THEY TURN OUT TO
BE ACTUALLY PERIODIC PATTERNS,
BUT JUST WITH PENTAGONAL MOTIFS.
THIS IS SOMETHING REALLY
DIFFERENT.
THIS IS LEGITIMATELY A PATTERN
WHICH HAS AN INFINITE FIVE-FOLD
SYMMETRY.
NOW IF YOU LOOK AT THE PATTERN
OF LINES ALONG ANY ONE DIRECTION
YOU'LL DISCOVER SOMETHING ELSE
INTERESTING ABOUT THEM, WHICH IS
THAT THERE'S ONLY, WELL, THERE'S
NOT A SINGLE SEPARATION BETWEEN
THEM, BUT THERE'S ALSO NOT AN
INFINITE NUMBER OF THEM -

Now, diagonal black lines cross the picture.

He continues THERE'S ONLY TWO POSSIBLE
SEPARATIONS, WHAT WE CALL A LONG
OR SHORT SEPARATION.
OKAY?
SO EACH INTERVAL BETWEEN LINES
WHEN YOU PRODUCE THIS STRUCTURE
AS EITHER LONG OR SHORT, THE
RATIO OF THE LONG TO THE SHORT
LENGTHS, THE RATIO OF THOSE
LENGTHS IS AN IRRATIONAL
NUMBER.
SO THAT'S OUR FIRST HINT OF
QUASI PERIODICITY.
IT'S NOT SQUARE ROOT OF TWO, ITS
NOT SQUARE ROOT OF FIVE.
IT'S A NUMBER WHICH, WELL, CAN
BE WRITTEN AS ONE PLUS THE
SQUARE ROOT OF FIVE DIVIDE BY
TWO.

[Audience laughter]

He continues THAT MAY NOT
BE IMMEDIATELY RECOGNIZABLE TO
YOU.
BUT WHAT IF I TELL YOU...
WHAT IF I TELL YOU THAT NUMBER
IS THE GOLDEN RATIO?
OKAY.
THAT IS THE FAMOUS GOLDEN RATIO
THAT NATURE USES IN DESIGNING
THE SEEDS AND SUNFLOWERS, OR THE
SHAPE OF A NAUTILUS SHELL, OR
THAT THE GREEKS THOUGHT WAS THE
IDEAL RATIO OF THE HEIGHT TO
WIDTH OF A HUMAN BEING, IS USED
IN ART, AND ALSO APPEARS AS...
IT'S ALSO RELATED TO, UM, IF YOU
HAVE A PENTAGON, THAT'S RELATED
TO THE CIRCLE THAT...
AND SCRIBES THE PENTAGON
COMPARED TO THE EDGE LENGTH OF
PENTAGON, ETCETERA.

He walks back and forth on the stage.

He continues SO IT'S A FAMOUS RATIONAL
NUMBER.
NOW, SOMETHING ELSE INTERESTING.
IF YOU FOLLOW THE SEQUENCE OF
THOSE LONGS AND SHORTS, WELL,
THIS IS SORT OF TYPICAL A LITTLE
BIT, YOU SEE.
AT FIRST IT LOOKS LIKE THIS...
YOU HAVE SOME SORT OF REPEATING
PATTERN - LONG-SHORT-LONG-SHORT.
BUT THEN IT'S LONG-LONG.
JUST WHEN YOU THOUGHT YOU HAD
GOTTEN THE RHYTHM...
[Fingers snap]
IT BREAKS THE RHYTHM.
AND YOU'D FIND THAT IF WE'D MADE
THE PATTERN ARBITRARILY ALONG
THAT IT WOULD ALWAYS BE THE
CASE.
OKAY, IT LOOKS LIKE IT'S
SETTLING INTO A RHYTHM AND THEN
ALL OF A SUDDEN...
[Fingers snap]
SOMETHING HAPPENS TO BREAK THE
RHYTHM.
AS YOU STUDY THE PATTERN MORE
CLOSELY, DISCOVER THAT, IN FACT,
THERE'S SOMETHING QUITE SUBTLE
GOING ON.
IF YOU COUNT THE NUMBER OF LONGS
AS YOU GO ALONG AND THE NUMBER
OF SHORTS AS YOU GO ALONG, YOU
FIND THAT THEY FOLLOW A RATIO -
THERE'S A CERTAIN RATIO, WHICH
IS RATIO, OF COURSE, OF
INTEGERS, BUT FAMOUS INTEGERS,
INTEGERS KNOWN FIBONACCI
NUMBERS.
SO THOSE ARE A FAMOUS SET OF
INTEGERS THAT HAVE BEEN STUDIED
FOR OVER A THOUSAND YEARS, WHICH
HAVE THE PROPERTY THAT AS YOU
CHOOSE FIBONACCI NUMBERS OF
HIGHER AND HIGHER ORDER, THE
RATIO APPROACHES AN IRRATIONAL
NUMBER.
A RATIO OF INTEGERS WHICH IN A
SEQUENCE APPROACHES AN
IRRATIONAL NUMBER, AND NOT JUST
ANY IRRATIONAL NUMBER, BUT THE
GOLDEN RATIO.
OKAY?
SO WHAT DOES THAT MEAN?
THAT MEANS THAT THE FREQUENCY IN
WHICH L's APPEAR AND THE
FREQUENCY WITH WHICH S's APPEARS
ARE RELATIVELY IRRATIONAL -
THEIR RATIO IS IRRATIONAL.
AND, IN FACT, THEIR RATIO IS THE
GOLDEN RATIO.
SO THIS IS A PROOF IF YOU LIKE,
A GEOMETRICAL PROOF THAT WHAT
YOU'VE GOT IS SOMETHING WHICH IS
NOT CRYSTALLINE, IS FIVE-FOLD
SYMMETRIC, AND IS QUASI
PERIODIC, HAS TWO PERIODS, TWO
REGULAR INTERVALS WHOSE RATIO
REPEAT WITH RELATIVELY-- WHERE
THE FREQUENCY OF HIS RATIO IS AN
IRRATIONAL NUMBER.
NOW ONCE YOU UNDERSTAND THAT'S
WHAT'S REALLY GOING BEHIND THE
PENROSE TILING, IT TURNS OUT
IT'S PRETTY EASY TO SHOW THAT
YOU CAN NOW MAKE TILINGS WITH,
WELL, ARBITRARY SYMMETRY.
SO, HERE, FOR EXAMPLE, IS OUR
PENROSE TILING CONSISTING OF OUR
TWO TILES, FAT AND SKINNY TILES.
BUT HERE'S ONE WITH SEVEN-FOLD
SYMMETRY WHICH HAS-- IT HAS TO
HAVE THREE TILES, THREE
DIFFERENT RHOMBUS TILE SHAPES.
HERE'S ONE WITH, I THINK,
17-FOLD SYMMETRY.

Pictures show three intricate multi-coloured patterns.

He continues NOW, THESE ARE VERY PRETTY
PATTERNS.
ONE TO FOLD QUILT PATTERNS, FOR
EXAMPLE.
I HAVE A NICE QUILT OF ONE OF
THESE PATTERNS.
BUT, UM, I SHOULD SAY THAT WITH
THE PICTURE YOU'RE SEEING THEM
HERE IS A LITTLE BIT DECEIVING
BECAUSE I'VE INTENTIONALLY GONE
TO A PLACE IN THE PATTERN WHERE
THERE'S A LOCAL CENTER OF
SYMMETRY, SO IT'S EASY FOR YOU
TO SEE THE SYMMETRY.
IF YOU SORT OF FOCUS ON A CORNER
OUT HERE, YOU CAN IMAGINE THAT'S
ALL YOU SAW, THE THING LOOKS
VERY RANDOM.
YOU WOULDN'T GUESS THAT IT
REALLY HAD THIS HIDDEN QUASI
PERIODICITY AND SYMMETRY.
SO THESE THINGS ARE REALLY
QUITE, QUITE SUBTLE.
AND WHAT WE WERE ESPECIALLY
INTERESTED IN WAS
THREE-DIMENSIONAL PATTERNS
'CAUSE WE HAD IN MIND MAKING NEW
KINDS OF SOLIDS.
SO WE WERE PARTICULARLY
INTERESTED IN THE FACT THAT YOU
CAN ALSO MAKE THE
MOST
FORBIDDEN SYMMETRY, A
THREE-DIMENSIONAL STRUCTURE WITH
ICOSAHEDRAL SYMMETRY.
IN THIS CASE, INSTEAD OF NEEDING
TWO FUNDAMENTAL BUILDING BLOCKS
LIKE WE NEEDED FOR THE PENROSE
TILING, IT TURNS OUT THAT WE
NEED FOUR.
THEY'RE ACTUALLY A FAMOUS FAMILY
OF SHAPES - FAMOUS AT LEAST
AMONG GEOMETERS.
A SET OF SHAPES WHICH ARE KNOWN
AS ZONOHEDRON.
THEY GET THEIR NAME FOR THE
FOLLOWING REASON:
THEY TAKE THE FIRST ONE OF
THESE.

