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a portion of this video was sponsored by
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lastpass
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this video is about a pattern people
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thought was impossible
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and a material that wasn't supposed to
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exist
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the story begins over 400 years ago
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in prague i'm now in prague and the
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czech republic which is perhaps my
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favorite european city that i've visited
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so far
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i'm going to visit the kepler museum
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because he's one of the most famous
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scientists who lived and worked around
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prague
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i want to tell you five things about
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johannes kepler
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that are essential to our story
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number one kepler is most famous for
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figuring out that the shapes of
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planetary orbits are
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ellipses but before he came to this
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realization
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he invented a model of the solar system
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in which the planets were on nested
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spheres
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separated by the platonic solids
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what are the platonic solids well they
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are objects where
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all of the faces are identical and all
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of the vertices
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are identical which means you can rotate
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them through some
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angle and they look the same as they did
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before
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so the cube is an obvious example then
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you also have
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the tetrahedron the octahedron
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the dodecahedron which has 12 pentagonal
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sides
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and the icosahedron which has 20 sides
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and that's it there are just five
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platonic solids
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which was convenient for kepler because
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in his day they only knew
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about six planets so this allowed him to
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put a unique platonic solid between
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each of the planetary spheres
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essentially he used them as spacers
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he carefully selected the order of the
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platonic solids
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so that the distances between planets
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would match astronomical observations as
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closely as possible
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he had this deep abiding belief that
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there was some geometric regularity in
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the universe
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and of course there is just not this
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two kepler's attraction to geometry
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extended to more practical questions
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like
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how do you stack cannonballs so they
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take up the least space on a ship's deck
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by 1611 kepler had an answer hexagonal
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close packing
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and the face centered cubic arrangement
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are both equivalently
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and optimally efficient with cannonballs
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occupying about 74 percent of the volume
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they take up
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now this might seem like the obvious way
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to stack spheres i mean it is the way
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that oranges
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are stacked in the supermarket but
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kepler hadn't proved it he just
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stated it as fact which is why this
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became known as kepler's conjecture
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now it turns out he was right but it
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took around
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400 years to prove it the formal proof
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was only published in the journal form
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of mathematics
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in 2017. three
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kepler published his conjecture in a
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pamphlet called deniva sexangula
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on the six cornered snowflake in which
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he wondered
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there must be a definite cause why
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whenever snow begins to fall
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its initial formations invariably
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display the shape of a six cornered
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starlet
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for if it happens by chance why do they
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not fall just as well with five corners
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or with seven
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why always with six in kepler's day
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there was no real theory of atoms or
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molecules or how they self-arrange into
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crystals
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but kepler seemed to be on the verge of
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understanding this
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i mean he speculates about the smallest
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natural unit of a liquid like water
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essentially a water molecule and how
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these tiny units could
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stack together mechanically to form the
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hexagonal crystal
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not unlike the hexagonal close-packed
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cannonballs
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four kepler knew that regular hexagons
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can cover a flat surface perfectly with
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no gaps in mathematical jargon we say
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the hexagon tiles the plane
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periodically you know that a tiling is
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periodic
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if you can duplicate a portion of it and
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continue the pattern only through
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translation
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with no rotations or reflections
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periodic tilings can also have
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rotational symmetry
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a rhombus pattern has twofold symmetry
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because if you rotate it 180 degrees one
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half turn
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the pattern looks the same as it did
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before equilateral triangles have
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three-fold symmetry
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squares have four-fold symmetry and
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hexagons have six-fold symmetry but
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those
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are the only symmetries you can have two
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three four and six there is
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no fivefold symmetry regular pentagons
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do not tile the plane but that didn't
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stop kepler from trying
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see this pattern right here he published
