tonight she's going to tell us about

non-euclidean virtual reality and she

has a lot of cool demos and toys and so

if you want to stick around afterwards

and come down and chat and maybe give

things a try I think that's a

possibility tonight if you have

questions during the talk please try to

hold them until the end and then right

and I mean what's that you can't

interrupt oh okay you can interrupt

professor Matsumoto and then at the end

if you have a question Brian and I will

run around the microphone to you so just

raise your hand and then wait because

it's very hard for the speaker in the

front of the room to hear your questions

the acoustics sort of work one direction

all right without further ado all right

Thank You Edie well thank you all for

coming out to this public lecture so I'm

a professor here in the physics

department I have a group and we use

geometry to make new materials so

basically our motto is making materials

with math so I'm gonna be telling you

about some of the cool things that we've

discovered along the way so currently

I've got students working doing virtual

reality 3d printing we're looking at

textiles fabrics knits things like that

so we have a pretty eclectic group and I

have plenty of my students are here up

front so if you're interested you can

come talk to them afterwards too so

today I'm gonna be giving a bit of a

non-conventional lecture so at the

beginning I'm gonna tell you a bit about

curvature and how you can think about

shape and curvature from the patterns

that you use to make clothing and then

from there we're gonna take that idea

abstract it a little bit to make it more

mathematically rigorous and then from

there we're gonna start taking our first

steps into understanding what non

Euclidean spaces so what curved spaces

so you may have heard things about

gravitational waves recently if you've

heard of gravitational lensing a lot of

these phenomenon in cosmology are really

based on the idea that

space is intrinsically curved so this is

this is going to extend this idea and

try to create spaces that are not

physically possible to make um so

basically what we're gonna do after that

is I'm gonna get the virtual your

virtual reality running and I'm gonna

take volunteers from the audience

and we're gonna do some exercises with

the virtual reality to try to give us a

gut intuition about how curved spaces

work and what sorts of phenomenon you

would see in curved spaces that you just

don't see in regular old everyday

Euclidean world okay

so to begin with we're gonna talk about

curvature so curvature is a geometric

principle that's very near and dear to

my heart but let's start let's start out

with something really really simple I

mean we've got Marilyn Monroe and this

beautiful flowy dress but we're not

going to start quiet there we're gonna

start with something a little more

simple so we're gonna start with making

a man's dress shirt okay so I mean I

guess everyone here is wearing clothes

how many of you have ever thought about

what sorts of pieces make up your

clothes what this would look like before

it was sewn in okay so I see maybe 10%

of you holding up your hands so I'm

gonna show some of the rest of you what

this might be like so I mean you might

think okay well I've got a t-shirt it's

clearly just got two pieces of fabric

cut out like this and I saw them take

out there and I'm good to go but if you

actually look at the pattern that's used

to make this fresh shirt you find that

you get this ridiculous crazy thing if

you look at these so there is a fold

which gives you the only straight line

in this entire piece everything else you

have all of these intricate little bits

fold zigzags curves all of these shapes

are there to construct take something

that is flat and construct it

to a curved surface like no matter how

skinny you are or you think you are you

you have curves every human body

intrinsically has a curvature to it and

it's impossible to take something flat

and wrap it around a curved surface

without either adding curved lines or as

we're going to look in a second

what happens when you add seams so this

is an example of something that's very

commonly used in shaping women's wear I

guess I don't have anything I don't have

any darts on this dress but most women's

clothing have what are called darts so

you take a little triangular fold and

you sew it off so what happens is that

instead of laying flat you find out that

the fabric has folded up into this

conical shape so here's the little point

of the triangle and then the rest of it

is hidden inside this fold so this is

something that we use to make what are

called

what's called positive curvature so

we're removing a little bit of area and

we make positive curvature so this has

maybe a little counterintuitive but just

just bear with me on this okay so what

happens this this is something you can

see in my skirt here so I've got I've

got this sort of roughly tulip skirt and

this is something that you see here and

these things are called go days so this

is kind of the antique dark so I take an

extra wedge fabric I slit open a seam on

my skirt and I sew that in so instead of

taking area away from my sample and from

my dress I'm now adding new area into

that so this is something that's going

to give you what's called negative

curvatures and this is gonna be

something that's like a saddle so my

skirt here it curves out down here

but in along this direction so this is

something that has what's called

negative curvature things that have

positive curvature are things like the

surface

of the sphere so it's bending down in

both directions or from the bottom

bending up in both directions all right

so here I have let's make this a little

more precise wet fabric so here I got

this lovely fabric that has a honeycomb

pattern on it and I have cut out exactly

1/6 of this pattern so I have sixty

degrees here and 300 degrees around here

and then what I'm gonna do is I'm going

to join this edge to this edge with a

seam so I get now this clinical shape

where I've taken out exactly 60 degrees

of angle surrounding this point so

that's one way we can measure curvature

is by looking the angular deficit around

this point so how much I'll go we've

taken away okay so now I've got this

extra wedge of fabric so what do you

think I'm gonna do with it okay I'm

gonna stick it into another piece of

fabric so I'm gonna take this cut it

open along here and put this in and see

what we're good I got okay so now I have

this perfectly regular hexagon in the