A close-up reveals four geometric objects, a red one, a blue one, a yellow one and a white one. He places them on a white surface.

He picks the red one and says OKAY, THIS IS CALLED, THIS
PARTICULAR ONE, IS CALLED THE
RHOMBIC TRIACONTAHEDRON.
OKAY, SO, NOW THAT YOU GOT
YOUR-- YOU'RE QUICK ON YOUR
GREEK, YOU HAVE ALREADY REALIZED
THAT "HEDRON" MEANS FACE, AND
"TRIACONTRON" MEANS 30, SO THIS
HAS 30 RHOMBIC FACES.
IT'S A RHOMBIC TRIACONTAHEDRON.
ANOTHER INTERESTING CUTE THING
ABOUT IT IS IF YOU HOLD IT BY
ITS TIPS, WHAT YOU'LL FIND IS
THAT THERE'S A BAND OF RHOMBUSES
THAT RUN AROUND ITS CENTER THAT
KIND OF GO UP AND DOWN.
IF YOU IMAGINE CUTTING OUT THAT
BAND OF RHOMBUSES AND GLUING
THE TWO PIECES TOGETHER, YOU
WOULD END UP WITH T[HIS
SHAPE, OKAY?

He grabs the blue object and says THIS SHAPE IS KNOWN AS RHOMBIC
ICOSAHEDRON.
SO, REMEMBER, ICOSA MEANS 20, SO
IT'S A 20-FACED RHOMBIC SURFACE.
AND AGAIN, IF YOU HOLD IT BY ITS
TIPS YOU WILL SEE THERE'S A BAND
OF RHOMBUSES THAT RUN THROUGHOUT
IT, IT'S BELT.
AND IF YOU IMAGINE CUTTING THOSE
OUT AND GLUING THEM TOGETHER
YOU'LL GET TO THIS SHAPE, WHICH
IS THE RHOMBIC DODECAHEDRON.

Holding the yellow object, he continues AND, YES, IF YOU HOLD IT BY ITS
TIPS AND IMAGINE CUTTING OUT THE
CENTRAL BAND OF RHOMBUSES,
YOU'LL GET TO THIS SHAPE WHICH
IS JUST A SIMPLE RHOMBAHEDRON.
HERE'S THE POINT, HERE'S THE
INTERESTING THING ABOUT THESE
PARTICULAR SHAPES.
WE DON'T JUST HAVE THESE SHAPES,
BUT YOU NOTICE, YOU CAN PROBABLY
SEE ON THERE THAT THEY HAVE
THESE VARIOUS SLITS AND NOTCHES
IN THEM, AND THEY'RE NOT ALL
IDENTICAL - A LOT OF THEM ARE
IDENTICAL - BUT SOME OF THEM ARE
NOT.
OKAY, LET'S SEE IF I CAN GET
ONE.
IT'S NOT, OKAY?
SOME OF THEM ARE NOT.
THAT MEANS I CAN JOIN ACTUALLY
SOME OF THESE FACETS TOGETHER
BUT NOT OTHERS.
THE INTERESTING THING IS THAT IF
YOU WANTED TO FILL THIS ROOM
WITH UNITS LIKE THIS, YOU COULD
DO IT, BUT THE ONLY WAY YOU
COULD DO IT, THE ONLY POSSIBLE
WAY YOU COULD DO IT IS IN A
NONPERIODIC ICOSAHEDRALLY
SYMMETRIC QUASICRYSTAL LATTICE.
SO THIS IS THE THREE-DIMENSIONAL
ANALOG OF WHAT WE SAW FROM THE
PENROSE TILING.
MUCH MORE SUBTKE THE WAY IT
WORKS, BUT IT'S THE
THREE-DIMENSIONAL ANALOG OF THE
PENROSE TILING.
MUCH MORE SUBTLE IN THE WAY IT
WORKS, BUT IT'S A
THREE-DIMENSIONAL ANALOG OF
THE PENROSE TILE.
AND I'M NOT GONNA FILL THIS
ROOM, BUT JUST SO YOU CAN GET AN
IMPRESSION OF WHAT ONE OF THESE
THINGS-- WHAT IT WOULD LOOK
LIKE...
CAN YOU SEE?
OKAY.
HERE'S A SURFACE OF WHAT IT
WOULD LOOK LIKE.

He grabs a round piece made up of the geometrical objects mentioned.

He continues SO IF YOU ACTUALLY FILL THIS
ROOM, LOOKED DOWN TO FIVE-FOLD
SYMMETRY AXIS, WHAT YOU'D FIND
IS THAT IT FORMS LAYERS - NO TWO
OF WHICH ARE IDENTICAL - WHICH
WOULD FORM A KIND OF CRINKLY
SURFACE WHICH COMES IN THE
JOININGS OF THESE ELEMENTS.
IF YOU LOOKED AT THIS THING -
STARE AT IT A LITTLE BIT - YOU
MIGHT EVEN NOTICE THAT THERE
ARE-- THAT ACTUALLY, THESE
LITTLE SLOTS I'VE PUT INTO THIS
TO JOIN THEM UP, ACTUALLY FORM
LINES THAT RUN THROUGHOUT THE
STRUCTURE.
AND IF YOU HAVE A GOOD
IMAGINATION YOU CAN IMAGINE
LOOKING THROUGH IT AND EVEN
SEEING PLANES RUNNING THROUGH
THE STRUCTURE.
AND IN FACT, THAT IDEA IS
EXACTLY RIGHT.
THE WAY THESE WERE OBTAINED WAS
JUST AS BEFORE FOR THE PENROSE
TILES WE HAD LITTLE LINE
SEGMENTS WHICH WE HAD TO MATCH
UP IN ORDER TO FORCE THEM TO
JOIN TOGETHER.
HERE, THE ANALOG IN THREE
DIMENSIONS HAPPEN TO BE PLANES,
AND THE WAY WE'VE SHOWN THE
PLANES IS THROUGH THOSE SLIDES.
WELL IN 1984 WE HAD MADE THIS
STRUCTURE AND WE HAD EVEN
FIGURED OUT HOW YOU WOULD DETECT
THAT A MATERIAL HAD THE
STRUCTURE.
SEE, WE HAD THE IDEA THAT IF YOU
COULD MAKE UNITS WHICH HAD
FORCES, JOINTS ON THEM WHICH CAN
FORCE THE STRUCTURE, MAYBE YOU
CAN HAVE ATOMS OR MOLECULES
WHICH MAKE THE SAME STRUCTURE,
AND THEN YOU COULD HAVE A NEW
KIND OF MATERIAL, A NEW KIND OF
MATTER, NEW KIND OF SOLID, WHICH
WOULD HAVE THIS ICOSAHEDRAL
QUASICRYSTAL SYMMETRY.
SO WHAT WE DID IS WE FIGURED OUT
WHAT WOULD HAPPEN IF YOU WOULD
DIFFRACT, FOR EXAMPLE, ELECTRONS
WITH SUCH A MATERIAL, WHAT IT
WOULD LOOK LIKE.
IN THE FALL
OF '84 I WENT TO IBM RESEARCH TO
TRY TO CONVINCE PEOPLE TO LOOK,
TRY TO SEE IF THEY CAN MAKE
MATERIALS WITH THIS STRUCTURE.
IT WAS A HARD GO BECAUSE THIS IS
JUST A HYPOTHETICAL IDEA BASED
ON THIS KIND OF TOYS.
IT'S A HARD EFFORT TO TRY TO
MAKE A MATERIAL OR TO IMAGINE
MAKING A MATERIAL WITH THIS
STRUCTURE.
BUT WHILE I WAS AT IBM I HAD A
NICE VISIT FROM A FORMER
COLLABORATOR, DAVID NELSON FROM
HARVARD UNIVERSITY, AND HE
HAPPENED TO BE COMING BY TO IBM.
AND HE KNEW I WAS WORKING ON
SOMETHING IN THIS GENERAL AREA,
ALTHOUGH HE DIDN'T KNOW QUITE
WHAT I WAS DOING, BUT HE SAID
THAT-- HE CALLED ME UP AND HE
SAID "I THINK I HAVE A PAPER
THAT MIGHT INTEREST YOU."
AND SO HE CAME BY MY OFFICE, AND
I SAID "BUT I HAVE SOMETHING I
WANNA SHOW YOU."
HE SAID "NO, NO.
I WANNA SHOW YOU THIS PAPER."
I SAID "NO, NO.
I WANNA SHOW YOU THIS
CALCULATION.
SO, UM, WE WERE JOKING AROUND
FOR A BIT, AND HE WON, UM...
SO HE GOT TO SHOW ME THE PAPER,
AND I LOOK AT THIS PAPER WHICH
WAS THIS PAPER FROM DAN
"SHEKTWAN" SHOWING THOSE QUASI...
THAT MATERIAL I SHOWED YOU
BEFORE, THOSE GRAINS I SHOWED
YOU BEFORE.
AND I WAS FLIPPING THROUGH IT
AND FLIPPING THROUGH IT, MY EYES
WERE GETTING WIDER AND WIDER,
AND AT SOME POINT, I THINK I
MUST HAVE JUST JUMPED UP IN THE
AIR, AND I DON'T KNOW WHAT HE
WAS THINKING AT THE TIME, UM,
WHY I DID THAT.
BUT BASICALLY, THE REASON WHY I
DID THAT WAS BECAUSE, WELL, THE
PATTERN ON THE RIGHT IS THE
PATTERN FROM THAT MATERIAL I
SHOWED YOU, AND WHAT I HAD
WANTED TO SHOW HIM WAS OUR
CALCULATION.