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it in his book harmonics mundi or
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harmony of the world it has a certain
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five-fold symmetry
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but not exactly and it's not entirely
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clear
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how you would continue this pattern to
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tile the whole plane
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there are an infinite number of shapes
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that can tile the plane periodically
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the regular hexagon can only tile the
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plane periodically
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there are also an infinite number of
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shapes that can tile the plane
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periodically
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or non-periodically for example
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isosceles triangles can tile the plane
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periodically
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but if you rotate a pair of triangles
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well then the pattern is no longer
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perfectly periodic
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a sphinx tile can join with another
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rotate it at 180 degrees
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and tile the plane periodically but a
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different arrangement of these
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same tiles is non-periodic
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this raises the question are there some
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tiles that can
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only tile the plane non-periodically
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well in 1961 how wang was studying
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multi-colored square tiles the rules
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were
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touching edges must be the same color
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and you can't rotate or reflect tiles
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only slide them around now the question
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was
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if you're given a set of these tiles can
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you tell
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if they will tile the plane wang's
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conjecture
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was that if they can tile the plane well
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they can do so periodically
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but it turned out wang's conjecture was
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false
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his student robert berger found a set of
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20
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426 tiles that could tile the plane
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but only non-periodically
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think about that for a second here we
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have a finite set of tiles okay it's a
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large number
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but it's finite and it can tile all the
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way out to infinity
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without ever repeating the same pattern
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there's no way even to force them to
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tile periodically
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and a set of tiles like this that can
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only tile the plane
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non-periodically is called an aperiodic
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tiling
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and mathematicians wanted to know were
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there aperiodic tilings that required
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fewer tiles well robert berger himself
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found a set with only 104 donald knuth
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got the number down to 92
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and then in 1969 you had raphael
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robinson who came up with
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six tiles just six that could tile the
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entire plane without ever repeating
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then along came roger penrose who would
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ultimately get the number down to two
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penrose started with a pentagon he added
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other pentagons around it
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and of course noticed the gaps but this
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new shape
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could fit within a larger pentagon which
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gave penrose an idea what if he took the
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original pentagons
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and broke them into smaller pentagons
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well now some of the gaps start
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connecting up into
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rhombus shapes other gaps have three
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spikes
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but penrose didn't stop there he
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subdivided the pentagons again
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now some of the gaps are large enough
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that you can use pentagons to fill in
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part of them
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and the remaining holes you're left with
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are just rhombuses
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stars and a fraction of a star that
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penrose calls a justice cap
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you can keep subdividing indefinitely
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and you will only ever find
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these shapes so with just these pieces
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you can tile the plane
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aperiodically with an almost five-fold
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symmetry
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the fifth thing about johannes kepler is
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that if you take his pentagon pattern
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and you overlay it on top of penrose's
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the two match up perfectly
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[Music]
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once penrose had his pattern he found
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ways to simplify the tiles
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he distilled the geometry down to just
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two tiles a thick rhombus
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and a thin rhombus the rules for how
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they can come together can be enforced
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by bumps and notches
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or by matching colors and the rules
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ensure that these
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two single tiles can only tile the plane
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non-periodically
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just two tiles go all the way out to
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infinity without
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ever repeating now one way to see this
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is to print up two copies of the same
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penrose pattern
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and one on a transparency and overlay
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them on top of each other
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now the resulting interference you get
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is called a moire
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pattern where it is dark the patterns
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are
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not aligned you can see there are
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also some light spots and that's where
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the patterns do
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match up and as i rotate around you can
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see the
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light spots move in and get smaller and
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then at a certain point
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they move out and get bigger
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and what i want to do is try to enlarge
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one of these bright spots
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and see how big of a matching section i
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can find
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oh yes yes it's like