middle so this had 360 degrees around at

this point and I'm gonna add in 60

degrees so that's gonna give me

something with 420 degrees and when I do

that I end up with this structure that's

all ruffled so this is something that

cannot lie flat in the plate no matter

how hard I try I can sort of smooth some

parts out and that looks flat but then

I've forced it to bunch up somewhere

else so this cannot be smooth no matter

what and some people find this a little

bit a little bit strange when they're

playing with material like that

they'd like to be able to neatly fold it

up the same way you pulled up here

t-shirt and put it in the drawer after

you wash it so here's a dress that's

based on this idea so this is called the

five six seven dress so it was made by

my friend Andrea

who is a clothing designer and Robin

Sullinger who is a professor of

theoretical physics at Kent State so the

idea here is that this dress is made

entirely out of these regular polygons

we look at hexagons Pentagon's and

hexagons and despite how it looks each

of the side angles of these are the same

are the same length the edges are the

same length so what's going on here is

that we can use the number of edges on

each of these polygons to control the

curvature in the dress so places where I

would have positive curvature I want to

take angle out so I'm going to use I'm

going to use my 5 so those have a little

bit less angle so I've got a 5 here and

that's going to be part of the bust so

places where I'd not truly have positive

curvature whereas I've got some 7 so

I've got one in the center and a ring of

sevens around the waist so these are

there to add in extra angle so these are

going to sort of let the material flare

out and this is going to give us

something with negative curvature so we

can use this idea of adding an angle or

taking away angle to prescribe curvature

to a garment ok so what do you think

these things have in common so I've got

this beautiful Couture wedding dress and

this jumble of human intestines does

anyone have a guess they're curved okay

that's a that's a definitely true but

would you believe me if I told you that

the mechanism that caused the physical

mechanism that causes this wedding dress

to ruffle the tulle in the skirt to

ruffle is exactly the same as the

mechanism that causes every human and

toss

to ruffle I mean there's a reason that

if you looked at my intestines the

Wiggles in them are gonna be roughly the

same as yours it's not like someone took

a rope and just jammed them inside our

abdomens they there is a pattern to it

and the pattern has everything to do

with this particular type of elasticity

okay so both of these have negative

curvature so let's look at this

mechanism so this is what happens in the

dress so I take a stretchy piece of

material that's cut short and I've got a

long piece of a stiff material so in

this case it's called it's called boning

and in a lot of textile applications so

it's this sort of thick stiff but still

curva below material and what I do is

I'm gonna take this stretchy material

I'm gonna stretch it out as wide as the

tool sorry that's the boning and I'm

gonna sew them together so I've got a

huge amount of strain in in my material

but there's this constraint that it has

to be as long along the seam as this top

stiff piece of fabric and what happens

is it relaxes it buckles out of the

plane into this beautiful regular

ruffled pattern this is what makes up

what makes up this skirt so it turns out

that human intestines grow in a very

similar way so what happens when you're

while you're developing while you're an

embryonic and embryo is that you have

you've got a gut tube which is what's

going to become your digestive tract and

you've got under all tube which is going

to become your spine in your back and

and your entire spinal cord so what

happens is that there's actually a

membrane that connects your gut tube to

your neural tube so it's very thin

membrane if that connects them and what

happens is as you grow is like your gut

tube grows longer

stur than your than your spinal tube and

your neural tube does and what happens

then is the material that's going to

make up your small intestines is growing

faster than the membrane that's

connecting it to the neural tube and

it's forced because of geometry and

elasticity to ruffle informally is very

regular Wiggles with negative curvature

so we do see so we do see negative

curvature in a huge amount of huge

number of physical systems so this is a

micrograph of the mitochondrial

membranes inside the retina of a tree

shrew and this forms these beautiful

intricate what are called triply

periodic minimal surfaces so this this

this texture forms

entirely naturally at self-assembles and

you can get this to form in a lot of

systems so this is entirely based on

with the geometry of the molecules at a

larger length scale so films try to try

to minimize surface area and it turns

out that that naturally leads them to

have these saddle-like structures so

here is just a plain little saddle here

but on this side this is this is a

framework that would give us a triple a

periodic minimal surface so the soaps

soap film that spans this should be

exactly the same as one of the

structures in the mitochondria in the

eye mitochondria and then we know that

the universe is curved

we don't happen to know at the largest

scales what the curvature is it seems

like it's pretty flat but this is still

an open question since we don't have any

information about the boundary yet okay

so we also see this this idea of

negative curvature in

so we've got in a top panel here this is

a new - prank that's swimming so this is

a marine invertebrate and it's got these

ruffles that are running down running

down its Mansell and those travelling

waves it uses to propel itself so I can

change the frequency on one side and

turn and so on and so forth um you see

it in things like coral and kill so

what's going on here is that the plant

wants to maximize or the or I guess

corals an animal but it wants to

maximize the surface area that is

accessible to resources so it's gonna do

this by ruffling and bending and this is

going to give you a gotten negative

curvature and the calla lily that's my

favorite flower so I figured I'd throw

it in for four kicks okay so okay so now

we're gonna switch gears a little bit

and try to make some of these notions

coming from a clothing design coming

from biology and make them a little bit

more mathematically rigorous okay so