Two pictures appear on the screen. On the left, a one shows white dots against a blue surface. On the right, another one shows larger white dots against a black surface.

He continues WE'D JUST CALCULATED WHAT THE
DIFFRACTION PATTERN FROM AN
IDEAL QUASICRYSTAL SHOULD LOOK
LIKE.
AND THE THING ON THE LEFT WAS
THE THING I WANTED TO SHOW HIM.
AND I SHOULD POINT OUT THE
THEORETICAL CALCULATION SHOWS
MORE DOTS ON THE LEFT THAN THE
ONE ON THE RIGHT, AND THAT'S
BECAUSE THE THEORETICAL
CALCULATION ASSUMES AN
EXPERIMENT WHICH IS MUCH MORE
ACCURATE AND WITH A MATERIAL
WHICH IS MUCH MORE PERFECT THAN
THE MATERIAL ON THE RIGHT.
BUT TO THE LEVEL AT WHICH ONE
CAN COMPARE THEM, THEY ARE ON
ONE TO ONE, CORRESPONDS WITH ONE
ANOTHER.
AND SO... [Unclear]...THE SUGGESTION BY US THAT, IN
FACT, MAYBE WHAT THE
EXPLANATION, THE NATURAL
EXPLANATION FOR WHAT THEY FOUND
IS THAT IT IS INDEED A
QUASICRYSTAL.
THE FIRST EXAMPLE, THE
QUASICRYSTAL EVER SEEN BY
HUMANS.
NOW, AT FIRST THIS IDEA WAS...
GRABBED ONTO, BUT AFTER A FEW
MONTHS PEOPLE BEGAN TO BECOME
INCREASINGLY SCEPTICAL.
THEY POINTED OUT, FOR EXAMPLE,
THAT ALTHOUGH THERE SEEMS TO BE
PRETTY GOOD AGREEMENT, I MEAN,
AT LEAST SUPERFICIALLY THE EYE
THERE'S PRETTY GOOD AGREEMENT.
IN FACT, YOU'LL NOTICE THAT
THESE MATERIALS ARE NOT
BEAUTIFULLY FACETTED LIKE
CRYSTALS; THEY'RE THESE HIGHLY
DISORDERED...
FEATHERY-LIKE THINGS.
AND ALTHOUGH SHORTLY AFTER THIS
FIRST EXAMPLE IS FOUND, PEOPLE
BEGAN TO MAKE MANY OTHER
QUASICRYSTALS.
ONCE THEY FOUND ONE THEY BEGAN
TO MAKE OTHER ALLOYS WHICH HAD
ICOSAHEDRAL SYMMETRY AND GAVE A
SIMILAR DIFFRACTION PATTERN.
ALL OF THEM WERE THESE KINDS OF
FEATHERY LOOKING AFFAIRS.
AND FURTHERMORE, ALL OF THEM ARE
WHAT WE CALL METASTABLE, THAT IS
TO SAY THAT, UH, IF YOU COOL
THINGS QUICKLY YOU CAN MAKE
THEM, BUT IF YOU GIVE THE ATOMS
ENOUGH TIME TO RELAX THEY TEND
TO FORM CRYSTALS.
AND SO IT BEGAN TO MAKE PEOPLE
SCEPTICAL THAT QUASICRYSTALS
REALLY CAN BE ACHIEVED.
MAYBE YOU CAN GET CLOSE TO THEM,
MAYBE THAT'S WHAT'S HAPPENED
HERE, BUT TRUE QUASICRYSTALS
WOULD HAVE THIS TRUE BEAUTIFUL
ORDER MAY, IN FACT, BE
IMPOSSIBLE.
NOT ONLY DID THEY HAVE THE FACT
THAT THEY COULDN'T MAKE THEM IN
THE LABORATORY, BUT THEY HAD TWO
PRETTY SOLID THEORETICAL
ARGUMENTS.
THE FIRST WAS THAT IN ORDER TO
GROW ON OF THESE STRUCTURES, IN
ORDER TO ACTUALLY FILL THE ROOM
WITH THOSE TILES, IT LOOKS LIKE
YOU NEED SOME KIND OF NON-LOCAL
INTERACTIONS.
BUT I MEAN BY NON-LOCAL
INTERACTIONS IS THE FOLLOWING.
SUPPOSE I LEFT YOU WITH-- YOU
COME UP AFTERWARDS AND YOU TRIED
TO MAKE THE PENROSE TILE WITH
THOSE CHICKEN TILES, OKAY?
NOW YOU CAN DO IT IN PRINCIPLE,
BUT WHAT YOU'RE GONNA FIND IN
PRACTICE IS WHEN YOU START TO
PUT, OH, FIVE OR TEN OF THEM
TOGETHER, YOU'RE GONNA FIND THAT
YOU RUN INTO A CONFLICT, A
DISAGREEMENT - WHAT YOU TILED ON
ONE END WILL DISAGREE WITH WHAT
YOU TILED ON THE OTHER END, AND
YOU'RE GONNA HAVE TO TAKE SOME
AWAY AND MAKE A DIFFERENT CHOICE
AND START AGAIN.
IN FACT, YOU'LL FIND YOU'LL HAVE
TO DO THIS OVER AND OVER AND
OVER AGAIN.
THE ONLY WAY YOU CAN AVOID THAT
IS IF YOU HAD IN MIND THE GLOBAL
PICTURE OF WHAT THE IDEAL TILING
IS LIKE AND COULD LOOK AHEAD AND
MAKE SURE WHEN YOU PUT A TILE
OVER HERE IT WASN'T GONNA
CONFLICT WITH A TILE THAT YOU
WERE LAYING OVER HERE, WHICH
IMPLIES SOME KIND OF LONG RANGE
INTERACTION BETWEEN THE TWO.
YOU'RE USING YOUR EYE IN THIS
CASE, BUT WHAT ARE THE ATOMS AND
MOLECULES DOING WHEN THEY MAKE
THIS STRUCTURE?
ARE THEY SOMEHOW LOOKING OVER
LONG DISTANCES IN ORDER TO
DECIDE WHETHER TO FIT HERE OR
HERE IN THIS WAY?
WELL, THE ATOMS WE'RE TALKING
ABOUT ARE VERY SIMPLE ATOMS,
METAL ATOMS, WHICH, AS FAR AS WE
KNOW, DON'T HAVE THESE LONG
RANGE INTERACTIONS.
SO THE FACT THAT THEY SEEM TO
REQUIRE THESE LONG INTERACTIONS
IS A SERIOUS PROBLEM SUGGESTING
THAT MAYBE YOU CAN'T REALLY GET
TO THESE PERFECT
QUASICRYSTALS.
A SECOND ARGUMENT IS THAT THIS
STRUCTURE'S VERY COMPLICATED -
YOU CAN ALREADY SEE THAT AS YOU
SEE THE PIECES GOING AROUND, THE
WAY THEY JOIN TOGETHER IS RATHER
COMPLICATED, IT NEEDS TO HAVE
ALL UNITS.
HOW DO THESE ATOMS FIGURE OUT TO
MAKE TWO KINDS OF UNITS THAT
ONLY FIT TOGETHER IN CERTAIN
WAYS?
HOW DO THEY FIGURE OUT HOW TO DO
THAT?
SEEMS TOO COMPLICATED COMPARED
TO A CRYSTAL WHICH JUST HAS A
SINGLE REPEATING UNIT.
THESE ARE GOOD ARGUMENTS, AND I
NOW WANT TO TURN TO THEM BECAUSE
IT TURNS OUT THAT THE SUBJECT OF
QUASICRYSTALS DIDN'T JUST HAVE
ONE SURPRISE - THE ONE I'VE
ALREADY PRESENTED YOU, THE FACT
THAT THEY EXIST - BUT IN THESE
ISSUES THERE ARE FURTHER
SURPRISES TO BE FOUND.
SO LET ME FIRST OF ALL TALK
ABOUT THIS ISSUE OF THE
NON-LOCAL INTERACTIONS.