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all of a sudden everything is
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illuminated i love it
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so these patterns are perfectly matching
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up here here here here and here but
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not along these radial lines
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and that is why they look dark so what
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this shows us
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is that you can't ever match any section
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perfectly to one
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beneath it there will always be some
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difference
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so my favorite penrose pattern is
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actually made out of
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these two shapes which are called kites
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and darts and they have these very
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particular angles
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and the way they're meant to match up is
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based on these two curves you can see
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there's a curve on each piece
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and so you have to connect them so that
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the curves are continuous
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and that's the rule that allows you to
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build an aperiodic tiling from these two
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pieces
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so uh i laser cut thousands
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of these pieces and oh i'm gonna
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try to put them together and make a huge
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penrose oh man
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come on
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if you stare at a pattern of kites and
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darts you'll start to notice all kinds
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of
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regularities like stars and suns
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but look closer and they don't quite
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repeat in the way you'd expect them to
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these two tiles create an ever-changing
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pattern that extends out to infinity
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without repeating does this mean
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there is only one pattern of kites and
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darts
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and every picture that we see is just a
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portion
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of that overall singular pattern
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well the answer is no there are actually
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an uncountably infinite number of
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different patterns of kites and darts
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that tile the entire plane
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and it gets weirder if you were on any
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of those tilings
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you wouldn't be able to tell which one
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it is
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i mean you might try to look further and
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further out gather more and more data
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but
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it's futile because any finite region
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of one of these tilings appears
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infinitely many times
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in all of the other versions of those
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tilings
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i mean don't get me wrong those tilings
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are also different in
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an infinite number of ways but it's
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impossible to tell that unless you could
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see the whole pattern
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which is impossible
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there's this kind of paradox to penrose
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tilings
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where there's an uncountable infinity
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of different versions but just by
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looking at them
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you could never tell them apart
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now what if we count up all the kites
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and darts in this pattern
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well i get 440 kites and 272
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darts does that ratio ring any bells
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well if you divide one by the other you
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get 1.618
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that is the golden ratio
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so why does the golden ratio appear in
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this pattern
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well as you know it contains a kind of
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five-fold symmetry
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and of all the irrational constants the
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golden ratio
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phi is the most five-ish of the
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constants i mean you can express the
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golden ratio as
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0.5 plus 5 to the power of 0.5
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times 0.5 the golden ratio is also
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heavily associated with pentagons
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i mean the ratio of the diagonal to an
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edge
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is the golden ratio and the kite and
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dart pieces themselves
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are actually sections of pentagons same
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with the rhombuses
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so they actually have the golden ratio
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built right
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into their construction the fact that
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the ratio of kites to darts approaches
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the golden ratio an irrational number
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provides evidence that the pattern can't
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possibly be periodic
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if the pattern were periodic then the
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ratio of kites to darts
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could be expressed as a ratio of two
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whole numbers
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the number of kites to darts in each
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periodic segment
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and it goes deeper if you draw on the
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tiles
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not curves but these particular straight
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lines
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well now when you put the pattern
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together you see something interesting
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they all connect up perfectly into
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straight lines
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there are five sets of parallel lines
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this is a kind of proof of the five-fold
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symmetry of the pattern
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but it is not perfectly regular
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take a look at any one set of parallel
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lines you'll notice
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there are two different spacings call
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them long and short
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from the bottom we have long short long
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short long
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long wait that breaks the pattern
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these gaps don't follow a periodic
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pattern either
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but count up the number of longs and
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shorts in any section
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here i get 13 shorts and 21 longs
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and you have the fibonacci sequence
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1 1 2 3 5 8 13 21
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34 and so on and the ratio of
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one fibonacci number to the previous one