what I've got here are three of the five

platonic solids and these are ones that

are made only with triangles so I've got

so I've got a tetrahedron which is made

out of four triangles but if you look at

each vertex there are three triangles

around every vertex and every vertex is

the same as every other vertex and this

is what makes it a platonic solid here

I've got an octahedron and the only

difference between these is now each of

my vertices has four equilateral

triangles around them um and then and

the far far right over here I've got an

icosahedron so this has five equilateral

triangles around every vertex okay so

what happens if I do six what do you

guys think

yep six triangles around every vertex is

it gonna be a sphere what do you think

I'll be flat okay so let's see what's

going on so I'm gonna skip that so we if

we have six triangles we do in fact end

up with something flat so this is a

regular triangular packing of space the

circles in here make a honeycomb

honeycomb lattice so I've got you see do

I have well this is squares there should

be one somewhere that's trying okay so

I've got this guy it has six triangles

around every vertex and so it's flat so

let's continue this pattern what happens

if I put seven so I've got another

another little doily and I can get seven

triangles you can verify that there are

indeed seven triangles around every

single one of those holes you verify

okay excellent

um and then I can do the same thing with

eight and you'll notice these get more

and more ruffled as we go so getting

back to what this has to do with

curvature so these are tilings are sort

of ways of fitting regular structures

onto surfaces so if you imagine I take

my tetrahedron I put a balloon inside

and sort of inflate it out to the

surface of the sphere I will get

something that tiles the sphere so this

is what would happen if I did this with

the icosahedron so I have a triangle

this sort of weird curvy triangle here

and there's notice for weird curvy

triangles that meet every vertex so this

is this is a tiling of the sphere so we

discussed that this tiling with six

triangles around every vertex that gives

us a plane but what about this

I mean I've got this model they're

really roughly I mean I can go on for a

while but at some point I'm gonna got

stock this is not something that

intrinsically can fit into our euclidean

split space in fact there's a theorem

that you cannot embed the entirety of

this structure into Euclidean space so

what is this so let's let's think for a

second um so this is this is something

that's called the hyperbolic plane this

I told you that you can't fit it in all

of Euclidean space but then I've got

this drawing right here this looks like

hey BAM I've got it I can draw it on the

board we're all good but let's think

about maths for a second

so maps from a curved surface to a flat

surface always introduce distortion so

we're pretty familiar with this idea of

the Mercator projection here this is the

standard high school elementary school

mapping your classroom and you'll notice

that Greenland here is maybe a little

bit larger event Africa and if you look

at the globe on the other hand here's

Greenland and here's Africa I you can

fit something like seven or eight copies

of Greenland and inside Africa for I

forgot the exact number but this is just

showing you that anytime I take some

curved object and try to smoosh it down

on to the plane I'm always going to

introduce distortion so the same way

we've got Mercator projections and all

sorts of other types of of maps you

might find like there's now a new more

Geo politically correct map that tries

to keep the area of the countries as

close to what they actually are in real

life not something that they're there's

a movement to move put that into schools

now we'd like to do the same thing with

this hyperbolic plan so here is this is

the most common map

the hyperbolic plane this is called the

punker a disc model and here is one of

the triangles so if I follow this around

I've got a triangle here and a triangle

here a triangle here four five six seven

triangle so this is indeed this case

where I have seven equilateral triangles

that meet around every vertex but you'll

notice this doesn't look like an

equilateral triangle this is kind of

stretched in similar ways and I have two

straight lines and one curved line the

you know this is kind of a funky looking

triangle but this is the distortion

that's coming from the map I've got

another map this is called the Klein

model here this looks a lot more like an

equilateral triangle I'm happy that

being the edges are straight and this is

where the distortion and these screens

might come up but if you look closely

you'll notice that these angles are not

60 degrees like you would expect in an

equilateral triangle so here we're

getting angular distortion but we're

keeping straight lines straight lines so

no matter what model you you choose

something's gotta give somehow there's

one more this is called the upper

half-plane model I'm not going to go

into details but there's this lovely

print by Anne Rice Hagerman and Saul

slammer that shows how all of these

different models are related to one

another so this is a model that's on

what's called the hemisphere model if I

take a light at the North Pole of that

and look at how the light ray has passed

through it I get a projection down here

this is my plunker hska model this is

the first one we saw the second one is

generated by taking the light source and

moving it infinitely far away like the

Sun I'm projecting straight down this

gives us the Klein model is the second

model and then if I put a light at the

equator and cast it on the wall this is

the last model the

or half-plane model so these are these

are models you might have seen before

how many of you are familiar with this

artwork okay so there's a number of you

so this is this is a circle limit for I

think this is a lithograph by MC usher

and what MC Usher

wanted to do was he wanted a way to see

infinity so he had a conversation with

the mathematician coxeter who told him

about this notion of the hyperbolic

plane and in the plonker a disc map and

he's created this this beautifully

intricate interconnecting series of

demons and angels so let's have a closer

look at this so I've got three demons in

the center interlocking with three

angels now let's look at their wingtips

so I've got wingtips that meet along

here so I've got six six figures going

across and then let's look at each

wingtip so I've got one two three four