A blue slate reads "Reasons to be skeptical. Requires non-local interactions in order to grow? Two or more repeating units with complex rules for how to join: Too complicated?"

Paul continues LET ME JUST
FIRST OF ALL, LET ME JUST
CONVINCE YOU THAT THERE REALLY
IS A PROBLEM HERE, AND WE'LL DO
IT BY DOING, NOT A PROBLEM OF
TILING THE PLANE 'CAUSE THAT
WOULD TAKE A LOT OF SPACE, WE'LL
THINK ABOUT A MUCH SIMPLER
PROBLEM WHICH IS THE PROBLEM OF
MAKING A CHAIN IN ONE DIMENSION,
WHAT WE'LL CALL A FIBONACCI
CHAIN.

The screen shows two columns of vertical lines, a series that reads "L, S, L, L, S, L, S, L, L, S, L, L, S, L, S, L, L, S, L, S, L" and horizontal lines of different lengths.

He continues A CHAIN OF LONG AND SHORT
LENGTHS WHICH FOLLOWS THAT SAME
SEQUENCE OF LONGS AND SHORTS
THAT WE SAW IN THE PENROSE
PATTERN, IN WHICH THE RATIO OF
LONGS TO SHORTS FOLLOWS A RATIO
OF FIBONACCI NUMBERS.
HERE I'VE SHOWN YOU A GOOD PIECE
OF THE PATTERN THAT CAN BE
CALCULATED-- THERE'S A PIECE OF
THAT INFINITE PATTERN, AND
THAT'S THE IDEAL THAT YOU'RE
SHOOTING FOR.
HOWEVER, THE WAY WE WANT TO GROW
OUR CHAIN IS NOT TO SIMPLY COPY
THAT PATTERN, 'CAUSE THAT WOULD
BE EASY.
WHAT WE WANNA DO IS MAKE A SET
OF RULES.
WE WANT YOU TO DEVELOP...
WHAT WE WANNA DO IS DEVELOP A
SET OF RULES WHICH TELLS US WHEN
TO ADD A LONG AND WHEN TO ADD A
SHORT SUCH THAT WE'LL GET THIS
IDEAL CHAIN.
AND FURTHERMORE, WE WANNA MAKE
IT THAT THESE RULES DON'T
REQUIRE THAT I HAVE TO LOOK ALL
THE WAY BACK ALONG THE CHAIN.
I SHOULD ONLY HAVE TO LOOK BACK
ONE, TWO, MAYBE TEN ELEMENTS,
BUT A FINITE NUMBER OF ELEMENTS.
NOW I CAN USE THE IDEAL PATTERN
TO HELP GUIDE MY THINKING, BUT
I'M NOT ALLOWED TO JUST SIMPLY
COPY IT 'CAUSE THAT'S USING
GLOBAL INFORMATION.
I HAVE TO ONLY USE INFORMATION
ABOUT THE END OF THE CHAIN.
TAKE THIS CHAIN FOR EXAMPLE.
LONG-SHORT-LONG-LONG-SHORT.
CAN I FIGURE OUT WHAT TO ADD
NEXT, LONG OR SHORT?
WELL, YES I CAN BECAUSE IF I
LOOK AT THE IDEAL PATTERN - I
CAN USE THAT AS A GUIDE - I
NOTICE THAT THE SHORT IS ALWAYS
SURROUNDED ON BOTH SIDES BY
LONGS.
YOU NEVER HAVE TWO SHORTS IN A
ROW.
SO I KNOW IN THIS CASE WHICH ONE
TO DO.
JUST BY HAVING A SHORT AT THE
END I KNOW THE NEXT ELEMENT MUST
BE LONG.
SO THAT'S AN EXAMPLE OF WHAT I
WOULD CALL A LOCAL RULE.
IF YOU THINK ABOUT IT AS ATOMS
IT WOULD ONLY REQUIRE LOCAL
INFORMATION ABOUT WHAT'S
HAPPENING AT THE END OF THE
CHAIN TO DECIDE WHETHER TO ADD
AN ATOM OF ONE TYPE OR AN ATOM
OF ANOTHER TYPE.
OR, IN THE TWO DIMENSIONS, A
TILE OF ONE TYPE OR A TILE AT
ANOTHER TYPE.
BUT HERE'S THE PROBLEM.
THE PROBLEM IS YOU CAN ALWAYS
MAKE CHAINS LIKE THIS ONE SHOWN
HERE.
IN THIS CHAIN SHOWN HERE, AT THE
VERY END OF THE CHAIN, I HAVE A
LONG AND I'M NOW FACED WITH A
DECISION.
AM I SUPPOSED TO ADD A LONG OR
AM I SUPPOSED TO ADD A SHORT,
AND IF I ADD A SHORT IT'S GONNA
BE FOLLOWED BY A LONG.
SO I LOOK AT THE IDEAL PATTERN
FOR GUIDANCE, AND WHAT DO I
FIND?
I FIND, HERE'S LONG TO LONG,
THAT'S ALLOWED.
AND HERE'S LONG-SHORT-LONG.
SO ACTUALLY BOTH OF THEM ARE
ALLOWED.
I CAN'T MAKE A DECISION JUST BY
KNOWING THAT THERE'S A LONG IN
THE CHAIN WHICH ONE TO ADD.
SO I'LL GO BACK TWO ELEMENTS -
SHORT-LONG.
SO I CAN EITHER HAVE
SHORT-LONG-LONG; OR
SHORT-LONG-SHORT-LONG.
WHAT HAPPENS UP HERE?
WELL, OKAY, HERE'S
SHORT-LONG-LONG; AND HERE'S
SHORT-LONG-SHORT-LONG.
SO AGAIN, I CAN'T DECIDE ON THE
BASIS OF HAVING SHORT-LONG AT
THE END, LOOKING BACK TWO
ELEMENTS WHICH ONE TO MAKE.
AND I WON'T WALK YOU THROUGH THE
WHOLE THING.
WE COULD GO BACK THREE AND FOUR
AND FIVE AND SIX AND SEVEN...
UNTIL WE GET TO THE END.
AND THEN ALL OF A SUDDEN IT
MAKES A BIG DIFFERENCE WHICH ONE
WE CHOOSE.
IN FACT, THAT ONE IS NOT
ALLOWED.
AND WHY?
BECAUSE, IF I MAKE THAT CHOICE
THERE, THE BOTTOM CHOICE THERE,
WHAT'LL HAPPEN IS I'LL END UP
WITH A CHAIN WHICH GOES
LONG-SHORT-LONG-SHORT-LONG,
LONG-SHORT-LONG-SHORT-LONG.
IT'LL REPEAT THE SAME MOTIF
THREE TIMES, AND THAT YOU'LL
NEVER FIND IN THE IDEAL LATTICE.
SO THAT IS DISALLOWED.
BUT THE ONLY WAY I COULD
DISCOVERY THIS WAS DISALLOWED
WAS TO GO ALL THE WAY BACK TO
THE END OF THE CHAIN AND
RECONSTRUCT IT.
SO I LOOK BACK ALL THOSE MANY
ELEMENTS, AND, I CAN MAKE BIGGER
CHAINS, AND BIGGER CHAINS AND
BIGGER CHAINS WHICH HAD THE SAME
PROPERTY.
WE HAVE TO LOOK ALL THE WAY BACK
TO THE END OF THE CHAIN.
SO IN FACT, I CAN MAKE A CHAIN,
WHICH [Unclear] YOU HAVE TO LOOK
BACK ARBITRARILY FAR IN ORDER TO
MAKE THE RIGHT DECISION.
SO THERE'S A PROOF, AN ABSOLUTE
PROOF - I'VE JUST DONE IT BY
HAND WAVING - BUT IT'S AN
ABSOLUTE-- IT CAN BE FORMULATED
INTO AN ABSOLUTE MATHEMATICAL
PROOF THAT YOU NEED INFINITE
RANGE INFORMATION IN ORDER TO
ENSURE THAT YOU CAN BUILD AN
INFINITE CHAIN PERFECTLY.
AND, OF COURSE, IN TWO
DIMENSIONS OR THREE DIMENSIONS,
WE HAVE A PROBLEM THAT'S MUCH
WORSE, BECAUSE WHEN WE BUILD A
TILING, A PENROSE TILING, WE'RE
BUILDING A LATTICE OF LINES
WHICH FOLLOWS THE SEQUENCE IN
FIVE DIFFERENT DIRECTIONS AT THE
SAME TIME.
SO THIS LOOKS REALLY IMPOSSIBLE.
AND IN FACT, IF I BEGIN TILING
THE PLANE ACCORDING TO PENROSE'S
RULES -