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approaches
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the golden ratio
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now the question penrose faced from
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other scientists was
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could there be a physical analog for
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these patterns
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do they occur in nature perhaps in
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crystal structure
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penrose thought that was unlikely the
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very nature of a crystal is that it is
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made up of repeating units
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just as the fundamental symmetries of
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the shapes that tile the plane had been
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worked out much
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earlier the basic unit cells that
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compose
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all crystals were well established there
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are 14 of them
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and no one had ever seen a crystal that
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failed to fit
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one of these patterns and there was
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another problem
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crystals are built by putting atoms and
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molecules together
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locally whereas penrose tilings well
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they seem to require some sort of
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long range coordination take this
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pattern for example
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you could put a dart over here
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and continue to tile out to infinity no
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problems
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or you could put a kite
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over here on the other side again no
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problems
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but if you place the kite
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and dart in here simultaneously
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well then this pattern will not work
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i mean you can keep tiling for a while
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but when you get to somewhere around
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here
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well it's not gonna work you can put a
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dart in there which completes the
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pattern nicely but then you get this
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really awkward shape there which is
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actually the shape of another dart
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but if you put that one in there then
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the
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lines don't match up the pattern doesn't
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work so how could this work as a crystal
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i mean both of these tiles obey the
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local rules
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but in the long term they just don't
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work in the early 1980s
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paul steinhardt and his students were
00:16:49
using computers to model how atoms come
00:16:52
together into condensed matter
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that is essentially solid material at
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the smallest scales
00:16:58
and he found that locally they like to
00:17:00
form icosahedrons
00:17:02
but this was known to be the most
00:17:04
forbidden shape because it is full
00:17:06
of five-fold symmetries so the question
00:17:09
they posed was
00:17:10
how big can these zykosahedrons get they
00:17:13
thought maybe 10 atoms or
00:17:14
100 atoms but inspired by penrose
00:17:17
tilings
00:17:18
they designed a new kind of structure a
00:17:20
3d analog of penrose tilings now known
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as a quasi-crystal
00:17:24
and they simulated how x-rays would
00:17:26
diffract off such a structure
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and they found a pattern with rings of
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10 points reflecting the five-fold
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symmetry
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just a few hundred kilometers away
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completely unaware of their work another
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scientist dan
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schechtman created this flaky material
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from
00:17:41
aluminum and manganese and when he
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scattered electrons off his material
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this is the picture he got it
00:17:49
almost perfectly matches the one made by
00:17:52
steinhardt
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so if penrose tilings require long
00:18:00
range coordination then how do you
00:18:02
possibly make
00:18:03
quasi crystals well i was talking to
00:18:06
paul steinhardt about this and he told
00:18:07
me
00:18:08
if you just use the matching rules on
00:18:10
the edges
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those rules are not strong enough and if
00:18:14
you apply them locally you run into
00:18:15
problems like this you misplace
00:18:17
tiles but he said if you have rules for
00:18:20
the vertices the way the vertices can
00:18:22
connect with each other those rules are
00:18:25
strong enough
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locally so that you never make a mistake
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and the pattern
00:18:29
can go on to infinity one of the seminal
00:18:33
papers on quasi crystals
00:18:34
was called deniva quinquangula
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on the pentagonal snowflake in a shout
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out
00:18:41
to kepler now not everyone was delighted
00:18:44
at the announcement of quasi crystals a
00:18:45
material that up until then people
00:18:47
thought
00:18:47
totally defied the laws of nature double
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nobel prize winner
00:18:52
linus pauling famously remarked there
00:18:54
are no quasi crystals
00:18:56
only quasi scientists
00:19:01
but uh schechtman got the last laugh he
00:19:03
was awarded the nobel prize for
00:19:05
chemistry in 2011
00:19:07
and quasi crystals have since been grown
00:19:09
with beautiful dodecahedral shapes
00:19:12
they are currently being explored for
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applications from
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non-stick electrical insulation and
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cookware to
00:19:18
ultra durable steel and the thing about
00:19:20
this whole story that fascinates me the
00:19:22
most
00:19:23
is what exists that we just can't
00:19:25
perceive because it's considered
00:19:26
impossible
00:19:27
i mean the symmetries of regular
00:19:29
geometric shapes seemed
00:19:30
so obvious and certain that no one
00:19:33
thought to look beyond them
00:19:34
that is until penrose and what we found
00:19:37
are
00:19:38
patterns that are both beautiful and
00:19:40
counter-intuitive
00:19:41
and materials that existed all along
00:19:44
that we just couldn't see
00:19:45
for what they really are
00:19:53
hey this portion of the video was
00:19:54
sponsored by lastpass and
00:19:56
they have sponsored a number of my
00:19:57
videos in the past which tells me two
00:19:59
things
00:20:00
number one many of you have signed up so
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you don't have to remember your
00:20:03
passwords anymore or worry about getting
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locked out of your accounts
00:20:07
and number two some of you have not yet
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signed up
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hence this reminder that your accounts
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will be more secure and your brain
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less cluttered when you put your
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passwords on autopilot with lastpass
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they give you unlimited password storage
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free cross-device sync
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and even password sharing if you ever
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need to give someone else access to one
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of your accounts
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lastpass autofills your credentials on
00:20:28
mobile sites and apps for ios and
00:20:30
android
00:20:31
and it is so fast and easy when you open
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an app or site lastpass fills in your
00:20:35
username and password in an instant
00:20:37
now real talk do you have important
00:20:39
accounts with the same password
00:20:41
i used to and it's not secure but
00:20:43
remembering a random string of
00:20:45
characters is not something i expect my
00:20:47
brain to do
00:20:48
have you had to reset one of your
00:20:49
passwords recently it's annoying and the
00:20:52
time you waste
00:20:53
will just accumulate over your lifetime
00:20:55
there are small steps you can take today
00:20:57
that will improve every day for the rest
00:20:59
of your life and getting a great
00:21:01
password manager
00:21:02
is one of those steps so click the link
00:21:04
below and start using lastpass today
00:21:06
i want to thank lastpass for sponsoring
00:21:08
this portion of the video and i want to
00:21:09
thank you for watching