five six seven eight so I've got eight

figures that meet along each wingtip um

so what so what's going on here is that

you're supposed to imagine that this

demon in this under here is exactly the

same size and the same shape as this

demon up here and this demon over here

and this little tiny guy in the corner

right there if I could hold my hand

steady enough um

and that takes quite a lot of

imagination so let's let's look at this

again so if I look at these places where

all of the wind tips meet I can inscribe

a circle around each of those points so

what I end up with is I have a circle

and I have four other circles meeting it

at this point and if I as I go around

there are boundaries with six or

around that so what we'd like to do is

we'd like to be able to to see what it

would be like to live inside this space

so see what it would be like to be an

ant that's walking around in this

infinite space

um and the model we're gonna choose is

going to be a little bit different from

this but it's got a similar idea so here

I've got hex a sort of these hexagonal

circle bits that meet for around every

vertex I'm gonna swap that for here I've

got squares that meet six around uh pray

vertex and so I've got a blanket here

which is the same layout of of what you

might see there but here I've taken it

and I've tried to keep all of the edges

the same length in all of the squares of

the same size so in order to do this I

have to push things out of the plan

they're going to become very roughly um

but all of these models the looking at

the blanket looking at the usher

lithograph looking at the 3d print all

of these are models from outside the

outside the hyperbolic plane we really

want to see what it looks like inside

and so this is where virtual reality is

going to come in handy

so we're gonna do this using one

principle we're going to say that light

rays should follow geodesics so let me

unpack that for you so you can imagine

in our space the way I see things is

there's a photon that is coming out of

the projector there's one that's gonna

intersect my eye it's gonna excite some

molecules in my retina my brain

processes that and I say oh there is

some light coming out of the projector

so when the space is curved

we still want we still want photons to

to do the same thing but they're gonna

follow a different principle they're

gonna want to follow

the shortest distance between two points

so we can think about this from the

point of view of a globe so the shortest

distance between two points is part of a

great circle so this is why if you're

flying from San Francisco to New York it

seems like you're going very much

further north than you would expect or

if you're going to certain places in in

Europe or Russia you might end up flying

over the North Pole on depending on

where you leave so in curved spaces

light rays have to follow these lines of

shortest distance between two points and

these are called geodesics so this is

what what we're going to do so in order

to create a virtual world on our

computer we need four components so I'm

going to need a model of the space so

I'm going to need a way to associate

every point in my space to a number in

my computer I'm going to need a way to

draw those points on the screen and this

is this principle that ladies must

follow geodesics so this is telling me

that if I want to look at an object up

there you can imagine there is a

straight line in my space or a curved

line in curved space that's the shortest

distance possible going from my eye to

the from the projector to my eye I guess

makes more sense than me shooting light

beams out of my eyes I want a way to

move around

I want a way to move around this space

and so I've got some pretty nifty

equipment here so I'm gonna be using the

HTC vive so what you'll notice are I've

got these funny towers on my tripods and

those are those are called

lighthouses and what they're doing is

they're constantly sending out signals

so that my HUD set knows exactly where

it is in space so it's going to allow me

to walk around in the space and really

accurately track where I've moved and

then lastly I know

a set of landmarks to help the user

navigate space I mean I could put you in

there in a block screen and tell you who

this is curved space but you wouldn't

believe you'd say well your hello world

program broke so we're going to we're

gonna do this and we're gonna do this

with with tilings so the first space

we're going to do is we're going to do a

space that's called mathematically

speaking it's called h2 cross so this is

gonna be what would happen if I took

this blanket made it infinitely large

and then extruded it into the Z

direction so this is like I'm taking the

entire hyperbolic plane in the X Y

direction and then it's just normal

regular Euclidean space up and down okay

so I think I'm gonna skip this um so

okay so let's see this is the time for

my first demo so let me switch over and

I switch over to this screen so for all

of the demos I'm gonna want some

audience volunteers so please keep in

mind keep that in mind I think I'm gonna

take two for this and then I think

there's probably room for four or five

volunteers and we're basically gonna

spend the rest of the time talking about

how to use your body to move around in

in these hyperbolic spaces so um who can

I get to be my first demo oh my gosh

there's a lot of people who are excited

so okay

uh uh gray sweatshirt oh I thirst two of

you um I guess yes sorry I said what's

your name Jesse okay fantastic so what

I'm gonna ask you to do Jesse so I'll

have you put on the headset I'm gonna

have you stand right here and I'm gonna

narrate white what you're gonna do so

is gonna be I'm just gonna have you so

these are so the idea is that we are in

a space that is squares but this is a

weird space where there are six squares

that meet around every sort of vertical

pole so I'm gonna have you sort of walk

through that and try to try to show us

that but I'll narrate through what

you're gonna do I'm gonna put my hands

on your shoulders and sort of put you

into okay perfect um so right now you're

just seeing the well she's seeing

something that's a lot cooler than what

you guys are seeing right now she's

seeing this sort of lobby of vive world

so what I'm gonna do is I'm gonna put

her into into now this is each to cross

II so I'm gonna have you look straight

forward into that sort of blue

sort of octagon in front of you so into

this and you're gonna just walk forward