Above a caption that reads "Penrose rules don’t guarantee a perfect tiling," a picture shows a blue and pink geometric pattern with numbers.

He approaches the screen and continues LET'S SAY I ADD THE
FIRST TILE, THEN I ADD A SECOND
TILE, FOLLOWING THE JOINING
RULE, AND A FOURTH AND A FIFTH,
ETCETERA, THEN JUST AS I
DESCRIBED FOR THE CHICKENS WHICH
YOU'LL TYPICALLY FIND JUST AFTER
A HANDFUL OR SO OF TILES, I'LL
RUN INTO A PROBLEM.
OKAY?
AND THE ONLY WAY TO GET RID OF
THAT PROBLEM IS EITHER TO LOOK
AHEAD TO AVOID THE PROBLEM WITH
A LONGER RANGE INTERACTION, OR I
HAVE TO KEEP TAKING STUFF AWAY
AND REPLACING IT, AND TAKING
STUFF AWAY AND REPLACING IT.
SO THE FACT THAT ONE CAN NOT
MAKE THESE HIGHLY PERFECT,
BEAUTIFULLY FACETED, LARGE
QUASICRYSTALS SEEMS TO BE
UNDERSTOOD PERFECTLY FROM THIS
SIMPLE GEOMETRICAL ARGUMENT.
THE ONLY THING IS, THIS ARGUMENT
TURNS OUT TO BE WRONG.

[Audience murmuring]

Paul continues IT'S QUITE
SURPRISING THAT IT'S WRONG.
IT'S AN EXAMPLE, AGAIN, WHERE
JUST WHEN YOU THINK YOU
UNDERSTAND WHAT'S GOING ON IN
THIS SUBJECT - THIS WHAT I
ACTUALLY FIND SO ATTRACTIVE
ABOUT IT.
EVERY TIME I THINK I KNOW WHAT'S
GOING ON, SOMETHING SURPRISING
HAPPENS.
SO A FEW OF US DECIDED TO SAY...
TO ASK THE QUESTION, "WELL,
OKAY.
MAYBE WE HAVE TO MAKE LOTS OF
MISTAKES IN MAKING
QUASICRYSTALS.
MAYBE WE CAN REDUCE THEIR
FREQUENCY.
MAYBE WE CAN, INSTEAD OF MAKING
A MISTAKE ONCE EVERY TEN TILES,
AT LEAST WE CAN FIND A RULE THAT
MAKES IT ONCE EVERY, MAYBE A
HUNDRED TILES.
LET'S SEE HOW FAR WE CAN GO."
SO, WE MADE A RULE WHICH UNLIKE
THE PENROSE RULES WHICH
CONSTRAIN THE WAY TWO TILES CAN
JOIN ALONG AN EDGE, INSTEAD,
WELL, IT'S A SIMPLE VARIANT
IDEA, AND CONSTRAIN THE WAY YOU
ADD TILES TO A VERTEX
YOU THINK ABOUT ALL THE TILES
AROUND A VERTEX, NOT JUST THE
ONE AT THE NEAREST EDGE.
THAT'S STILL A FINITE DISTANCE,
IT'S A VERY CLOSE DISTANCE.
BUT IT'S JUST DIFFERENT THAN
PENROSE'S CONCEPT OF
CONSTRAINING THE WAY TILES JOIN
EDGE TO EDGE.
SO IN PARTICULAR, IF I HAVE SOME
TILING, LIKE THE ONE YOU'RE
GONNA MAKE DOWN HERE, AND THEN I
LOOK AT THE VERTICES ON THE
SURFACE, I CAN ALWAYS DIVIDE
THEM INTO WHAT I'LL CALL
UNFORCED AND FORCED.
UNFORCED MEANS THERE'S TWO OR
MORE CHOICES THAT YOU CAN USE TO
COMPLETE THE VERTEX; FORCED
MEANS THERE'S ONLY ONE CHOICE.
SO OUR RULE WAS VERY SIMPLY,
ONLY ADD TILES WHEN YOU HAVE
FORCED SIDES AND NEVER ADD THEM
WHEN YOU HAVE UNFORCED SIDES.
YOU CAN TELL IF IT'S FORCED AND
UNFORCED JUST BY LOOKING AT THAT
VERTEX.
YOU DON'T HAVE TO LOOK ANY
FURTHER, SO IT'S A LOCAL RULE.
SO WE MADE A LITTLE PROGRAM.
WE WANTED TO SEE HOW FAR THIS
WOULD GET US.
THAT BEGINNING FROM SEED, AND
I'VE JUST CHOSEN ONE HERE, WE'LL
SIMPLY ADD TILES AT FOUR SITES
AND WE'RE GONNA SEE HOW FAR WE
CAN GO BEFORE WE MAKE A MISTAKE.
SO THIS PROGRAM, THE WAY IT'S
DESIGNED, WILL STOP EITHER:
WHEN YOU RUN OUT OF FOUR SITES;
OR TWO, WHEN YOU'VE ACTUALLY
MADE A MISTAKE, WHEN YOU HAVE A
CONFLICT AND YOU HAVE TO GO BACK
AND START AGAIN.
LET'S SEE HOW FAR WE GET HERE.
OKAY, READY, SET...

He clicks and a clip shows a turquoise and pink tiling pattern that builds up from a small one.

[Audience murmuring]

Paul says OKAY, I...
I WILL...
I'LL RUN OUT OF MEMORY IF I LET
IT GO.

[Audience laughter]

Paul says AMAZINGLY,
WE'VE GONE FROM MAKING TILING
MISTAKES EVERY SO AND SO, TO
MAKING NO MISTAKES.
IN FACT, YOU CAN CONTINUE THIS
TILING FOREVER.
IT WILL CONTINUE TO GROW THIS
WAY FOREVER PERFECTLY OUT TO
INFINITY WITHOUT EVER MAKING A
MISTAKE.
SO SURPRISINGLY, SHOCKINGLY, THE
INTUITION THAT ONE HAD FROM THE
ONE DIMENSIONAL CHAIN, THE
INTUITION THAT ONE HAD FROM
THINKING ABOUT THE, UH, TWO
DIMENSIONAL TILING WITH
PENROSE'S RULES, ALL THAT
INTUITION IS ACTUALLY DECEPTIVE.
WITH ANOTHER SET OF RULES YOU
CAN ACTUALLY MAKE SOMETHING
INFINITELY BIG.
SO, IN FACT, THERE'S ACTUALLY NO
GOOD THEORETICAL REASON WHY YOU
CAN'T MAKE A PERFECT
QUASICRYSTAL.
AND SURE ENOUGH WITHIN A YEAR
PEOPLE BEGAN TO DO JUST THAT.
SUDDENLY, JUST BY PLAYING AROUND
WITH OTHER MATERIALS, THEY BEGAN
TO MAKE CRYSTALS WERE SUDDENLY
STABLE, AT LEAST AS FAR WAS CAN
TELL WE'RE STABLE.
AND COULD BE GROWN TO MILLIMETRE
OR EVEN, UM...
A CENTIMETRE SIZE, WHICH WERE
FACETED, WITH SMOOTH FACETS.
AND HERE'S ON EXAMPLE.
THIS PARTICULAR MATERIAL IS ONE
WE'LL BE LOOKING AT A LITTLE BIT
LATER.
IT CONTAINS
PLANES WHICH HAVE TEN-FOLD
SYMMETRY AND THEN THEY'RE
STACKED PERIODICALLY, AND THEY
FORM SPINDLES WHICH IF YOU
CALCULATE THE FACETS AROUND THE
SPIN-- WHEN YOU COUNT THE NUMBER
OF FACETS AROUND THE SPINDLE,
ARE TEN IDENTICAL FACETS.
SO THIS IS A PAT, THIS IS A
STRUCTURE WHICH [Unclear]...
ACTUALLY, A MIXTURE OF CRYSTAL
AND QUASICRYSTAL.
QUASICRYSTALS ON THE PLANE,
PERIODIC OUT OF THE PLANE, WITH
TEN-FOLD SYMMETRY.
SO, IN FACT, THE FIRST OBJECTION
FOR THAT THESE ARE
QUASICRYSTALS, THE IDEA THAT THE
REQUIRE THESE LONG LOCAL
INTERACTIONS TURNS OUT TO BE
MISLEADING, WE DISCOVERED, AND
NOW WE KNOW WE CAN ACTUALLY MAKE
THESE THINGS, OR AT LEAST IT
LOOKS LIKE WE CAN MAKE THESE
THINGS IN NATURE.
NEVERTHELESS, SCEPTICISM STILL
REMAIN DUE TO THIS FACT:
THAT IT STILL SEEMS LIKE THEY'RE
AWFULLY COMPLICATED.
THE IDEA THAT YOU NEED TO HAVE
TWO SHAPES, OR MORE SHAPES, THAT
INTERLOCK IN SOME WAY TO DO THIS
SEEMS AWFULLY COMPLICATED.
SO, INTERESTINGLY ENOUGH, WE'VE
LEARNED RECENTLY THAT THIS TOO
IS NOT NECESSARILY REQUIRED.
CERTAINLY SEEMS TO BE REQUIRED.
WE'RE SUPPOSED TO GET THESE TWO
LENGTHS WHO'S RATIO IS
IRRATIONAL.
HOW ARE WE GONNA DO THAT IF WE
ONLY HAVE A SINGLE SHAPE?
WELL, IT TURNS OUT THERE'S
ANOTHER TRICK WHICH IS POSSIBLE.
AND THIS IS THE IDEA OF WHAT WE
CALL COVERINGS, INSTEAD OF
TILINGS.
THE COVERING YOU HAVE A
REPEATING ELEMENT WHICH I'VE
CALLED HERE A TILE, BUT ACTUALLY
IT'S MORE-- WHAT WE HAVE IN MIND
IS NOT THAT-- UNLIKE TILES WHICH
JOINED TOGETHER EDGE ON EDGE,
YOU MAY IMAGINE THAT THESE
REPEATING UNITS ARE ALLOWED TO
OVERLAP ONE ANOTHER.
INSTEAD OF HAVING RULES FOR
WHERE THEY JOIN EDGE TO EDGE, WE
HAVE RULES FOR THE WAY THEY CAN
OVERLAP ONE ANOTHER.

A slate under the title "Quasi-unit cell picture" appears. On the left, a red and blue decagon made up of different shapes reads "Gummelt Tile." On the right, two smaller sets of decagons combined read "A" and "B" respectively.

He continues WE'RE GONNA MAKE THE RULE THAT
WHEN YOU JOIN THESE TWO TILES UP
ONE ANOTHER-- IF YOU ALLOW ONE
TO OVERLAP THE OTHER, THEY CAN
ONLY OVERLAP SO THAT RED OVER
LAPS RED AND BLUE OVERLAPS BLUE.
THAT TURNS OUT, ALLOWS TWO
POSSIBLE JOININGS.
ONE IN WHICH THEY ARE FAR AND
SEPARATED, "A"; ONE IN WHICH
THEY'RE MORE CLOSELY SEPARATED,
"B."
AND WOULD IT SURPRISE YOU TO
KNOW THAT THE RATIO OF THOSE
DISTANCES IS THE GOLDEN RATIO.
HOPE NOT.
OKAY?
THAT'S HOW YOU CAN GENERATE TWO
LENGTHS FROM A SINGLE ELEMENT,
BY CONSTRAINING THE WAY THEY'RE
ALLOWED TO OVERLAP.
OF COURSE, IF YOU WERE TRYING TO
BUILD A TILING FOR YOUR
BATHROOM, YOU WOULD NOT WANT TO
HAVE TILES WHICH OVERLAP ONE
ANOTHER, PLEASE.
BUT, IF WE'RE THINKING ABOUT
ATOMS, IF YOU THINK ABOUT THIS
AS REPRESENTING A CLUSTER OF
ATOMS, THEN WHAT THIS OVERLAP
WOULD MEAN IS SIMPLY THAT TWO
CLUSTERS ARE SHARING ATOMS, THAT
YOU'RE GETTING TWO CLUSTERS FOR
LESS THAN THE PRICE OF TWO BY
ALLOWING THEM TO SHARE SOME
ATOMS AND SORT OF HOOK ONTO ONE
ANOTHER.
AND IT'S JUST THAT THE
ARRANGEMENT ALLOWS TWO WAYS FOR
THEM TO HOOK ONTO ONE ANOTHER,
ONE WHICH BRINGS THEM CLOSE AND
ONE OF WHICH MAKES THEM MORE
DISTANT, AND WHERE THE RATIO IS
THE GOLDEN RATIO.
NOW IT TURNS OUT, IF YOU FOLLOW
THIS RULE WITH JUST THIS SINGLE
ELEMENT, YOU CAN CONSTRUCT A
SPACE-FILLING TILING...
WHICH IS FIVE-FOLD SYMMETRIC,
QUASI PERIODIC.
IN FACT, YOU CAN ACTUALLY MAP
THIS DIRECTLY INTO A PENROSE
TILING - IT'S WHAT WE CALL
ISOMORPHIC MATHEMATICALLY TO A
PENROSE TILING.
AND FURTHERMORE, NATURE SEEMS TO
DO THIS.
IT'S WHAT I'VE SAID.
THERE'S A RING OF TEN,
SURROUNDED BY A RING OF TEN,
SURROUNDED BY A RING OF THREE.
THAT RING OF THREE BREAKS THE
TEN-FOLD SYMMETRY OF THE
DECAGON.
BUT OF COURSE, WHEN WE PAINTED
THEM WITH RED AND BLUE SPOTS, WE
ALSO BROKE THE TEN-FOLD SYMMETRY
OF THE DECAGON.
WE USED THAT AS PART OF THE
JOINING RULES.
BUT THE INTERESTING THING IS
THAT YOU BREAK THE SYMMETRY OF
THE DECAGON IN THE EXACTLY THE
SAME WAY AS YOU DO WITH SYMMETRY
OF THE...
RED AND BLUE.
THAT IS TO SAY, YOU BREAK THE
TEN-FOLD SYMMETRY AND JUST SORT
OF A TWO-FOLD REFLECTING
SYMMETRY.
WHICH MEANS IF I ROTATE MY
DECAGON BY A MULTIPLE OF 36
DEGREES, THE OUTSIDE COMES INTO
ITSELF, BUT OF COURSE, THE
INSIDE DOES NOT.
SO AS I MOVE ALONG THIS, A
LATTICE OF THE DECAGON, SO I SEE
THESE CENTRAL MOTIFS SWITCHING
DIRECTION.
AND IF COMPARE-- SORRY.
IF I COMPARE IN THE IDEAL
PATTERN THE WAY, AS I GO FROM
ONE DECAGON TO ANOTHER, THE
DIRECTION SHIFT FOR THE IDEAL
PATTERN, TO THE WAY THE ACTUAL
ATOMS SHIFT, I FIND THAT THEY
ARE IN ONE-TO-ONE
CORRESPONDENCE.
SO THIS THING IS FOLLOWING THE
RULES AS CLOSELY AS YOU CAN
POSSIBLY IMAGINE.
FURTHERMORE, WE CAN THEN USE
THIS IMAGE TO HELP DETERMINE THE
STRUCTURE OF THE QUASICRYSTAL
WHICH HAS BEEN ONE OF THE
LONGSTANDING MYSTERIES - WE'VE
NEVER QUITE DECIPHERED UNTIL NOW
WHAT THE ATOMIC STRUCTURE IS