just straight forward into that so there

are I will make sure you don't crash

into anything okay okay stop there and

can you take like a step back so you're

just in the centre of that sort of blue

room okay so if you don't put your hands

over it sometimes that loses tracking so

look straight down so just for her

everyone looking straight down

everything looks kind of normal straight

down but if she looks side-to-side

everything looks totally different okay

so okay so now turn 90 degrees to your

left Jessie so me okay so now you're

looking into a sort of light green room

normally I give these sort of ice cream

cone names

I guess this is MintChip room um so you

can walk straight into that

okay stop and turn 90 degrees to your

left a gun

oh now you're going into the rainbow

sherbert room so you can walk straight

into that okay now turn 90 degrees to

your left again so now you're going into

the strawberry cheesecake room okay so

that's the that's the I guess fifth room

she's gone through so if this was

Euclidean space she'd be back where she

started okay now she's gonna turn I

guess

90 degrees to her right again and you're

gonna look into a dark red I don't your

me rotate you turn 180 degrees around

okay so now you're now you should be

facing a dark red room okay so the dark

red chair is in chocolate room you can

walk into that um sorry those things

caught around your foot okay perfect

okay um and then turn 90 degrees to your

left a gun and then walk into the black

forest cake room okay and then turn 90

degrees to your left again and you

should be basically looking at what you

started when you first walked into the

space so take off your headset and see

where you're facing so so you started

out so for everyone Jesse started out

standing where I'm standing facing you

guys facing the audience and what she

did is she she walked around in this big

circuit and ended up standing here

facing away from you so thank you so

much Jesse

okay so let's have um let's have a

little look at what Jesse did sort of

from the top-down perspective not from

not from the her her eyes perspective um

so this would be Jesse here and this is

what she would see if she wasn't busy

looking around at all the cool stuff in

there um so this is an animation so she

first walked into the blueberry room and

then turn 90 degrees to the left

walked into the MintChip room turned 90

degrees to her left walked into the

rainbow sherbert room walked 90 degrees

to her left walked into the strawberry

cheesecake room turned 90 degrees to her

left a gun walked into the cherries and

chocolate room turned 90 degrees to her

left a gun walked into the black forest

cake room where she started turn 90

degrees to left again and she's looking

at the blueberry room again um so you'll

notice in hyperbolic space she would

have walked around a complete loop but

you guys all watched her she started in

the corner here and then walked around

did six 90-degree turns and ended up as

one would predict was it six times 90 is

what five five what 540 is that the the

snowboarder skateboarder trick so she

turned as many as many times as you

would expect a nuke Linnaean space and

these two things don't match up and this

is this is

actually a mathematical a consequence of

living in curved space is that this is a

concept that is called hollow no me so

this is something we might be familiar

with so imagine I'm standing on the

surface of a sphere my right arm is

pointing to the North Pole so this is

the earth say I'm standing at the

equator my right arm is gonna be

pointing towards towards the North Pole

and my left arm is gonna be pointed east

okay so what I'm gonna do is I'm gonna

start walking so I walk a quarter of the

way around the world I walk up to the

North Pole and I walk back down and I

haven't moved my arms and orientation so

I'm walking kind of like this so I'm

fixing my orientation I'm fixing the

position of my arms and when I get back

around I find that I've rotated so that

my right arm which was so my right arm

which was originally pointed towards the

North Pole is now pointing to the west

and my left arm which was originally

pointed to the East is now pointing

north so I've got around a closed loop

on this curved space and have come back

rotated so this is what's called holla

at me so let's have a look at what what

this means for Jessi

so imagine instead of turning 90 degrees

she walked like this so I guess just

keeping her orientation fixed she's

always looking straight forward at you

the audience and this is what's gonna

happen as she walks around so she's

facing along the blue arrow and her

right arm is pointing along the red

arrow so she starts here in the first

room moves walks

le'me sorry

okay hopefully that won't flicker at you

anymore

unless it's gonna fall off the desk yeah

it's a it should still be able to see

this one oh I'll put it on the floor

though um anyway so she starts in this

room and walks forward into the next

room to the left into the next room

backwards into the fourth room to the

right into the fifth room forward into

the sixth room and backwards into the

room she started and upwards into the

first room and you'll notice that she

started facing this direction and she

ends up facing in this direction so she

ended up coming back as having rotated

by 180 degrees and this is what in fact

you guys saw so she started over here

facing you and ended up in an entirely

different place but facing the opposite

direction so this is this is another way

of understanding Haleh know me all right

so can I have one more volunteer for

this space I think I saw you first I

guess I have to pick from the side of

the room next time what's your name

Lucy alright so you'll put on a headset

not too tight okay

so Lucy you're gonna face any one of the

octagonal rooms you like and do you see

like you've got a pretty regular octagon

in front of you and through that you've

got this like tall skinny octagon behind

that and then you see this sort of

string of skinnier and skinnier octagon

so look at the second one away and keep

your eyes fixed on that and walk forward

towards it so what's happening to it

it's getting bigger what directions are

getting bigger in it's widening out so

it's it's changing its aspect ratio so

do you want to find maybe turn to a

different side and okay um I'm gonna

you're gonna get a little close to the

edge but I'm gonna keep an eye on you

um so as she's walking towards you

you'll notice that the aspect ratio is

changing so thank you so much Lucy

[Applause]