He opens his hands and says BECAUSE WHEN YOU THINK ABOUT
THEM AS BEING COMPOSED OF TWO
ELEMENTS WHICH JOIN IN DIFFERENT
WAYS, WE DON'T HAVE ENOUGH
INFORMATION TO DECIDE WHICH
ATOMS BELONG TO WHICH ELEMENT.
BUT IF WE JUST HAVE A SINGLE
ELEMENT WHICH DESCRIBES THE
ENTIRE STRUCTURE AND WE HAVE
THIS KIND OF IMAGING, AND WE
KNOW ENOUGH ABOUT THE RATIO OF
THE ELEMENTS IN THERE, THAT
GIVES US MORE THAN ENOUGH
INFORMATION TO ACTUALLY DECIPHER
THE STRUCTURE.
SO IN FACT, USING THIS
REASONING, WE ACTUALLY DECIPHER.
WE HAD A BEST-FIT...
FIT TO WHAT THE STRUCTURE SHOULD
LOOK LIKE AND IT'S A VERY
COMPLEX STRUCTURE - NOT THE
FIRST THING THAT YOU WOULD COME
TO MIND, BUT THAT SEEMED TO BE
THE UNIQUE SOLUTION THAT WE
FOUND.
AND RECENTLY, PEOPLE HAVE BEEN
ABLE TO DO A MUCH HIGHER
RESOLUTION IMAGING WHICH NOW
REVEALS THE ALUMINUM AS WELL AS
THE NICKEL AND COBALT WITH MUCH
GREATER ACCURACY, AND AGREE SPOT
ON WITH WHAT WE HAD PREDICTED
BASED ON THE EARLY ANALYSIS.

A series of pictures show colourful abstract patterns.

He continues SO WE NOW HAVE OUR FIRST EXAMPLE
OF WHAT I CALL A SOLVED
QUASICRYSTAL, WHERE WE ACTUALLY
UNDERSTAND ITS STRUCTURE, WE
UNDERSTAND HOW IT CAN BE GOING
FROM A SINGLE ELEMENT BY THESE
OVERLAP RULES, AND WE ALSO
UNDERSTAND, ACTUALLY, WHERE THE
ATOMS SIT IN THIS STRUCTURE.
AND THIS IS WHAT YOU NEED TO
SORT OF REALLY OPEN THE DOOR UP
TO USING THESE MATERIALS AND
TECHNOLOGY.
IT TAKES A LONG TIME BETWEEN
WHEN YOU HAVE A SORT OF
HYPOTHETICAL IDEA OF A
QUASICRYSTAL OR THE FIRST
EXAMPLES OF QUASICRYSTAL, AND
YOU CAN ACTUALLY BEGIN TO USE
THEM BECAUSE YOU HAVE TO BE
ABLE TO CONTROL...
HAVE ENOUGH CONTROL THAT YOU CAN
GROW MATERIALS WITH THE KINDS OF
ATOMS, WITH THE KINDS OF
STRUCTURES THAT YOU WANT FOR
YOUR APPLICATION.
BUT NOW WE UNDERSTAND THAT WE...
NOW THAT WE UNDERSTAND A MUCH
SIMPLER WAY, THE STRUCTURE OF
THESE SOLIDS AND HOW THEY GROW
AND THAT WE CAN GROW THEM
PERFECTLY, WE CAN NOW BEGIN TO
FOCUS ON THE UNIQUE PHYSICAL
PROPERTIES THAT DERIVE FROM
THEIR UNIQUE SYMMETRY
PROPERTIES.
WE'VE ALREADY SEEN THAT THEY
HAVE UNIQUE DIFFRACTION
PROPERTIES AND UNIQUE FACETING
PROPERTIES BECAUSE OF THEIR
UNUSUAL SYMMETRIES, BUT WE'VE
ALSO LEARNED FROM-- BOTH WITH
THEORETICAL ANALYSIS AND FROM
EXPERIMENT, IS THAT THEY HAVE
UNUSUAL ELASTIC AND ELECTRONIC
PROPERTIES.
THEY ARE, FOR EXAMPLE, HARDER
THAN CRYSTALS MADE OF SIMILAR
MATERIALS, SIMILAR ATOMS.
THEY ARE MUCH MORE RESISTANT AT
LOW TEMPERATURES THAN CRYSTALS,
AND TEND TO TRAP ELECTRONS MUCH
MORE EASILY AND LOCALIZE SITES
MUCH MORE EASILY THAN ORDINARY
CRYSTALS WHICH CAN BE USEFUL FOR
SOME APPLICATIONS.
YOU, YOURSELF, MIGHT FIND SOME
PRACTICAL APPLICATION FOR
QUASICRYSTALS.

A picture of shows a set of saucepans and pots under the title "Cybernox Collection." A caption
reads "The speaker has NO fiducial interest whatsoever in Cybernox."

He continues TURNS OUT ONE OF THE PROPERTIES
THAT THEY HAVE THAT WE ACTUALLY
DON'T UNDERSTAND THEORETICALLY
IS THAT SOME OF THEM ARE
REMARKABLY SLIPPERY.
IN FACT, SOME OF THEM ARE
CLAIMED TO BE THE SECOND-MOST
SLIPPERY MATERIAL COMPARED TO
TEFLON.
THE ONLY THING IS WHEN YOU COAT
A PAN WITH TEFLON, UM, YOU KNOW
TEFLON'S KIND IF DELICATE.
YOU WOULDN'T WANNA TAKE A KNIFE
TO YOUR TEFLON PAN.
BUT WHEN YOU COAT A PAN WITH
QUASICRYSTAL AS THESE ARE, YOU
CAN
ACTUALLY TAKE A KNIFE
TO THAT PAN WITHOUT DAMAGING THE
SURFACE 'CAUSE IT'S JUST A METAL
ALLOY, A RATHER HARD METAL
ALLOY.
I SHOULD WARN YOU, IF YOU BUY
ONE OF THESE THINGS AND TAKE A
KNIFE TO IT, YOU WILL FIND THAT
THERE ARE SCRATCHES.
HOWEVER THE SCRATCHES ARE NOT A
SCRATCH IN THE QUASICRYSTAL
ALLOY.
WHAT'S HAPPENED IS THE ALLOY IS
SO HARD, SOME OF YOUR KNIFE HAS
RUBBED OFF ON THE PAN.

[Audience laughter]