okay for me well let me try holding this

um so what's going on is that in this

particular space the horizontal

direction behaves very differently than

the vertical direction so the vertical

direction is just standard Euclidean

space so we're all kind of used to how

distances change we've got a vanishing

point and as things move away from you

they decrease linearly in size so this

is very prominent in classical art in

this space we've got something really

strange going on so in hyperbolic space

we have we've got an exponentially

growing radius so we've got an

exponential number of more rooms as you

go further and further out so what's

going on is that as you move a short

distance the the width between the

windows that you see is growing

exponentially in this direction whereas

it's growing linearly in the in the

vertical direction so the idea here is

that is that the and I sought your P if

the space is giving you this really

strange impression that rigid objects

are flexible it's like living in a house

of mirrors where the mirrors are fixed

but it makes everything else look like

it's changing in size okay so we're

gonna move on to our other space so this

is giving us our first taste of

hyperbolic space but but after that we

are going to this is full-on

three-dimensional hyperbolic space every

direction you look is hyperbolic in this

space okay so so what we'd like to do is

we'd like to go from this idea of

two-dimensional space to

three-dimensional space so in this space

here I have

have a set of squares and I have four

squares that meet around every corner so

this is this is a standard like floor

tiles so I guess I will tell you what

these numbers mean here so the first one

is the number of sides of the polygon

and the second one is the number of

polygons that meet at a vertex so I'm

gonna take this idea and let's go up the

dimension let's go into

three-dimensional space so I've got

another tiling here so the first two

numbers mean the same thing as they do

here so I've got now squares that meet

three around every corner so can anyone

tell me what shape that would give me a

cube exactly I have squares I mean I

guess I would pick a normal room but

there aren't really right angles here so

in the corner of a normal room I'd have

you know one wall another wall and the

ceiling or the floor and that would give

me the corner of a cube so I've got

three squares around each each corner

and then the last number here is the

number of these cells these cubes around

every edge so here I've got four so you

can imagine this is like the standard

minecraft tiling of space you've got

cubes and you stack them together and

you can create your world that way so

there are four cubes you can stack

around every vertical edge so now we're

gonna go to hyperbolic space so let's

think about how we extend this um here

so we've got this is our tiling here

we've got four comma five so this is the

cells are squares and we've got five

around every every vertex so this is

something that if we took actual squares

and tried to fit them in the plan this

would ruffle out of the plane so this

gives us some tiling of

hyperbolic plane but when we go up a

dimension let's see what happens so

we've got again four three is gonna give

us a cubit now we've got five five cubes

around every single edge in this so it's

like we took the minecraft tiling and on

every edge we opened up space and we

shoved in another cube and we've done

that everywhere in all of space and

we're gonna do this again we're gonna

make this four three six so this is

cubes that meet six around every edge so

this is our analog of taking squares

that meet for around sorry six around

every vertical pillar and this is the

this is what's called the palang crate

ball model this is what happens if we

take the pawn crate disk and move it

into three-dimensional space and we have

here's a cube I've added another layer

of cubes so those inner ones should have

six around every edge and they do it

again and this is just I don't know this

looks like Sonic the Hedgehog had a

really bad hair day so so let's try to

understand what's going on in here and

to do that I've got this beautiful 3d

print that was made in collaboration

with Royce Nelson so this is going to be

the actual space so we're walking around

in in virtual reality so the idea is

that we've again taken cubes but instead

of regular cubes I've cut off the

corners so I end up with these sort of

octagonal windows and this is just

drawing the thickened edges so this is

what our space would look like in the

plunker 8ball model so if you hold your

eye up right to the center of it you

might start to get an idea of what it

looks like but it doesn't actually

substitute for the real thing okay so so

I'm going to do a bunch of demos in this

space so let's see

I need a volunteer but I need a

volunteer who's willing to be really

silly okay huh you seem super excited

about being silly so come on down so

what's your name

Dean okay so I'm gonna teach you the

moves to the hall anomie dance okay so

this is this is bad middle-school

dancing so you're gonna keep your feet

planted to the floor and you're gonna

move your head around a lot okay so the

first move is gonna be called the hula

hoop so I'm gonna plant my feet and I'm

gonna move my head in as big a circle as

I possibly can okay you got that one

okay the next one is called the TV set

so you gotta you know get ready you're

gonna go over down over up so you're

gonna just keep doing that motion okay

so this one's the TV set and the last

one is called the bicycle so just wanna

haven't try to do in a skirt but we'll

see if the SyFy fall over or not um so

you're gonna change up your stance a

little bit so you know you're light on

your feet um and you're gonna go like

this so you're gonna go forward down

back up okay okay so that's that's the

bicycle that's the last one so I'm gonna

put you in the headset and get you

switched over to three-dimensional

hyperbolic space so put it on like you

put on ski goggles it might be a little

tight you want me to loosen it

let's try that

you got them microphone jack on so you

might want to pull it down a little bit

further in the back okay is that

comfortable or is that too loose now let

me tighten it a little bit okay okay so

I'm gonna switch you over to um fully

three-dimensional hyperbolic space okay

so here we are in the sort of standard

looking rooms I'm gonna change your view

a little bit so you actually are seeing

what it's like to live in a cube okay so

which was your favorite dance move the

hula hoop okay so so we're standing

facing you're standing facing a magenta

cube so keep your eyes fixed straight

forward and now do the hula hoop so

remember the bigger the circle with your

head the butter alright okay stop

so you started out looking at the

magenta cube and now what's going on

you're and you're facing a blue cube so

as Dean's making these clothes circles

with his head what he's doing is he's

changing his angle with respect to the

curved space so this is kala notice you

can stand there with your feet planted

move your head around and try to just

move space just by moving your head okay

so what's your second favorite dance

move the TV set okay so let's let's do

the TV set so now you're facing square

on to the to the blue the blue room okay

so now let's try the TV set and see what

happens this one you don't have to go

very far to see what's going on so now

instead of rotating like this he's

rotating

like this okay so okay so the last one

and the bicycle

so what before you start what do you

guys think is gonna happen okay where

I'm seeing a lot of people doing this so

I'm gonna rotate around around through

this axis okay so Dean give me the

bicycle so this one's a little harder to

sweep out area uh you're getting it

pretty close okay

now stand up and see what's happened so

where's the blue room that you were

staring at so it's way up there I

rotated around so this is this is this

is the whole enemy down so everyone

Thank You Dean

all right so we're running a little bit

late so I've got time for one last

volunteer hey I saw your hand first all

right what's your name

Margo Margo yeah okay great so paragraph

why don't you put on the put on the

headset I'm gonna tighten it all all

that on this time okay so why don't you

turn around to face face the audience

okay so I'm gonna change your view back

to the original view sorry can you take

a step back so okay so take two more

steps back so you're nowhere near the

wall okay so now we we've got you sort

of zoomed out so you can see see this is

our original of you so what I'm gonna do

now is I'm gonna change his view so that

I remove the pillars in between in

between those triangular windows um and

this is going to make it look like he's

um sort of in some sort of outer space

scene with a lot of jewels in it

so find um a triangular window and go

and stick your head in it I think I'm

getting close to them okay so stick your

head all the way inside it so it looked

on the outside I look kind of like a

regular closed um polyhedron so you can

sort of walk around inside there and see

what happens

so so look like this closed polyhedron

this closed an icosahedron or something

like that um but now that he's inside

he's in this

sort of strange well done as as

paragraph walks around he's gonna see

that the that the triangles will follow

him around everywhere so would you guys

believe that if I told you that this is

the Euclidean plane says the regular

Euclidean space now I'm getting some

nose so let me see if I can show it to

you based on what we talked about with

the triangles at the beginning

so can you let's see what's a good place

for you to so take a few steps forward

and then turn around

okay so part of is now looking looked

down a little bit perfect so you can see

that paragraph is looking at this

triangular pattern so these are the

triangles that are made from cutting the

corners off of the cubes these are just

regular equilateral triangles and if you

look at the vertex you can see that

there are six equilateral triangles so

there's a green one a magenta one a

purple one a pink wanna light pink one

and a yellow one so there's six of them

around here and at the very very

beginning of this lecture we pointed out

that there are there's a tiling of the

Euclidean plane that has six equilateral

triangles around of Reaver tucks so to

end I've got a question for you part of

how do you feel about monkeys okay so

you're not gonna mind if I stick you in

there with an infinite number of monkeys

aren't you okay so okay so step outside

go back through one of the triangular

windows you can sort of go down and duck

and now there are an infinite number of

monkeys the monkeys have two arms two

legs a head and a tail and they are all

holding each other's hands and feet

creating this infinite pattern of

monkeys so

so thank you so much parka

all right thank you for letting me run a

minute over so the last thing I'm gonna

point out is these are available on the

web if you happen to have an HTC vive at

home you can run the full simulation if

not you can go to these websites on your

smartphone and use the gyroscope inside

it to look around the space and if you

want to move forward you just touch the

screen so thank you so much for your

attention and thanks for letting me run

over oh and before you all go I would

like to point out that after we take a

couple of questions this is gonna be up

for like I guess in another hours I

think we're kicked out of here at 8:30

or something like that so this is around

you guys are free to hang out and try it

play with any of my toys up front do we

have any questions for the speaker

tonight it's a question down here

there's a microphone for you one of the

images we hello I think it's on anyway

one of the images look like you are

really looking into infinity versus the

one that you had labeled seeing infinity

yes so an it's D infinity and the one

labeled Matt but I yeah so that's a

that's a great point so uh sure was

trying to figure out a way of capturing

all of infinity into a finite space this

is if you lived in an and you shoot out

into the full area so you're actually

seeing those itty-bitty tiny little

figurines around the edges of space but

you are seeing them the way that you

know you you would see them if you live

there you're seeing them as you know

these real three-dimensional physical

objects and that's what makes it really

feel like you're looking into infinity

you've got depth cues you've got spatial

cues

whereas for uh sure it's just trying to

get it all into a like taking all of

infinity and putting it in a finite

space you we have a question over here

hi so I was wondering when you actually

like wrote the software for these

negative curved spaces like does it just

extend indefinitely and it just keeps

friend during or is these complete

spaces that wrap around that's a great

question so we did actually do it as

phases that wrap around so you're really

inside a single cube and then we've

drawn it so that it looks like you can

see that cube for you know sums step

size distance like translational

distance away from you but as you walk

through the cube once you pass the

window you teleport back to where you

started and we've got some code that

changes the colors so it makes it look

like it's continuous so that's why

you're seeing some things flickering in

and out

well we care I can think of a few

practical like applications of this on

my own but I was wondering if you could

tell us what some practical applications

that I might not be able to think of on

my own would be so I mean there's a ton

of practical applications of virtual

reality in general so we were really

interested in the sort of connection

between between these mathematical

spaces and our spatial perception so

it's sort of interesting that we have

all of these cubes that are sort of

built into our existence like we know we

know that as something moves away it's

gonna get linearly smaller but trying to

adapt to spaces where you can move your

head around and things spin it's sort of

an interesting way of looking at how

perception and how you deal with that

so my grad student Brian who's kindly

helping out with the microphone

he's his PhD project is to make um a

game engine that runs in here so you can

not just walk around in the space but

you can interact with like you could

play tenants in this space or you can

sort of move around or do different

activities hi

does it so given that when you turn 90

degrees your inner air knows you're

turning 90 degrees but visually you're

not turning the whole 90 degrees does it

get its location doesn't get nauseating

so it turns out that there's there's two

reasons why it doesn't so one is that

I'm I have the state of the art

equipment and I'm running it on a gtx

1080 so that the refresh rate is 120

frames a second and so any tiny motion

of my head is getting tracked before my

brain can detect that there's a problem

the other answer to your question which

is the geometrical answer is that it is