Paul continues OKAY, I
REALIZE I'VE RUN A LITTLE BIT
OVER, BUT I CAN'T HELP BUT END
UP WITH ONE SIMPLE SHORT
EPILOGUE.
EXAMPLE OF ONE LAST TOY I WANNA
SHOW YOU.
I CAN'T HELP BUT SHOW IT TO YOU
BECAUSE TO ME THIS-- WELL, WHAT
YOU'RE ABOUT TO SEE IS WHAT I
LIKE TO CALL...
THE RAREST OBJECT IN THE GALAXY.
IT'S AN EXAMPLE OF WHAT WAS
CALLED A PHOTONIC QUASICRYSTAL.
ITS A DEVICE WHICH WAS DESIGNED
TO TRAP LIGHT - THE WORD
"PHOTONIC" IMPLIES LIGHT - WAS
DESIGNED TO TRAP LIGHT OF
CERTAIN FREQUENCIES.
PHOTONIC CRYSTALS IS ACTUALLY A
VERY BIG INDUSTRY.
PEOPLE MANUFACTURE - SCIENTISTS
AND ENGINEERS MANUFACTURE BY
HAND DEVICES MADE OF RODS MADE
OF DIFFERENT METALS AND
ELEMENTS-- IN A CRYSTALLINE
OF RAYS.
BECAUSE IT TURNS OUT THE WAVE OF
LIGHT PROPAGATES THROUGH ONE OF
THESE ARRAYS, IS THAT IT BOUNCES
OFF OF DIFFERENT SURFACES AND
INTERFERES WITH ITSELF SO THAT
SOME FREQUENCIES GET TRAPPED
WHILE OTHERS PASS THROUGH.
NOW THESE PHOTONIC CRYSTALS IN
ORDER TO TRAP VISIBLE LIGHT HAVE
TO BE MADE MICROSCOPICALLY
SMALL.
HERE WE WANNA PLAY THE SAME GAME
BUT USING A QUASICRYSTAL.
WHY?
BECAUSE ONE OF THE PROBLEMS WITH
A PHOTONIC CRYSTAL IS IT'S NOT
VERY SPHERICALLY SYMMETRIC.
SO IF LIGHT HAPPENS TO HIT ALONG
ONE DIRECTION IT MIGHT GET
TRAPPED, BUT THE SAME LIGHT THAT
HITS SLIGHTLY OFF MAY NOT BE
TRAPPED.
SO YOU LIKE TO HAVE SOMETHING
WHICH IS MORE SPHERICALLY
SYMMETRICAL.
BUT THE MOST SPHERICALLY
SYMMETRICAL SYMMETRY YOU COULD
HAVE IS ICOSAHEDRAL SYMMETRY.
SO WHY NOT MAKE A PHOTONIC
CRYSTAL OUT OF-- WHICH IS
ICOSAHEDRAL RATHER THAN A
CRYSTAL?
AND THAT WAS OUR NOTION.
THE ONLY PROBLEM IS IT'S VERY
EXPENSIVE TO MAKE ONE OF THESE
MICROSCOPIC DEVICES.
SO INSTEAD, WHAT WE DID IS WE
MADE ONE OUT OF PLASTIC WHICH
WAS A SORT OF HANDHELD SIZE.

He places a delicate white ball made up of geometrical pieces.

He continues NOW, THIS DEVICE, BECAUSE IT'S
MUCH BIGGER, BECAUSE THE SPACES
ARE MUCH BIGGER, CAN'T TRAP
ORDINARY LIGHT.
YOU'RE NOT SEEING ORDINARY LIGHT
TRAPPED IN THERE.
WHAT IT TRAPS IS MICROWAVE
RADIATION.
AND, BECAUSE IT'S MORE
SPHERICALLY SYMMETRIC, IN FACT,
IT TURNS OUT THAT IT TRACKS IT
ALONG ALL POSSIBLE, ESSENTIALLY
ALL POSSIBLE DIRECTIONS OF WHICH
THE MICROWAVES MIGHT HIT IT.
AND THE NICE THING IS, THAT IF
YOU SOLVE THIS PROBLEM FOR
MICROWAVES, YOU CAN THEN THINK
ABOUT-- ONCE YOU'VE PERFECTED
THE DESIGN, YOU COULD THEN
SHRINK IT DOWN TO MICROSCOPIC
SIZE SO THEN IT CAN WORK FOR
LIGHT.
SO, OUR NOTION WAS TO BEGIN BIG
AND THEN WORK OUR WAY SMALL.
NOW WHAT MAKES THIS THING SO
RARE...
WELL, FIRST OF ALL, NO ONE WOULD
EVER THINK TO MAKE A DEVICE LIKE
THIS ORDINARILY TO TRAP
MICROWAVE LIGHT.
THAT'S NOT THE KIND OF LIGHT YOU
WANNA TRAP; IT'S NOT DENSE
ENOUGH, IT DOESN'T CARRY ENOUGH
INFORMATION.
SO, NO ORDINARY SOCIETY OUT
THERE WOULD TRY TO MAKE THIS
INTENTIONALLY.
WHY'D WE MAKE IT?

[Audience laughter]

Paul continues WELL...
[Chuckles]
WELL, WE HAD THIS INTERESTING
COINCIDENCE.
FIRST OF ALL, WE HAD THIS REALLY
NEAT DEVICE AT ONE END OF OUR
CAMPUS WHICH WAS CALLED A 3D
PRINTER.
YOU CAN PUT ANY DESIGN INTO IT
LIKE ONE OF OUR COMPLICATED
QUASICRYSTAL DESIGNS, AND IT
WILL TURN IT INTO PLASTIC FOR
YOU.
SO THAT WAS THE FIRST ELEMENT.
SECONDLY, WE HAD PEOPLE LIKE ME
ON THE CAMPUS WHO WERE
INTERESTED IN BOTH COSMOLOGY AND
QUASICRYSTALS.
AND WHY WAS THAT IMPORTANT?
BECAUSE IT TURNS OUT THAT MY
NEIGHBOUR IN COSMOLOGY HAS A
NICE MICROWAVE BACKGROUND
LABORATORY DESIGNED TO LOOK AT
MICROWAVE BACKGROUND-- MICROWAVE
RADIATION FROM THE EARLY
UNIVERSE.
SO HE HAD A BEAUTIFUL LAB WHICH
GENERATED MICROWAVE RADIATION
AND WHICH YOU COULD PUT THIS
DEVICE IN THE MIDDLE OF IT AND
LOOK AT THE SCATTERING PATTERN.
I THINK THERE'S ALMOST NOWHERE
ELSE IN THE GALAXY, MAYBE THE
VISIBLE UNIVERSE, WE'D HAVE THIS
WEIRD COINCIDENCE.
PEOPLE WHO WORK WITH PLASTICS,
SOME GUYS, SOME THEORIST WHO
WORKS WITH QUASICRYSTALS AND
COSMOLOGY, AND THEN THIS, THIS
COSMOLOGIST ALL WORKING IN THE
SAME PLACE.
I CAN'T THINK OF ANY PLACE,
CERTAINLY ON THIS PLANET, WHERE
THAT WOULD OCCUR.
AND THAT'S HOW THIS DEVICE CAME
OUT.

He grabs the ball and says SO, I LEAVE THIS TO YOU AS AN
EXAMPLE OF ANOTHER KIND OF
FUTURE FOR THE APPLICATION OF
THESE IDEAS, ONE IN WHICH IN A
CONTROLLED WAY WE MAKE THESE
PATTERNS INTENTIONALLY.
IN THIS CASE, FOR A CONTROLLING,
MANIPULATING AND CHANNELLING
LIGHT.
TURNS OUT THE IDEA IS INDEED
SUCCESSFUL, SO WE ARE WORKING ON
YET OTHER PATENTS AND OTHER USES
OF THIS IDEA FOR VARIOUS
PURPOSES.
AND, UM, I'M GONNA LEAVE THIS
ONE UP HERE.
YOU'RE WELCOME TO COME UP AND
LOOK AT IT AND HANDLE IT VERY
GINGERLY 'CAUSE IT'S THE ONLY
ONE...
IN THE GALAXY.

[Audience laughter]

Paul concludes THANK YOU.

[Applause]

The caption changes to "Andrew Moodie."

Andrew says AH, THE GOLDEN
RATIO MAKES YET ANOTHER
SURPRISING APPEARANCE.
IT'S
INTERESTING HOW A SIMPLE
EXPLORATION CAN LEAD TO THE
DISCOVERIES OF COMPLEX
QUASICRYSTAL AND THE DEVELOPMENT
OF NEW MATTER.
I HAVE TO ADMIT, I HAVE SOME
CONCERNS ABOUT THE EFFECTS OF
NEW MATERIALS ON THE ENVIRONMENT
AND ON THE HUMAN BODY, BUT AT
THE SAME TIME, I'M GIDDY WITH
CHILDISH EXCITEMENT ABOUT THE
POSSIBILITIES OF THE FUTURE.
I MEAN, MATERIALS THAT ARE
SLIPPERY AND YET STRONG AS STEAL
THAT OBSERVE MICROWAVES, OR
PERHAPS FORM THE OUTER SHELL OF
THE MILLENNIUM FALCON.
NOW, IF YOU WANNA RECEIVE WEEKLY
UPDATES ABOUT OUR UPCOMING
PROGRAMS, DROP US AN e-mail AT
BIGIDEAS@TV.ORG.
I'M ANDREW MOODIE.

He spins the Starwars pistol and sticks his tongue out.

He says SEE YOU NEXT TIME.

[Theme music plays]

The end credits roll.

bigideas@tvo.org

416-484-2746

Big Ideas. Producer, Wodek Szemberg.

Producers, Lara Hindle, Mike Miner, Gregg Thurlbeck.

Logos: Unifor, Canadian Media Guild.

A production of TVOntario. Copyright 2006, The Ontario Educational Communications Authority.

Watch: Paul Steinhardt on Impossible Crystals