true that angles move differently but

there's a theorem in differential

geometry that says that if you get a

small and

patch of curved space it's always gonna

look Euclidean so Center it at your head

it basically seems Euclidean to you so

moving a tiny amount it's imperceptible

very close to your head but far away

from your head there's a large

difference so that's something that your

brain Connick hopes with I mean you can

try it afterwards and it's I mean I've

not had anyone see that they've been

Ozzie ated I've had a couple people who

have balance problems really feel the

vertigo and the first one but but not

not nausea and I probably had a thousand

people through this

that was it that was a great talk I'm

really curious one of your volunteers

you asked you know which way did it get

bigger in so how do these spaces like

change communicating discovery like if

somebody's in these virtual spaces like

how do you communicate what you're

seeing and actually like it potential

insight when you start looking at these

complex like these weird projections of

space so a lot of it the best thing

about in particular the HTC vive is that

you're really using your body you are

navigating using proprioception you know

where you're sort of hands are you know

where your body is and you know how your

body interacts with this space you can

kind of use what you see and sort of try

to describe it in turn I mean we only

know Euclidian terms like we don't have

a language for well I mean enough

mathematicians have a language for us

but you know as as sort of physical

beings we don't have a language for how

our interactions would change so yeah so

gestures gestures are great trying to do

like very controlled motion like moving

your head in a circle is something you

can control and moving your head in a

big circle or a small circle you can

sort of change that and you're sort of

using the scientific method to explore

how simple emotions change how you're

you're seeing things thank you question

over here so so I gather

um you were telling us any any set of

triangles like less than six triangles

on a verse vertices it's going to create

a Euclidean space oh and if it's more

than that it's going to become flat or

ruffled so less than six is gonna be a

spherical space so like a like an

icosahedron like it's sort of a a closed

fearful shape exactly six is flat and

more than six is ruffled in this

hyperbolic space

okay just just check making sure what

over here is this really related to

fractals in some way um so fractals have

this notion of being self similar but

self symbol are on different scales so I

have you know a triangle like this and I

take that and put it over here and then

each of those I put it over there so

it's self similar at different scales

this is self-similar at the same scale

but because of how the optics play

tricks on you it really looks fractal

and like you are seeing a larger and

larger number of of squares or triangles

or rooms or whatever as you go further

out so that gives you sort of the

illusion then it's fractal but you could

also say like well I don't really care

that this is three dimensional all I

care about is that 2d image and so you

could use this to create a sort of

fractal and pattern in 2d yeah thanks a

lot for the great talk the phenomenon in

which someone does the hula hoop and

then ends up having turned is this

analogous to a barre face that people

talk about in that's a great question

so yes it is so various phasers are

called geometric faces precisely because

when you go around and you're sort of

measuring something about having gone

entirely around in a circle it turns out

that the to think about that you have to

think about slightly different space

than Euclidean space you have to get

that curvature coming from basically

coming from Energon except the quantum

mechanics so you are seeing you are

seeing that notion it's just that the

curved space isn't directly in front of

you it's a little bit more subtle and

just so why didn't we see it in 2d when

you only have the hyperbolic

space so you do see some of it in 2d so

what you can try it out so the hula-hoop

one you see you will see things rotate

around but the the other ones you're not

gonna see and that's because the two

directions the hyperbolic direction and

the Euclidean direction commute so I'm

not gonna pick up any anything is

between them one over here hi so what

are the applications of the hyperbolic

plane like how is it being used today or

where is it being used okay so this is

sort of an interesting question so you

do see some of these notions in biology

there's reasons why you might want to

create things that have very large

surface areas and things like that and

you can't we know you can't fit all of

the this entire space into into our

Euclidean space so what biology does is

it sort of makes an approximation so it

gets like as close to this as possible

and then takes it and repeats it in

these facial chunks that can fill a

Euclidean space so that's something that

is very common in like lipid membranes

and all sorts of sort of growth patterns

and you can imagine doing this as a way

of using the space as a way to

understand really complicated data

structures things that have nested

folders and them and instead of

remembering this weird linear path

through space you can have this zoom

feature so you can look at all of the

network connectivity and move back and

forth between it I was sort of a

top-down spatial view so that's sort of

on the UI side you could also imagine if

you can create these structures that

have really high surface area you could

get better data storage per unit volume

um so you're not stuck with you know

layered 2d systems you could in you can

sort of

use the curvature of these spaces to to

get a higher density of bits so we have

been studying some of these minimal

surface models so one of my students

who's not here today has studied studied

a way of making a chiral minimal surface

so for that we've only currently 3d

printed it the idea is that if you put

it down to sub micron scales you get

features that are about the size of a

wavelength of light and you can use that

to control photonic properties in detail

so about one last question and then

maybe if you still have questions you

can come down to the front that way we

have a little bit of time for people to

play around was there another question

over here in the back I think over here

I was just wondering this might be a

dumb question but why does this only

work with triangles like could you make

an oddly Euclidean space with like

squares or other types of shapes

absolutely it doesn't just work with

triangles it works with any any geometry

so that dress I showed at the beginning

that was made out of hexagons and

heptagons and Pentagon's that's just

using the idea that you want to either

add a little bit of area or take a

little bit of area away so basically

anytime you're sort of changing that you

are inducing curvature but it doesn't it

doesn't have to be a tiling it doesn't

have to be regular it doesn't have to be

anything so that's how these curved

patterns in clothes work to is instead

of having these sharp corners it sort of

spreads that that wedge out over a

longer period that tell us have princess

seams work so you could like tell people

that like you're wearing like a

non-euclidean skirt or something

absolutely I am wearing a non-euclidean

skirt okay

thanks thank you all right thank you

everyone so much for for coming by and

please feel free to come up and play

with VR